Knight's tour: Difference between revisions
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===Functional=== |
===Functional=== |
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Revision as of 05:04, 28 June 2019
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Problem: you have a standard 8x8 chessboard, empty but for a single knight on some square. Your task is to emit a series of legal knight moves that result in the knight visiting every square on the chessboard exactly once. Note that it is not a requirement that the tour be "closed"; that is, the knight need not end within a single move of its start position.
Input and output may be textual or graphical, according to the conventions of the programming environment. If textual, squares should be indicated in algebraic notation. The output should indicate the order in which the knight visits the squares, starting with the initial position. The form of the output may be a diagram of the board with the squares numbered according to visitation sequence, or a textual list of algebraic coordinates in order, or even an actual animation of the knight moving around the chessboard.
Input: starting square
Output: move sequence
- Related tasks
- A* search algorithm
- N-queens problem
- Solve a Hidato puzzle
- Solve a Holy Knight's tour
- Solve a Hopido puzzle
- Solve a Numbrix puzzle
- Solve the no connection puzzle
360 Assembly
<lang 360asm>* Knight's tour 20/03/2017 KNIGHT CSECT
USING KNIGHT,R13 base registers
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R13,R15 set addressability
MVC PG(20),=CL20'Knights tour ..x..'
L R1,NN n
XDECO R1,XDEC edit
MVC PG+14(2),XDEC+10 n
MVC PG+17(2),XDEC+10 n
XPRNT PG,L'PG print buffer
LA R0,1 1
ST R0,X x=1
ST R0,Y y=1
SR R0,R0 0
ST R0,TOTAL total=0
LOOP EQU * do loop
L R1,X x
BCTR R1,0 -1
MH R1,NNH *n
L R0,Y y
BCTR R0,0 -1
AR R1,R0 (x-1)*n+y-1
SLA R1,1 ((x-1)*n+y-1)*2
LA R0,1 1
STH R0,BOARD(R1) board(x,y)=1
L R2,TOTAL total
LA R2,1(R2) total+1
STH R2,DISP(R1) disp(x,y)=total+1
ST R2,TOTAL total=total+1
L R1,X x
L R2,Y y
BAL R14,CHOOSEMV call choosemv(x,y)
C R0,=F'0' until(choosemv(x,y)=0)
BNE LOOP loop
LA R2,KN*KN n*n
IF C,R2,NE,TOTAL THEN if total<>n*n then
XPRNT =C'error!!',7 print error
ENDIF , endif
LA R6,1 i=1
DO WHILE=(C,R6,LE,NN) do i=1 to n
MVC PG,=CL128' ' init buffer
LA R10,PG pgi=0
LA R7,1 j=1
DO WHILE=(C,R7,LE,NN) do j=1 to n
LR R1,R6 i
BCTR R1,0 -1
MH R1,NNH *n
LR R0,R7 j
BCTR R0,0 -1
AR R1,R0 (i-1)*n+j-1
SLA R1,1 ((i-1)*n+j-1)*2
LH R2,DISP(R1) disp(i,j)
XDECO R2,XDEC edit
MVC 0(4,R10),XDEC+8 output
LA R10,4(R10) pgi+=4
LA R7,1(R7) j++
ENDDO , enddo j
XPRNT PG,L'PG print buffer
LA R6,1(R6) i++
ENDDO , enddo i
L R13,4(0,R13) restore previous savearea pointer
LM R14,R12,12(R13) restore previous context
XR R15,R15 return_code=0
BR R14 exit
- ------- ---- ----------------------------------------
CHOOSEMV EQU * choosemv(xc,yc)
ST R14,SAVEACMV save return point
ST R1,XC store xc
ST R2,YC store yc
MVC MM,=F'9' m=9
L R1,XC xc
LA R1,1(R1)
L R2,YC yc
LA R2,2(R2)
BAL R14,TRYMV call trymv(xc+1,yc+2)
L R1,XC xc
LA R1,1(R1)
L R2,YC yc
SH R2,=H'2'
BAL R14,TRYMV call trymv(xc+1,yc-2)
L R1,XC xc
BCTR R1,0
L R2,YC yc
LA R2,2(R2)
BAL R14,TRYMV call trymv(xc-1,yc+2)
L R1,XC xc
BCTR R1,0
L R2,YC yc
SH R2,=H'2'
BAL R14,TRYMV call trymv(xc-1,yc-2)
L R1,XC xc
LA R1,2(R1)
L R2,YC yc
LA R2,1(R2)
BAL R14,TRYMV call trymv(xc+2,yc+1)
L R1,XC xc
LA R1,2(R1)
L R2,YC yc
BCTR R2,0
BAL R14,TRYMV call trymv(xc+2,yc-1)
L R1,XC xc
SH R1,=H'2'
L R2,YC yc
LA R2,1(R2)
BAL R14,TRYMV call trymv(xc-2,yc+1)
L R1,XC xc
SH R1,=H'2'
L R2,YC yc
BCTR R2,0
BAL R14,TRYMV call trymv(xc-2,yc-1)
L R4,MM m
IF C,R4,EQ,=F'9' THEN if m=9 then
LA R0,0 return(0)
ELSE , else
MVC X,NEWX x=newx
MVC Y,NEWY y=newy
LA R0,1 return(1)
ENDIF , endif
L R14,SAVEACMV restore return point
BR R14 return
SAVEACMV DS A return point
- ------- ---- ----------------------------------------
TRYMV EQU * trymv(xt,yt)
ST R14,SAVEATMV save return point
ST R1,XT store xt
ST R2,YT store yt
SR R10,R10 n=0
BAL R14,VALIDMV
IF LTR,R0,Z,R0 THEN if validmv(xt,yt)=0 then
LA R0,0 return(0)
B RETURTMV
ENDIF , endif
L R1,XT
LA R1,1(R1) xt+1
L R2,YT
LA R2,2(R2) yt+2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+1,yt+2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
LA R1,1(R1) xt+1
L R2,YT
SH R2,=H'2' yt-2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+1,yt-2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
BCTR R1,0 xt-1
L R2,YT
LA R2,2(R2) yt+2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-1,yt+2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
BCTR R1,0 xt-1
L R2,YT
SH R2,=H'2' yt-2
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-1,yt-2)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
LA R1,2(R1) xt+2
L R2,YT
LA R2,1(R2) yt+1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+2,yt+1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
LA R1,2(R1) xt+2
L R2,YT
BCTR R2,0 yt-1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt+2,yt-1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
SH R1,=H'2' xt-2
L R2,YT
LA R2,1(R2) yt+1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-2,yt+1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
L R1,XT
SH R1,=H'2' xt-2
L R2,YT
BCTR R2,0 yt-1
BAL R14,VALIDMV
IF C,R0,EQ,=F'1' THEN if validmv(xt-2,yt-1)=1 then
LA R10,1(R10) n=n+1;
ENDIF , endif
IF C,R10,LT,MM THEN if n<m then
ST R10,MM m=n
MVC NEWX,XT newx=xt
MVC NEWY,YT newy=yt
ENDIF , endif
RETURTMV L R14,SAVEATMV restore return point
BR R14 return
SAVEATMV DS A return point
- ------- ---- ----------------------------------------
VALIDMV EQU * validmv(xv,yv)
C R1,=F'1' if xv<1 then
BL RET0
C R1,NN if xv>nn then
BH RET0
C R2,=F'1' if yv<1 then
BL RET0
C R2,NN if yv>nn then
BNH OK
RET0 SR R0,R0 return(0)
B RETURVMV
OK LR R3,R1 xv
BCTR R3,0
MH R3,NNH *n
LR R0,R2 yv
BCTR R0,0
AR R3,R0
SLA R3,1
LH R4,BOARD(R3) board(xv,yv)
IF LTR,R4,Z,R4 THEN if board(xv,yv)=0 then
LA R0,1 return(1)
ELSE , else
SR R0,R0 return(0)
ENDIF , endif
RETURVMV BR R14 return
- ---- ----------------------------------------
KN EQU 8 n compile-time NN DC A(KN) n fullword NNH DC AL2(KN) n halfword BOARD DC (KN*KN)H'0' dim board(n,n) init 0 DISP DC (KN*KN)H'0' dim disp(n,n) init 0 X DS F Y DS F TOTAL DS F XC DS F YC DS F MM DS F NEWX DS F NEWY DS F XT DS F YT DS F XDEC DS CL12 PG DC CL128' ' buffer
YREGS
END KNIGHT</lang>
- Output:
Knight's tour 8x 8 1 4 57 20 47 6 49 22 34 19 2 5 58 21 46 7 3 56 35 60 37 48 23 50 18 33 38 55 52 59 8 45 39 14 53 36 61 44 51 24 32 17 40 43 54 27 62 9 13 42 15 30 11 64 25 28 16 31 12 41 26 29 10 63
Ada
First, we specify a naive implementation the package Knights_Tour with naive backtracking. It is a bit more general than required for this task, by providing a mechanism not to visit certain coordinates. This mechanism is actually useful for the task Solve a Holy Knight's tour#Ada, which also uses the package Knights_Tour.
<lang Ada>generic
Size: Integer;
package Knights_Tour is
subtype Index is Integer range 1 .. Size;
type Tour is array (Index, Index) of Natural;
Empty: Tour := (others => (others => 0));
function Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty) return Tour;
-- finds tour via backtracking
-- either no tour has been found, i.e., Get_Tour returns Scene
-- or the Result(X,Y)=K if and only if I,J is visited at the K-th move
-- for all X, Y, Scene(X,Y) must be either 0 or Natural'Last,
-- where Scene(X,Y)=Natural'Last means "don't visit coordiates (X,Y)!"
function Count_Moves(Board: Tour) return Natural;
-- counts the number of possible moves, i.e., the number of 0's on the board
procedure Tour_IO(The_Tour: Tour; Width: Natural := 4);
-- writes The_Tour to the output using Ada.Text_IO;
end Knights_Tour;</lang>
Here is the implementation:
<lang Ada>with Ada.Text_IO, Ada.Integer_Text_IO;
package body Knights_Tour is
type Pair is array(1..2) of Integer;
type Pair_Array is array (Positive range <>) of Pair;
Pairs: constant Pair_Array (1..8)
:= ((-2,1),(-1,2),(1,2),(2,1),(2,-1),(1,-2),(-1,-2),(-2,-1));
-- places for the night to go (relative to the current position)
function Count_Moves(Board: Tour) return Natural is
N: Natural := 0;
begin
for I in Index loop
for J in Index loop if Board(I,J) < Natural'Last then N := N + 1; end if; end loop;
end loop;
return N;
end Count_Moves;
function Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty)
return Tour is
Done: Boolean;
Move_Count: Natural := Count_Moves(Scene);
Visited: Tour;
-- Visited(I, J) = 0: not yet visited
-- Visited(I, J) = K: visited at the k-th move
-- Visited(I, J) = Integer'Last: never visit
procedure Visit(X, Y: Index; Move_Number: Positive; Found: out Boolean) is
XX, YY: Integer;
begin
Found := False;
Visited(X, Y) := Move_Number;
if Move_Number = Move_Count then
Found := True;
else
for P in Pairs'Range loop
XX := X + Pairs(P)(1);
YY := Y + Pairs(P)(2);
if (XX in Index) and then (YY in Index)
and then Visited(XX, YY) = 0 then
Visit(XX, YY, Move_Number+1, Found); -- recursion
if Found then
return; -- no need to search further
end if;
end if;
end loop;
Visited(X, Y) := 0; -- undo previous mark
end if;
end Visit;
begin
Visited := Scene;
Visit(Start_X, Start_Y, 1, Done);
if not Done then
Visited := Scene;
end if;
return Visited;
end Get_Tour;
procedure Tour_IO(The_Tour: Tour; Width: Natural := 4) is
begin
for I in Index loop
for J in Index loop
if The_Tour(I, J) < Integer'Last then Ada.Integer_Text_IO.Put(The_Tour(I, J), Width); else for W in 1 .. Width-1 loop Ada.Text_IO.Put(" "); end loop; Ada.Text_IO.Put("-"); -- deliberately not visited end if;
end loop;
Ada.Text_IO.New_Line;
end loop;
end Tour_IO;
end Knights_Tour;</lang>
Here is the main program:
<lang Ada>with Knights_Tour, Ada.Command_Line;
procedure Test_Knight is
Size: Positive := Positive'Value(Ada.Command_Line.Argument(1));
package KT is new Knights_Tour(Size => Size);
begin
KT.Tour_IO(KT.Get_Tour(1, 1));
end Test_Knight;</lang>
For small sizes, this already works well (< 1 sec for size 8). Sample output:
>./test_knight 8 1 38 55 34 3 36 19 22 54 47 2 37 20 23 4 17 39 56 33 46 35 18 21 10 48 53 40 57 24 11 16 5 59 32 45 52 41 26 9 12 44 49 58 25 62 15 6 27 31 60 51 42 29 8 13 64 50 43 30 61 14 63 28 7
For larger sizes we'll use Warnsdorff's heuristic (without any thoughtful tie breaking). We enhance the specification adding a function Warnsdorff_Get_Tour. This enhancement of the package Knights_Tour will also be used for the task Solve a Holy Knight's tour#Ada. The specification of Warnsdorff_Get_Tour is the following. <lang Ada>
function Warnsdorff_Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty)
return Tour;
-- uses Warnsdorff heurisitic to find a tour faster -- same interface as Get_Tour</lang>
Its implementation is as follows.
<lang Ada> function Warnsdorff_Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty) return Tour is
Done: Boolean;
Visited: Tour; -- see comments from Get_Tour above
Move_Count: Natural := Count_Moves(Scene);
function Neighbors(X, Y: Index) return Natural is
Result: Natural := 0;
begin
for P in Pairs'Range loop
if X+Pairs(P)(1) in Index and then Y+Pairs(P)(2) in Index and then
Visited(X+Pairs(P)(1), Y+Pairs(P)(2)) = 0 then
Result := Result + 1;
end if;
end loop;
return Result;
end Neighbors;
procedure Sort(Options: in out Pair_Array) is
N_Bors: array(Options'Range) of Natural;
K: Positive range Options'Range;
N: Natural;
P: Pair;
begin
for Opt in Options'Range loop
N_Bors(Opt) := Neighbors(Options(Opt)(1), Options(Opt)(2));
end loop;
for Opt in Options'Range loop
K := Opt;
for Alternative in Opt+1 .. Options'Last loop
if N_Bors(Alternative) < N_Bors(Opt) then
K := Alternative;
end if;
end loop;
N := N_Bors(Opt);
N_Bors(Opt) := N_Bors(K);
N_Bors(K) := N;
P := Options(Opt);
Options(Opt) := Options(K);
Options(K) := P;
end loop;
end Sort;
procedure Visit(X, Y: Index; Move: Positive; Found: out Boolean) is
Next_Count: Natural range 0 .. 8 := 0;
Next_Steps: Pair_Array(1 .. 8);
XX, YY: Integer;
begin
Found := False;
Visited(X, Y) := Move;
if Move = Move_Count then
Found := True;
else
-- consider all possible places to go
for P in Pairs'Range loop
XX := X + Pairs(P)(1);
YY := Y + Pairs(P)(2);
if (XX in Index) and then (YY in Index)
and then Visited(XX, YY) = 0 then
Next_Count := Next_Count+1;
Next_Steps(Next_Count) := (XX, YY);
end if;
end loop;
Sort(Next_Steps(1 .. Next_Count));
for N in 1 .. Next_Count loop
Visit(Next_Steps(N)(1), Next_Steps(N)(2), Move+1, Found);
if Found then
return; -- no need to search further
end if;
end loop;
-- if we didn't return above, we have to undo our move
Visited(X, Y) := 0;
end if;
end Visit;
begin
Visited := Scene;
Visit(Start_X, Start_Y, 1, Done);
if not Done then
Visited := Scene;
end if;
return Visited;
end Warnsdorff_Get_Tour;</lang>
The modification for the main program is trivial: <lang Ada>with Knights_Tour, Ada.Command_Line;
procedure Test_Fast is
Size: Positive := Positive'Value(Ada.Command_Line.Argument(1));
package KT is new Knights_Tour(Size => Size);
begin
KT.Tour_IO(KT.Warnsdorff_Get_Tour(1, 1));
end Test_Fast;</lang>
This works still well for somewhat larger sizes:
>./test_fast 24 1 108 45 52 3 112 57 60 5 62 131 144 7 64 147 170 9 66 187 192 11 68 71 190 46 51 2 111 56 53 4 113 130 59 6 63 146 169 8 65 186 215 10 67 188 191 12 69 107 44 109 54 123 114 129 58 61 132 145 168 143 148 185 214 171 198 225 216 193 70 189 72 50 47 122 115 110 55 140 133 128 167 142 149 184 213 172 199 226 255 246 197 224 217 194 13 43 106 49 124 139 134 127 166 141 150 183 212 173 200 227 254 247 242 223 256 245 196 73 218 48 121 116 135 126 165 138 151 182 211 174 201 228 253 248 241 290 263 304 243 222 257 14 195 105 42 125 164 137 152 181 210 175 202 229 252 249 240 289 264 329 308 291 262 303 244 219 74 120 117 136 153 180 163 176 203 230 267 250 239 288 265 328 309 334 345 330 305 292 221 258 15 41 104 119 160 177 204 231 268 209 238 287 266 251 310 335 344 357 332 307 346 261 302 75 220 118 159 154 205 162 179 208 237 286 269 324 311 336 327 438 333 418 347 356 331 306 293 16 259 103 40 161 178 207 232 285 270 323 312 337 326 483 416 343 422 437 358 419 298 349 260 301 76 158 155 206 233 284 271 236 313 338 325 482 415 342 439 484 417 420 423 348 355 360 299 294 17 39 102 157 272 235 314 339 322 481 414 341 492 497 514 421 440 485 436 359 424 297 350 77 300 156 273 234 315 276 283 478 413 340 493 480 513 530 491 498 515 452 441 454 435 354 361 18 295 101 38 275 282 397 412 321 494 479 512 557 496 543 534 529 490 499 486 451 442 425 296 351 78 274 279 316 277 320 477 410 511 570 495 554 535 556 531 542 533 516 453 444 455 434 353 362 19 37 100 281 398 411 396 575 476 567 558 561 544 553 536 521 528 489 500 487 450 443 426 79 352 280 317 278 319 402 409 510 569 560 571 566 555 550 541 532 537 522 517 460 445 456 433 20 363 99 36 389 378 399 576 395 574 475 568 559 562 545 552 525 520 527 488 501 462 449 364 427 80 94 379 318 401 388 403 408 509 572 565 474 551 540 549 538 523 518 461 446 459 432 457 366 21 35 98 93 390 377 400 573 394 375 508 563 546 373 524 519 526 371 502 463 466 365 448 81 428 380 95 382 385 404 387 376 407 564 473 374 507 548 539 372 503 464 467 370 447 458 431 22 367 383 34 97 92 391 32 405 90 393 30 547 88 471 28 505 86 469 26 465 84 369 24 429 82 96 381 384 33 386 91 392 31 406 89 472 29 506 87 470 27 504 85 468 25 430 83 368 23
ALGOL 68
<lang algol68># Non-recursive Knight's Tour with Warnsdorff's algorithm #
- If there are multiple choices, backtrack if the first choice doesn't #
- find a solution #
- the size of the board #
INT board size = 8;
- directions for moves #
INT nne = 1, nee = 2, see = 3, sse = 4, ssw = 5, sww = 6, nww = 7, nnw = 8;
INT lowest move = nne; INT highest move = nnw;
- the vertical position changes of the moves #
- nne, nee, see, sse, ssw, sww, nww, nnw #
[]INT offset v = ( -2, -1, 1, 2, 2, 1, -1, -2 );
- the horizontal position changes of the moves #
- nne, nee, see, sse, ssw, sww, nww, nnw #
[]INT offset h = ( 1, 2, 2, 1, -1, -2, -2, -1 );
MODE SQUARE = STRUCT( INT move # the number of the move that caused #
# the knight to reach this square #
, INT direction # the direction of the move that #
# brought the knight here - one of #
# nne, nee, see, sse, ssw, sww, nww #
# or nnw - used for backtracking #
# zero for the first move #
);
- the board #
[ board size, board size ]SQUARE board;
- initialises the board so there are no used squares #
PROC initialise board = VOID:
FOR row FROM 1 LWB board TO 1 UPB board
DO
FOR col FROM 2 LWB board TO 2 UPB board
DO
board[ row, col ] := ( 0, 0 )
OD
OD; # initialise board #
INT iterations := 0;
INT backtracks := 0;
- prints the board #
PROC print tour = VOID: BEGIN
print( ( " a b c d e f g h", newline ) ); print( ( " +--------------------------------", newline ) );
FOR row FROM 1 UPB board BY -1 TO 1 LWB board
DO
print( ( whole( row, -3 ) ) );
print( ( "|" ) );
FOR col FROM 2 LWB board TO 2 UPB board
DO
print( ( " " ) );
print( ( whole( move OF board[ row, col ], -3 ) ) )
OD;
print( ( newline ) )
OD
END; # print tour #
- determines whether a move to the specified row and column is possible #
PROC can move to = ( INT row, INT col )BOOL:
IF row > 1 UPB board
OR row < 1 LWB board
OR col > 2 UPB board
OR col < 2 LWB board
THEN
# the position is not on the board #
FALSE
ELSE
# the move is legal, check the square is unoccupied #
move OF board[ row, col ] = 0
FI;
- used to hold counts of the number of moves that could be made in each #
- direction from the current square #
[ lowest move : highest move ]INT possible move count;
- sets the elements of possible move count to the number of moves that #
- could be made in each direction from the specified row and col #
PROC count moves in each direction from = ( INT row, INT col )VOID:
FOR move direction FROM lowest move TO highest move DO
INT new row = row + offset v[ move direction ];
INT new col = col + offset h[ move direction ];
IF NOT can move to( new row, new col )
THEN
# can't move to this square #
possible move count[ move direction ] := -1
ELSE
# a move in this direction is possible #
# - count the number of moves that could be made from it #
possible move count[ move direction ] := 0;
FOR subsequent move FROM lowest move TO highest move
DO
IF can move to( new row + offset v[ subsequent move ]
, new col + offset h[ subsequent move ]
)
THEN
# have a possible subsequent move #
possible move count[ move direction ] +:= 1
FI
OD
FI
OD;
- update the board to the first knight's tour found starting from #
- "start row" and "start col". #
- return TRUE if one was found, FALSE otherwise #
PROC find tour = ( INT start row, INT start col )BOOL: BEGIN
initialise board;
BOOL result := TRUE;
INT move number := 1; INT row := start row; INT col := start col;
# the tour will be complete when we have made as many moves #
# as there squares on the board #
INT final move = ( ( ( 1 UPB board ) + 1 ) - 1 LWB board )
* ( ( ( 2 UPB board ) + 1 ) - 2 LWB board )
;
# the first move is to place the knight on the starting square # board[ row, col ] := ( move number, lowest move - 1 ); # start off with an unknown direction for the best move # INT best direction := lowest move - 1;
# attempt to find a sequence of moves that will reach each square once #
WHILE
move number < final move AND result
DO
iterations +:= 1;
# count the number of moves possible from each possible move #
# from this square #
count moves in each direction from( row, col );
# find the direction with the lowest number of subsequent moves #
IF best direction < lowest move
THEN
# must find the best direction to move in #
INT lowest move count := highest move + 1;
FOR move direction FROM lowest move TO highest move
DO
IF possible move count[ move direction ] >= 0
AND possible move count[ move direction ] < lowest move count
THEN
# have a move with fewer possible subsequent moves #
best direction := move direction;
lowest move count := possible move count[ move direction ]
FI
OD
ELSE
# following a backtrack - find an alternative with the same #
# lowest number of possible moves - if there are any #
# if there aren't, we will backtrack again #
INT lowest move count := possible move count[ best direction ];
WHILE
best direction +:= 1;
IF best direction > highest move
THEN
# no more possible moves with the lowest number of #
# subsequent moves #
FALSE
ELSE
# keep looking if the number of moves from this square #
# isn't the lowest #
possible move count[ best direction ] /= lowest move count
FI
DO
SKIP
OD
FI;
IF best direction <= highest move
AND best direction >= lowest move
THEN
# we found a best possible move #
INT new row = row + offset v[ best direction ];
INT new col = col + offset h[ best direction ];
row := new row;
col := new col;
move number +:= 1;
board[ row, col ] := ( move number, best direction );
best direction := lowest move - 1
ELSE
# no more moves from this position - backtrack #
IF move number = 1
THEN
# at the starting position - no solution #
result := FALSE
ELSE
# not at the starting position - undo the latest move #
backtracks +:= 1;
move number -:= 1;
INT curr row := row;
INT curr col := col;
best direction := direction OF board[ curr row, curr col ];
row -:= offset v[ best direction ];
col -:= offset h[ best direction ];
# reset the square we just backtracked from #
board[ curr row, curr col ] := ( 0, 0 )
FI
FI
OD;
result
END; # find tour #
main:(
# get the starting position #
CHAR row; CHAR col;
WHILE
print( ( "Enter starting row(1-8) and col(a-h): " ) );
read ( ( row, col, newline ) );
row < "1" OR row > "8" OR col < "a" OR col > "h"
DO
SKIP
OD;
# calculate the tour from that position, if possible #
IF find tour( ABS row - ABS "0", ( ABS col - ABS "a" ) + 1 )
THEN
# found a solution #
print tour
ELSE
# couldn't find a solution #
print( ( "Solution not found - iterations: ", iterations
, ", backtracks: ", backtracks
, newline
)
)
FI
)</lang>
- Output:
Enter starting row(1-8) and col(a-h): 5d
a b c d e f g h
+--------------------------------
8| 51 18 53 20 41 44 3 6
7| 54 21 50 45 2 5 40 43
6| 17 52 19 58 49 42 7 4
5| 22 55 64 1 46 57 48 39
4| 33 16 23 56 59 38 29 8
3| 24 13 34 63 30 47 60 37
2| 15 32 11 26 35 62 9 28
1| 12 25 14 31 10 27 36 61
ANSI Standard BASIC
ANSI BASIC doesn't allow function parameters to be passed by reference so X and Y were made global variables.
<lang ANSI Standard BASIC>100 DECLARE EXTERNAL FUNCTION choosemove 110 ! 120 RANDOMIZE 130 PUBLIC NUMERIC X, Y, TRUE, FALSE 140 LET TRUE = -1 150 LET FALSE = 0 160 ! 170 SET WINDOW 1,512,1,512 180 SET AREA COLOR "black" 190 FOR x=0 TO 512-128 STEP 128 200 FOR y=0 TO 512-128 STEP 128 210 PLOT AREA:x+64,y;x+128,y;x+128,y+64;x+64,y+64 220 PLOT AREA:x,y+64;x+64,y+64;x+64,y+128;x,y+128 230 NEXT y 240 NEXT x 250 ! 260 SET LINE COLOR "red" 270 SET LINE WIDTH 6 280 ! 290 PUBLIC NUMERIC Board(0 TO 7,0 TO 7) 300 LET X = 0 310 LET Y = 0 320 LET Total = 0 330 DO 340 LET Board(X,Y) = TRUE 350 PLOT LINES: X*64+32,Y*64+32; 360 LET Total = Total + 1 370 LOOP UNTIL choosemove(X, Y) = FALSE 380 IF Total <> 64 THEN STOP 390 END 400 ! 410 EXTERNAL FUNCTION choosemove(X1, Y1) 420 DECLARE EXTERNAL SUB trymove 430 LET M = 9 440 CALL trymove(X1+1, Y1+2, M, newx, newy) 450 CALL trymove(X1+1, Y1-2, M, newx, newy) 460 CALL trymove(X1-1, Y1+2, M, newx, newy) 470 CALL trymove(X1-1, Y1-2, M, newx, newy) 480 CALL trymove(X1+2, Y1+1, M, newx, newy) 490 CALL trymove(X1+2, Y1-1, M, newx, newy) 500 CALL trymove(X1-2, Y1+1, M, newx, newy) 510 CALL trymove(X1-2, Y1-1, M, newx, newy) 520 IF M=9 THEN 530 LET choosemove = FALSE 540 EXIT FUNCTION 550 END IF 560 LET X = newx 570 LET Y = newy 580 LET choosemove = TRUE 590 END FUNCTION 600 ! 610 EXTERNAL SUB trymove(X, Y, M, newx, newy) 620 ! 630 DECLARE EXTERNAL FUNCTION validmove 640 IF validmove(X,Y) = 0 THEN EXIT SUB 650 IF validmove(X+1,Y+2) <> 0 THEN LET N = N + 1 660 IF validmove(X+1,Y-2) <> 0 THEN LET N = N + 1 670 IF validmove(X-1,Y+2) <> 0 THEN LET N = N + 1 680 IF validmove(X-1,Y-2) <> 0 THEN LET N = N + 1 690 IF validmove(X+2,Y+1) <> 0 THEN LET N = N + 1 700 IF validmove(X+2,Y-1) <> 0 THEN LET N = N + 1 710 IF validmove(X-2,Y+1) <> 0 THEN LET N = N + 1 720 IF validmove(X-2,Y-1) <> 0 THEN LET N = N + 1 730 IF N>M THEN EXIT SUB 740 IF N=M AND RND<.5 THEN EXIT SUB 750 LET M = N 760 LET newx = X 770 LET newy = Y 780 END SUB 790 ! 800 EXTERNAL FUNCTION validmove(X,Y) 810 LET validmove = FALSE 820 IF X<0 OR X>7 OR Y<0 OR Y>7 THEN EXIT FUNCTION 830 IF Board(X,Y)=FALSE THEN LET validmove = TRUE 840 END FUNCTION</lang>
AutoHotkey
<lang AutoHotkey>#SingleInstance, Force
- NoEnv
SetBatchLines, -1
- Uncomment if Gdip.ahk is not in your standard library
- Include, Gdip.ahk
If !pToken := Gdip_Startup(){
MsgBox, 48, Gdiplus error!, Gdiplus failed to start. Please ensure you have Gdiplus on your system. ExitApp
}
- I've added a simple new function here, just to ensure if anyone is having any problems then to make sure they are using the correct library version
if (Gdip_LibraryVersion() < 1.30) { MsgBox, 48, Version error!, Please download the latest version of the gdi+ library ExitApp } OnExit, Exit tour := "a1 b3 d2 c4 a5 b7 d8 e6 d4 b5 c7 a8 b6 c8 a7 c6 b8 a6 b4 d5 e3 d1 b2 a4 c5 d7 f8 h7 f6 g8 h6 f7 h8 g6 e7 f5 h4 g2 e1 d3 e5 g4 f2 h1 g3 f1 h2 f3 g1 h3 g5 e4 d6 e8 g7 h5 f4 e2 c1 a2 c3 b1 a3 c2 "
- Knight's tour with maximum symmetry by George Jelliss, http://www.mayhematics.com/t/8f.htm
- I know, I know, but I followed the task outline to the letter! Besides, this path is the prettiest.
- Input
- starting square
InputBox, start, Knight's Tour Start, Enter Knight's starting location in algebraic notation:, , , , , , , , b3 i := InStr(tour, start) If i=0 { Msgbox Error, please try again. Reload }
- Output
- move sequence
Msgbox % tour := SubStr(tour, i) . SubStr(tour, 1, i-1)
- Animation
tour .= SubStr(tour, 1, 3) , CellSize := 30 ; pixels , Width := Height := 9*CellSize , TopLeftX := (A_ScreenWidth - Width) // 2 , TopLeftY := (A_ScreenHeight - Height) // 2 Gui, -Caption +E0x80000 +LastFound +AlwaysOnTop +ToolWindow +OwnDialogs Gui, Show, NA ; show board (currently transparent) hwnd1 := WinExist() ; required for Gdip OnMessage(0x201, "WM_LBUTTONDOWN") , hbm := CreateDIBSection(Width, Height) , hdc := CreateCompatibleDC() , obm := SelectObject(hdc, hbm) , G := Gdip_GraphicsFromHDC(hdc) , Gdip_SetSmoothingMode(G, 4)
Loop 1 ; remove '1' and uncomment next line to loop infinitely {
- Gdip_GraphicsClear(G) ; uncomment to loop infinitely
cOdd := "0xFFFFCE9E" ; create brushes , cEven := "0xFFD18B47" , pBrushOdd := Gdip_BrushCreateSolid(cOdd) , pBrushEven := Gdip_BrushCreateSolid(cEven)
Loop 64 ; layout board { Row := mod(A_Index-1,8)+1 , Col := (A_Index-1)//8+1 , Gdip_FillRectangle(G, mod(Row+Col,2) ? pBrushOdd : pBrushEven, Col * CellSize + 1, Row * CellSize + 1, CellSize - 2, CellSize - 2) } Gdip_DeleteBrush(pBrushOdd) ; cleanup memory , Gdip_DeleteBrush(pBrushEven) , UpdateLayeredWindow(hwnd1, hdc, TopLeftX, TopLeftY, Width, Height) ; update board
, pPen := Gdip_CreatePen(0x66FF0000, CellSize/10) ; create pen , Algebraic := SubStr(tour,1,2) ; get starting coordinates , x := (Asc(SubStr(Algebraic, 1, 1))-96+0.5)*CellSize , y := (9.5-SubStr(Algebraic, 2, 1))*CellSize
Loop 64 ; trace path { Sleep, 0.5*1000 xold := x, yold := y ; a line has start and end points , Algebraic := SubStr(tour,(A_Index)*3+1,2) ; get new coordinates , x := (Asc(SubStr(Algebraic, 1, 1))-96+0.5)*CellSize , y := (9.5-SubStr(Algebraic, 2, 1))*CellSize , Gdip_DrawLine(G, pPen, xold, yold, x, y) , UpdateLayeredWindow(hwnd1, hdc, TopLeftX, TopLeftY, Width, Height) ; update board } Gdip_DeletePen(pPen) } Return
GuiEscape: ExitApp
Exit: Gdip_Shutdown(pToken) ExitApp
WM_LBUTTONDOWN() { If (A_Gui = 1) PostMessage, 0xA1, 2 }</lang>
- Output:
For start at b3
b3 d2 c4 a5 b7 d8 e6 d4 b5 c7 a8 b6 c8 a7 c6 b8 a6 b4 d5 e3 d1 b2 a4 c5 d7 f8 h7 f6 g8 h6 f7 h8 g6 e7 f5 h4 g2 e1 d3 e5 g4 f2 h1 g3 f1 h2 f3 g1 h3 g5 e4 d6 e8 g7 h5 f4 e2 c1 a2 c3 b1 a3 c2 a1
... plus an animation.
AWK
<lang AWK>
- syntax: GAWK -f KNIGHTS_TOUR.AWK [-v sr=x] [-v sc=x]
- examples:
- GAWK -f KNIGHTS_TOUR.AWK (default)
- GAWK -f KNIGHTS_TOUR.AWK -v sr=1 -v sc=1 start at top left (default)
- GAWK -f KNIGHTS_TOUR.AWK -v sr=1 -v sc=8 start at top right
- GAWK -f KNIGHTS_TOUR.AWK -v sr=8 -v sc=8 start at bottom right
- GAWK -f KNIGHTS_TOUR.AWK -v sr=8 -v sc=1 start at bottom left
BEGIN {
N = 8 # board size
if (sr == "") { sr = 1 } # starting row
if (sc == "") { sc = 1 } # starting column
split("2 2 -2 -2 1 1 -1 -1",X," ")
split("1 -1 1 -1 2 -2 2 -2",Y," ")
printf("\n%dx%d board: starting row=%d col=%d\n",N,N,sr,sc)
move(sr,sc,0)
exit(1)
} function move(x,y,m) {
if (cantMove(x,y)) {
return(0)
}
P[x,y] = ++m
if (m == N ^ 2) {
printBoard()
exit(0)
}
tryBestMove(x,y,m)
} function cantMove(x,y) {
return( P[x,y] || x<1 || x>N || y<1 || y>N )
} function tryBestMove(x,y,m, i) {
i = bestMove(x,y) move(x+X[i],y+Y[i],m)
} function bestMove(x,y, arg1,arg2,c,i,min,out) {
- Warnsdorff's rule: go to where there are fewest next moves
min = N ^ 2 + 1
for (i in X) {
arg1 = x + X[i]
arg2 = y + Y[i]
if (!cantMove(arg1,arg2)) {
c = countNext(arg1,arg2)
if (c < min) {
min = c
out = i
}
}
}
return(out)
} function countNext(x,y, i,out) {
for (i in X) {
out += (!cantMove(x+X[i],y+Y[i]))
}
return(out)
} function printBoard( i,j,leng) {
leng = length(N*N)
for (i=1; i<=N; i++) {
for (j=1; j<=N; j++) {
printf(" %*d",leng,P[i,j])
}
printf("\n")
}
} </lang>
output:
8x8 board: starting row=1 col=1 1 50 15 32 61 28 13 30 16 33 64 55 14 31 60 27 51 2 49 44 57 62 29 12 34 17 56 63 54 47 26 59 3 52 45 48 43 58 11 40 18 35 20 53 46 41 8 25 21 4 37 42 23 6 39 10 36 19 22 5 38 9 24 7
BBC BASIC
<lang bbcbasic> VDU 23,22,256;256;16,16,16,128
VDU 23,23,4;0;0;0;
OFF
GCOL 4,15
FOR x% = 0 TO 512-128 STEP 128
RECTANGLE FILL x%,0,64,512
NEXT
FOR y% = 0 TO 512-128 STEP 128
RECTANGLE FILL 0,y%,512,64
NEXT
GCOL 9
DIM Board%(7,7)
X% = 0
Y% = 0
Total% = 0
REPEAT
Board%(X%,Y%) = TRUE
IF Total% DRAW X%*64+32,Y%*64+32 ELSE MOVE X%*64+32,Y%*64+32
Total% += 1
UNTIL NOT FNchoosemove(X%, Y%)
IF Total%<>64 STOP
REPEAT WAIT 1 : UNTIL FALSE
END
DEF FNchoosemove(RETURN X%, RETURN Y%)
LOCAL M%, newx%, newy%
M% = 9
PROCtrymove(X%+1, Y%+2, M%, newx%, newy%)
PROCtrymove(X%+1, Y%-2, M%, newx%, newy%)
PROCtrymove(X%-1, Y%+2, M%, newx%, newy%)
PROCtrymove(X%-1, Y%-2, M%, newx%, newy%)
PROCtrymove(X%+2, Y%+1, M%, newx%, newy%)
PROCtrymove(X%+2, Y%-1, M%, newx%, newy%)
PROCtrymove(X%-2, Y%+1, M%, newx%, newy%)
PROCtrymove(X%-2, Y%-1, M%, newx%, newy%)
IF M%=9 THEN = FALSE
X% = newx% : Y% = newy%
= TRUE
DEF PROCtrymove(X%, Y%, RETURN M%, RETURN newx%, RETURN newy%)
LOCAL N%
IF NOT FNvalidmove(X%,Y%) THEN ENDPROC
IF FNvalidmove(X%+1,Y%+2) N% += 1
IF FNvalidmove(X%+1,Y%-2) N% += 1
IF FNvalidmove(X%-1,Y%+2) N% += 1
IF FNvalidmove(X%-1,Y%-2) N% += 1
IF FNvalidmove(X%+2,Y%+1) N% += 1
IF FNvalidmove(X%+2,Y%-1) N% += 1
IF FNvalidmove(X%-2,Y%+1) N% += 1
IF FNvalidmove(X%-2,Y%-1) N% += 1
IF N%>M% THEN ENDPROC
IF N%=M% IF RND(2)=1 THEN ENDPROC
M% = N%
newx% = X% : newy% = Y%
ENDPROC
DEF FNvalidmove(X%,Y%)
IF X%<0 OR X%>7 OR Y%<0 OR Y%>7 THEN = FALSE
= NOT(Board%(X%,Y%))</lang>
Bracmat
<lang bracmat> ( knightsTour
= validmoves WarnsdorffSort algebraicNotation init solve
, x y fieldsToVisit
. ~
| ( validmoves
= x y jumps moves
. !arg:(?x.?y)
& :?moves
& ( jumps
= dx dy Fs fxs fys fx fy
. !arg:(?dx.?dy)
& 1 -1:?Fs
& !Fs:?fxs
& whl
' ( !fxs:%?fx ?fxs
& !Fs:?fys
& whl
' ( !fys:%?fy ?fys
& ( (!x+!fx*!dx.!y+!fy*!dy)
: (>0:<9.>0:<9)
|
)
!moves
: ?moves
)
)
)
& jumps$(1.2)
& jumps$(2.1)
& !moves
)
& ( init
= fields x y
. :?fields
& 0:?x
& whl
' ( 1+!x:<9:?x
& 0:?y
& whl
' ( 1+!y:<9:?y
& (!x.!y) !fields:?fields
)
)
& !fields
)
& init$:?fieldsToVisit
& ( WarnsdorffSort
= sum moves elm weightedTerms
. ( weightedTerms
= pos alts fieldsToVisit moves move weight
. !arg:(%?pos ?alts.?fieldsToVisit)
& ( !fieldsToVisit:!pos
& (0.!pos)
| !fieldsToVisit:? !pos ?
& validmoves$!pos:?moves
& 0:?weight
& whl
' ( !moves:%?move ?moves
& ( !fieldsToVisit:? !move ?
& !weight+1:?weight
|
)
)
& (!weight.!pos)
| 0
)
+ weightedTerms$(!alts.!fieldsToVisit)
| 0
)
& weightedTerms$!arg:?sum
& :?moves
& whl
' ( !sum:(#.?elm)+?sum
& !moves !elm:?moves
)
& !moves
)
& ( solve
= pos alts fieldsToVisit A Z tailOfSolution
. !arg:(%?pos ?alts.?fieldsToVisit)
& ( !fieldsToVisit:?A !pos ?Z
& ( !A !Z:&
| solve
$ ( WarnsdorffSort$(validmoves$!pos.!A !Z)
. !A !Z
)
)
| solve$(!alts.!fieldsToVisit)
)
: ?tailOfSolution
& !pos !tailOfSolution
)
& ( algebraicNotation
= x y
. !arg:(?x.?y) ?arg
& str$(chr$(asc$a+!x+-1) !y " ")
algebraicNotation$!arg
|
)
& @(!arg:?x #?y)
& asc$!x+-1*asc$a+1:?x
& str
$ (algebraicNotation$(solve$((!x.!y).!fieldsToVisit)))
)
& out$(knightsTour$a1);</lang>
a1 b3 a5 b7 d8 f7 h8 g6 f8 h7 g5 h3 g1 e2 c1 a2 b4 a6 b8 c6 a7 c8 e7 g8 h6 g4 h2 f1 d2 b1 a3 c2 e1 f3 h4 g2 e3 d1 b2 a4 c3 b5 d4 f5 d6 c4 e5 d3 f2 h1 g3 e4 c5 d7 b6 a8 c7 d5 f4 e6 g7 e8 f6 h5
C
For an animated version using OpenGL, see Knight's tour/C.
The following draws on console the progress of the horsie. Specify board size on commandline, or use default 8. <lang c>#include <stdio.h>
- include <stdlib.h>
- include <string.h>
- include <unistd.h>
typedef unsigned char cell; int dx[] = { -2, -2, -1, 1, 2, 2, 1, -1 }; int dy[] = { -1, 1, 2, 2, 1, -1, -2, -2 };
void init_board(int w, int h, cell **a, cell **b) { int i, j, k, x, y, p = w + 4, q = h + 4; /* b is board; a is board with 2 rows padded at each side */ a[0] = (cell*)(a + q); b[0] = a[0] + 2;
for (i = 1; i < q; i++) { a[i] = a[i-1] + p; b[i] = a[i] + 2; }
memset(a[0], 255, p * q); for (i = 0; i < h; i++) { for (j = 0; j < w; j++) { for (k = 0; k < 8; k++) { x = j + dx[k], y = i + dy[k]; if (b[i+2][j] == 255) b[i+2][j] = 0; b[i+2][j] += x >= 0 && x < w && y >= 0 && y < h; } } } }
- define E "\033["
int walk_board(int w, int h, int x, int y, cell **b) { int i, nx, ny, least; int steps = 0; printf(E"H"E"J"E"%d;%dH"E"32m[]"E"m", y + 1, 1 + 2 * x);
while (1) { /* occupy cell */ b[y][x] = 255;
/* reduce all neighbors' neighbor count */ for (i = 0; i < 8; i++) b[ y + dy[i] ][ x + dx[i] ]--;
/* find neighbor with lowest neighbor count */ least = 255; for (i = 0; i < 8; i++) { if (b[ y + dy[i] ][ x + dx[i] ] < least) { nx = x + dx[i]; ny = y + dy[i]; least = b[ny][nx]; } }
if (least > 7) { printf(E"%dH", h + 2); return steps == w * h - 1; }
if (steps++) printf(E"%d;%dH[]", y + 1, 1 + 2 * x); x = nx, y = ny; printf(E"%d;%dH"E"31m[]"E"m", y + 1, 1 + 2 * x); fflush(stdout); usleep(120000); } }
int solve(int w, int h) { int x = 0, y = 0; cell **a, **b; a = malloc((w + 4) * (h + 4) + sizeof(cell*) * (h + 4)); b = malloc((h + 4) * sizeof(cell*));
while (1) { init_board(w, h, a, b); if (walk_board(w, h, x, y, b + 2)) { printf("Success!\n"); return 1; } if (++x >= w) x = 0, y++; if (y >= h) { printf("Failed to find a solution\n"); return 0; } printf("Any key to try next start position"); getchar(); } }
int main(int c, char **v) { int w, h; if (c < 2 || (w = atoi(v[1])) <= 0) w = 8; if (c < 3 || (h = atoi(v[2])) <= 0) h = w; solve(w, h);
return 0; }</lang>
C++
Uses Warnsdorff's rule and (iterative) backtracking if that fails.
<lang cpp>#include <iostream>
- include <iomanip>
- include <array>
- include <string>
- include <tuple>
- include <algorithm>
using namespace std;
template<int N = 8> class Board { public:
array<pair<int, int>, 8> moves; array<array<int, N>, N> data;
Board()
{
moves[0] = make_pair(2, 1);
moves[1] = make_pair(1, 2);
moves[2] = make_pair(-1, 2);
moves[3] = make_pair(-2, 1);
moves[4] = make_pair(-2, -1);
moves[5] = make_pair(-1, -2);
moves[6] = make_pair(1, -2);
moves[7] = make_pair(2, -1);
}
array<int, 8> sortMoves(int x, int y) const
{
array<tuple<int, int>, 8> counts;
for(int i = 0; i < 8; ++i)
{
int dx = get<0>(moves[i]);
int dy = get<1>(moves[i]);
int c = 0;
for(int j = 0; j < 8; ++j)
{
int x2 = x + dx + get<0>(moves[j]);
int y2 = y + dy + get<1>(moves[j]);
if (x2 < 0 || x2 >= N || y2 < 0 || y2 >= N)
continue;
if(data[y2][x2] != 0)
continue;
c++;
}
counts[i] = make_tuple(c, i);
}
// Shuffle to randomly break ties
random_shuffle(counts.begin(), counts.end());
// Lexicographic sort
sort(counts.begin(), counts.end());
array<int, 8> out;
for(int i = 0; i < 8; ++i)
out[i] = get<1>(counts[i]);
return out;
}
void solve(string start)
{
for(int v = 0; v < N; ++v)
for(int u = 0; u < N; ++u)
data[v][u] = 0;
int x0 = start[0] - 'a';
int y0 = N - (start[1] - '0');
data[y0][x0] = 1;
array<tuple<int, int, int, array<int, 8>>, N*N> order;
order[0] = make_tuple(x0, y0, 0, sortMoves(x0, y0));
int n = 0;
while(n < N*N-1)
{
int x = get<0>(order[n]);
int y = get<1>(order[n]);
bool ok = false;
for(int i = get<2>(order[n]); i < 8; ++i)
{
int dx = moves[get<3>(order[n])[i]].first;
int dy = moves[get<3>(order[n])[i]].second;
if(x+dx < 0 || x+dx >= N || y+dy < 0 || y+dy >= N)
continue;
if(data[y + dy][x + dx] != 0)
continue;
++n;
get<2>(order[n]) = i + 1;
data[y+dy][x+dx] = n + 1;
order[n] = make_tuple(x+dx, y+dy, 0, sortMoves(x+dx, y+dy));
ok = true;
break;
}
if(!ok) // Failed. Backtrack.
{
data[y][x] = 0;
--n;
}
}
}
template<int N> friend ostream& operator<<(ostream &out, const Board<N> &b);
};
template<int N> ostream& operator<<(ostream &out, const Board<N> &b) {
for (int v = 0; v < N; ++v)
{
for (int u = 0; u < N; ++u)
{
if (u != 0) out << ",";
out << setw(3) << b.data[v][u];
}
out << endl;
}
return out;
}
int main() {
Board<5> b1;
b1.solve("c3");
cout << b1 << endl;
Board<8> b2;
b2.solve("b5");
cout << b2 << endl;
Board<31> b3; // Max size for <1000 squares
b3.solve("a1");
cout << b3 << endl;
return 0;
}</lang>
Output:
23, 16, 11, 6, 21 10, 5, 22, 17, 12 15, 24, 1, 20, 7 4, 9, 18, 13, 2 25, 14, 3, 8, 19 63, 20, 3, 24, 59, 36, 5, 26 2, 23, 64, 37, 4, 25, 58, 35 19, 62, 21, 50, 55, 60, 27, 6 22, 1, 54, 61, 38, 45, 34, 57 53, 18, 49, 44, 51, 56, 7, 28 12, 15, 52, 39, 46, 31, 42, 33 17, 48, 13, 10, 43, 40, 29, 8 14, 11, 16, 47, 30, 9, 32, 41 275,112, 19,116,277,604, 21,118,823,770, 23,120,961,940, 25,122,943,926, 27,124,917,898, 29,126,911,872, 31,128,197,870, 33 18,115,276,601, 20,117,772,767, 22,119,958,851, 24,121,954,941, 26,123,936,925, 28,125,912,899, 30,127,910,871, 32,129,198 111,274,113,278,605,760,603,822,771,824,769,948,957,960,939,944,953,942,927,916,929,918,897,908,913,900,873,196,875, 34,869 114, 17,600,273,602,775,766,773,768,949,850,959,852,947,952,955,932,937,930,935,924,915,920,905,894,909,882,901,868,199,130 271,110,279,606,759,610,761,776,821,764,825,816,951,956,853,938,945,934,923,928,919,896,893,914,907,904,867,874,195,876, 35 16,581,272,599,280,607,774,765,762,779,950,849,826,815,946,933,854,931,844,857,890,921,906,895,886,883,902,881,200,131,194 109,270,281,580,609,758,611,744,777,820,763,780,817,848,827,808,811,846,855,922,843,858,889,892,903,866,885,192,877, 36,201 282, 15,582,269,598,579,608,757,688,745,778,819,754,783,814,847,828,807,810,845,856,891,842,859,884,887,880,863,202,193,132 267,108,283,578,583,612,689,614,743,756,691,746,781,818,753,784,809,812,829,806,801,840,835,888,865,862,203,878,191,530, 37 14,569,268,585,284,597,576,619,690,687,742,755,692,747,782,813,752,785,802,793,830,805,860,841,836,879,864,529,204,133,190 107,266,285,570,577,584,613,686,615,620,695,684,741,732,711,748,739,794,751,786,803,800,839,834,861,528,837,188,531, 38,205 286, 13,568,265,586,575,596,591,618,685,616,655,696,693,740,733,712,749,738,795,792,831,804,799,838,833,722,527,206,189,134 263,106,287,508,571,590,587,574,621,592,639,694,683,656,731,710,715,734,787,750,737,796,791,832,721,798,207,532,187,474, 39 12,417,264,567,288,509,572,595,588,617,654,657,640,697,680,713,730,709,716,735,788,727,720,797,790,723,526,473,208,135,186 105,262,289,416,507,566,589,512,573,622,593,638,653,682,659,698,679,714,729,708,717,736,789,726,719,472,533,184,475, 40,209 290, 11,418,261,502,415,510,565,594,513,562,641,658,637,652,681,660,699,678,669,728,707,718,675,724,525,704,471,210,185,136 259,104,291,414,419,506,503,514,511,564,623,548,561,642,551,636,651,670,661,700,677,674,725,706,703,534,211,476,183,396, 41 10,331,260,493,292,501,420,495,504,515,498,563,624,549,560,643,662,635,650,671,668,701,676,673,524,705,470,395,212,137,182 103,258,293,330,413,494,505,500,455,496,547,516,485,552,625,550,559,644,663,634,649,672,667,702,535,394,477,180,397, 42,213 294, 9,332,257,492,329,456,421,490,499,458,497,546,517,484,553,626,543,558,645,664,633,648,523,666,469,536,393,220,181,138 255,102,295,328,333,412,491,438,457,454,489,440,459,486,545,518,483,554,627,542,557,646,665,632,537,478,221,398,179,214, 43 8,319,256,335,296,345,326,409,422,439,436,453,488,441,460,451,544,519,482,555,628,541,522,647,468,631,392,219,222,139,178 101,254,297,320,327,334,411,346,437,408,423,368,435,452,487,442,461,450,445,520,481,556,629,538,479,466,399,176,215, 44,165 298, 7,318,253,336,325,344,349,410,347,360,407,424,383,434,427,446,443,462,449,540,521,480,467,630,391,218,223,164,177,140 251,100,303,300,321,316,337,324,343,350,369,382,367,406,425,384,433,428,447,444,463,430,539,390,465,400,175,216,169,166, 45 6,299,252,317,304,301,322,315,348,361,342,359,370,381,366,405,426,385,432,429,448,389,464,401,174,217,224,163,150,141,168 99,250,241,302,235,248,307,338,323,314,351,362,341,358,371,380,365,404,377,386,431,402,173,388,225,160,153,170,167, 46,143 240, 5, 98,249,242,305,234,247,308,339,232,313,352,363,230,357,372,379,228,403,376,387,226,159,154,171,162,149,142,151, 82 63, 2,239, 66, 97,236,243,306,233,246,309,340,231,312,353,364,229,356,373,378,227,158,375,172,161,148,155,152, 83,144, 47 4, 67, 64, 61,238, 69, 96, 59,244, 71, 94, 57,310, 73, 92, 55,354, 75, 90, 53,374, 77, 88, 51,156, 79, 86, 49,146, 81, 84 1, 62, 3, 68, 65, 60,237, 70, 95, 58,245, 72, 93, 56,311, 74, 91, 54,355, 76, 89, 52,157, 78, 87, 50,147, 80, 85, 48,145
C#
<lang csharp>using System; using System.Collections.Generic;
namespace prog { class MainClass { const int N = 8;
readonly static int[,] moves = { {+1,-2},{+2,-1},{+2,+1},{+1,+2}, {-1,+2},{-2,+1},{-2,-1},{-1,-2} }; struct ListMoves { public int x, y; public ListMoves( int _x, int _y ) { x = _x; y = _y; } }
public static void Main (string[] args) { int[,] board = new int[N,N]; board.Initialize();
int x = 0, // starting position y = 0;
List<ListMoves> list = new List<ListMoves>(N*N); list.Add( new ListMoves(x,y) );
do { if ( Move_Possible( board, x, y ) ) { int move = board[x,y]; board[x,y]++; x += moves[move,0]; y += moves[move,1]; list.Add( new ListMoves(x,y) ); } else { if ( board[x,y] >= 8 ) { board[x,y] = 0; list.RemoveAt(list.Count-1); if ( list.Count == 0 ) { Console.WriteLine( "No solution found." ); return; } x = list[list.Count-1].x; y = list[list.Count-1].y; } board[x,y]++; } } while( list.Count < N*N );
int last_x = list[0].x, last_y = list[0].y; string letters = "ABCDEFGH"; for( int i=1; i<list.Count; i++ ) { Console.WriteLine( string.Format("{0,2}: ", i) + letters[last_x] + (last_y+1) + " - " + letters[list[i].x] + (list[i].y+1) );
last_x = list[i].x; last_y = list[i].y; } }
static bool Move_Possible( int[,] board, int cur_x, int cur_y ) { if ( board[cur_x,cur_y] >= 8 ) return false;
int new_x = cur_x + moves[board[cur_x,cur_y],0], new_y = cur_y + moves[board[cur_x,cur_y],1];
if ( new_x >= 0 && new_x < N && new_y >= 0 && new_y < N && board[new_x,new_y] == 0 ) return true;
return false; } } }</lang>
CoffeeScript
This algorithm finds 100,000 distinct solutions to the 8x8 problem in about 30 seconds. It precomputes knight moves up front, so it turns into a pure graph traversal problem. The program uses iteration and backtracking to find solutions. <lang coffeescript> graph_tours = (graph, max_num_solutions) ->
# graph is an array of arrays
# graph[3] = [4, 5] means nodes 4 and 5 are reachable from node 3
#
# Returns an array of tours (up to max_num_solutions in size), where
# each tour is an array of nodes visited in order, and where each
# tour visits every node in the graph exactly once.
#
complete_tours = []
visited = (false for node in graph)
dead_ends = ({} for node in graph)
tour = [0]
valid_neighbors = (i) ->
arr = []
for neighbor in graph[i]
continue if visited[neighbor]
continue if dead_ends[i][neighbor]
arr.push neighbor
arr
next_square_to_visit = (i) ->
arr = valid_neighbors i
return null if arr.length == 0
# We traverse to our neighbor who has the fewest neighbors itself.
fewest_neighbors = valid_neighbors(arr[0]).length
neighbor = arr[0]
for i in [1...arr.length]
n = valid_neighbors(arr[i]).length
if n < fewest_neighbors
fewest_neighbors = n
neighbor = arr[i]
neighbor
while tour.length > 0
current_square = tour[tour.length - 1]
visited[current_square] = true
next_square = next_square_to_visit current_square
if next_square?
tour.push next_square
if tour.length == graph.length
complete_tours.push (n for n in tour) # clone
break if complete_tours.length == max_num_solutions
# pessimistically call this a dead end
dead_ends[current_square][next_square] = true
current_square = next_square
else
# we backtrack
doomed_square = tour.pop()
dead_ends[doomed_square] = {}
visited[doomed_square] = false
complete_tours
knight_graph = (board_width) ->
# Turn the Knight's Tour into a pure graph-traversal problem
# by precomputing all the legal moves. Returns an array of arrays,
# where each element in any subarray is the index of a reachable node.
index = (i, j) ->
# index squares from 0 to n*n - 1
board_width * i + j
reachable_squares = (i, j) ->
deltas = [
[ 1, 2]
[ 1, -2]
[ 2, 1]
[ 2, -1]
[-1, 2]
[-1, -2]
[-2, 1]
[-2, -1]
]
neighbors = []
for delta in deltas
[di, dj] = delta
ii = i + di
jj = j + dj
if 0 <= ii < board_width
if 0 <= jj < board_width
neighbors.push index(ii, jj)
neighbors
graph = []
for i in [0...board_width]
for j in [0...board_width]
graph[index(i, j)] = reachable_squares i, j
graph
illustrate_knights_tour = (tour, board_width) ->
pad = (n) ->
return " _" if !n?
return " " + n if n < 10
"#{n}"
console.log "\n------"
moves = {}
for square, i in tour
moves[square] = i + 1
for i in [0...board_width]
s =
for j in [0...board_width]
s += " " + pad moves[i*board_width + j]
console.log s
BOARD_WIDTH = 8 MAX_NUM_SOLUTIONS = 100000
graph = knight_graph BOARD_WIDTH tours = graph_tours graph, MAX_NUM_SOLUTIONS console.log "#{tours.length} tours found (showing first and last)" illustrate_knights_tour tours[0], BOARD_WIDTH illustrate_knights_tour tours.pop(), BOARD_WIDTH </lang>
output <lang> > time coffee knight.coffee 100000 tours found (showing first and last)
1 4 57 20 47 6 49 22 34 19 2 5 58 21 46 7 3 56 35 60 37 48 23 50 18 33 38 55 52 59 8 45 39 14 53 36 61 44 51 24 32 17 40 43 54 27 62 9 13 42 15 30 11 64 25 28 16 31 12 41 26 29 10 63
1 4 41 20 63 6 61 22 34 19 2 5 42 21 44 7 3 40 35 64 37 62 23 60 18 33 38 47 56 43 8 45 39 14 57 36 49 46 59 24 32 17 48 55 58 27 50 9 13 54 15 30 11 52 25 28 16 31 12 53 26 29 10 51
real 0m29.741s user 0m25.656s sys 0m0.253s </lang>
Clojure
Using warnsdorff's rule <lang Clojure> (defn isin? [x li]
(not= [] (filter #(= x %) li)))
(defn options [movements pmoves n]
(let [x (first (last movements)) y (second (last movements))
op (vec (map #(vector (+ x (first %)) (+ y (second %))) pmoves))
vop (filter #(and (>= (first %) 0) (>= (last %) 0)) op)
vop1 (filter #(and (< (first %) n) (< (last %) n)) vop)]
(vec (filter #(not (isin? % movements)) vop1))))
(defn next-move [movements pmoves n]
(let [op (options movements pmoves n)
sp (map #(vector % (count (options (conj movements %) pmoves n))) op)
m (apply min (map last sp))]
(first (rand-nth (filter #(= m (last %)) sp)))))
(defn jumps [n pos]
(let [movements (vector pos)
pmoves [[1 2] [1 -2] [2 1] [2 -1]
[-1 2] [-1 -2] [-2 -1] [-2 1]]]
(loop [mov movements x 1]
(if (= x (* n n))
mov
(let [np (next-move mov pmoves n)]
(recur (conj mov np) (inc x)))))))
</lang>
- Output:
(jumps 5 [0 0])
[[0 0] [1 2] [0 4] [2 3] [4 4] [3 2] [4 0] [2 1] [1 3] [0 1] [2 0] [4 1] [3 3] [1 4] [0 2] [1 0] [3 1] [4 3] [2 4] [0 3] [1 1] [3 0] [4 2] [3 4] [2 2]]
(jumps 8 [0 0])
[[0 0] [2 1] [4 0] [6 1] [7 3] [6 5] [7 7] [5 6] [3 7] [1 6] [0 4] [1 2] [2 0] [0 1] [1 3] [0 5] [1 7] [2 5] [0 6] [2 7] [4 6] [6 7] [7 5] [6 3] [7 1] [5 0] [3 1] [1 0] [0 2] [1 4] [3 5] [4 7] [6 6] [7 4] [6 2] [7 0] [5 1] [7 2] [6 0] [4 1] [5 3] [3 2] [4 4] [5 2] [3 3] [5 4] [4 2] [2 3] [1 1] [3 0] [2 2] [0 3] [2 4] [4 3] [6 4] [4 5] [2 6] [0 7] [1 5] [3 4] [5 5] [7 6] [5 7] [3 6]]
(let [j (jumps 40 [0 0])] ;; are
(and (distinct? j) ;; all squares only once? and
(= (count j) (* 40 40)))) ;; 40*40 squares?
true
D
Fast Version
<lang d>import std.stdio, std.algorithm, std.random, std.range,
std.conv, std.typecons, std.typetuple;
int[N][N] knightTour(size_t N=8)(in string start) in {
assert(start.length >= 2);
} body {
static struct P { int x, y; }
immutable P[8] moves = [P(2,1), P(1,2), P(-1,2), P(-2,1),
P(-2,-1), P(-1,-2), P(1,-2), P(2,-1)];
int[N][N] data;
int[8] sortMoves(in int x, in int y) {
int[2][8] counts;
foreach (immutable i, immutable ref d1; moves) {
int c = 0;
foreach (immutable ref d2; moves) {
immutable p = P(x + d1.x + d2.x, y + d1.y + d2.y);
if (p.x >= 0 && p.x < N && p.y >= 0 && p.y < N &&
data[p.y][p.x] == 0)
c++;
}
counts[i] = [c, i];
}
counts[].randomShuffle; // Shuffle to randomly break ties.
counts[].sort(); // Lexicographic sort.
int[8] result = void;
transversal(counts[], 1).copy(result[]);
return result;
}
immutable p0 = P(start[0] - 'a', N - to!int(start[1 .. $])); data[p0.y][p0.x] = 1;
Tuple!(int, int, int, int[8])[N * N] order; order[0] = tuple(p0.x, p0.y, 0, sortMoves(p0.x, p0.y));
int n = 0;
while (n < (N * N - 1)) {
immutable int x = order[n][0];
immutable int y = order[n][1];
bool ok = false;
foreach (immutable i; order[n][2] .. 8) {
immutable P d = moves[order[n][3][i]];
if (x+d.x < 0 || x+d.x >= N || y+d.y < 0 || y+d.y >= N)
continue;
if (data[y + d.y][x + d.x] == 0) {
order[n][2] = i + 1;
n++;
data[y + d.y][x + d.x] = n + 1;
order[n] = tuple(x+d.x,y+d.y,0,sortMoves(x+d.x,y+d.y));
ok = true;
break;
}
}
if (!ok) { // Failed. Backtrack.
data[y][x] = 0;
n--;
}
}
return data;
}
void main() {
foreach (immutable i, side; TypeTuple!(5, 8, 31, 101)) {
immutable form = "%(%" ~ text(side ^^ 2).length.text ~ "d %)";
foreach (ref row; ["c3", "b5", "a1", "a1"][i].knightTour!side)
writefln(form, row);
writeln();
}
}</lang>
- Output:
23 16 11 6 21 10 5 22 17 12 15 24 1 20 7 4 9 18 13 2 25 14 3 8 19 63 20 3 24 59 36 5 26 2 23 64 37 4 25 58 35 19 62 21 50 55 60 27 6 22 1 54 61 38 45 34 57 53 18 49 44 51 56 7 28 12 15 52 39 46 31 42 33 17 48 13 10 43 40 29 8 14 11 16 47 30 9 32 41 275 112 19 116 277 604 21 118 823 770 23 120 961 940 25 122 943 926 27 124 917 898 29 126 911 872 31 128 197 870 33 18 115 276 601 20 117 772 767 22 119 958 851 24 121 954 941 26 123 936 925 28 125 912 899 30 127 910 871 32 129 198 111 274 113 278 605 760 603 822 771 824 769 948 957 960 939 944 953 942 927 916 929 918 897 908 913 900 873 196 875 34 869 114 17 600 273 602 775 766 773 768 949 850 959 852 947 952 955 932 937 930 935 924 915 920 905 894 909 882 901 868 199 130 271 110 279 606 759 610 761 776 821 764 825 816 951 956 853 938 945 934 923 928 919 896 893 914 907 904 867 874 195 876 35 16 581 272 599 280 607 774 765 762 779 950 849 826 815 946 933 854 931 844 857 890 921 906 895 886 883 902 881 200 131 194 109 270 281 580 609 758 611 744 777 820 763 780 817 848 827 808 811 846 855 922 843 858 889 892 903 866 885 192 877 36 201 282 15 582 269 598 579 608 757 688 745 778 819 754 783 814 847 828 807 810 845 856 891 842 859 884 887 880 863 202 193 132 267 108 283 578 583 612 689 614 743 756 691 746 781 818 753 784 809 812 829 806 801 840 835 888 865 862 203 878 191 530 37 14 569 268 585 284 597 576 619 690 687 742 755 692 747 782 813 752 785 802 793 830 805 860 841 836 879 864 529 204 133 190 107 266 285 570 577 584 613 686 615 620 695 684 741 732 711 748 739 794 751 786 803 800 839 834 861 528 837 188 531 38 205 286 13 568 265 586 575 596 591 618 685 616 655 696 693 740 733 712 749 738 795 792 831 804 799 838 833 722 527 206 189 134 263 106 287 508 571 590 587 574 621 592 639 694 683 656 731 710 715 734 787 750 737 796 791 832 721 798 207 532 187 474 39 12 417 264 567 288 509 572 595 588 617 654 657 640 697 680 713 730 709 716 735 788 727 720 797 790 723 526 473 208 135 186 105 262 289 416 507 566 589 512 573 622 593 638 653 682 659 698 679 714 729 708 717 736 789 726 719 472 533 184 475 40 209 290 11 418 261 502 415 510 565 594 513 562 641 658 637 652 681 660 699 678 669 728 707 718 675 724 525 704 471 210 185 136 259 104 291 414 419 506 503 514 511 564 623 548 561 642 551 636 651 670 661 700 677 674 725 706 703 534 211 476 183 396 41 10 331 260 493 292 501 420 495 504 515 498 563 624 549 560 643 662 635 650 671 668 701 676 673 524 705 470 395 212 137 182 103 258 293 330 413 494 505 500 455 496 547 516 485 552 625 550 559 644 663 634 649 672 667 702 535 394 477 180 397 42 213 294 9 332 257 492 329 456 421 490 499 458 497 546 517 484 553 626 543 558 645 664 633 648 523 666 469 536 393 220 181 138 255 102 295 328 333 412 491 438 457 454 489 440 459 486 545 518 483 554 627 542 557 646 665 632 537 478 221 398 179 214 43 8 319 256 335 296 345 326 409 422 439 436 453 488 441 460 451 544 519 482 555 628 541 522 647 468 631 392 219 222 139 178 101 254 297 320 327 334 411 346 437 408 423 368 435 452 487 442 461 450 445 520 481 556 629 538 479 466 399 176 215 44 165 298 7 318 253 336 325 344 349 410 347 360 407 424 383 434 427 446 443 462 449 540 521 480 467 630 391 218 223 164 177 140 251 100 303 300 321 316 337 324 343 350 369 382 367 406 425 384 433 428 447 444 463 430 539 390 465 400 175 216 169 166 45 6 299 252 317 304 301 322 315 348 361 342 359 370 381 366 405 426 385 432 429 448 389 464 401 174 217 224 163 150 141 168 99 250 241 302 235 248 307 338 323 314 351 362 341 358 371 380 365 404 377 386 431 402 173 388 225 160 153 170 167 46 143 240 5 98 249 242 305 234 247 308 339 232 313 352 363 230 357 372 379 228 403 376 387 226 159 154 171 162 149 142 151 82 63 2 239 66 97 236 243 306 233 246 309 340 231 312 353 364 229 356 373 378 227 158 375 172 161 148 155 152 83 144 47 4 67 64 61 238 69 96 59 244 71 94 57 310 73 92 55 354 75 90 53 374 77 88 51 156 79 86 49 146 81 84 1 62 3 68 65 60 237 70 95 58 245 72 93 56 311 74 91 54 355 76 89 52 157 78 87 50 147 80 85 48 145
Shorter Version
<lang d>import std.stdio, std.math, std.algorithm, std.range, std.typecons;
alias Square = Tuple!(int,"x", int,"y");
const(Square)[] knightTour(in Square[] board, in Square[] moves) pure @safe nothrow {
enum findMoves = (in Square sq) pure nothrow @safe =>
cartesianProduct([1, -1, 2, -2], [1, -1, 2, -2])
.filter!(ij => ij[0].abs != ij[1].abs)
.map!(ij => Square(sq.x + ij[0], sq.y + ij[1]))
.filter!(s => board.canFind(s) && !moves.canFind(s));
auto newMoves = findMoves(moves.back);
if (newMoves.empty)
return moves;
//alias warnsdorff = min!(s => findMoves(s).walkLength);
//immutable newSq = newMoves.dropOne.fold!warnsdorff(newMoves.front);
auto pairs = newMoves.map!(s => tuple(findMoves(s).walkLength, s));
immutable newSq = reduce!min(pairs.front, pairs.dropOne)[1];
return board.knightTour(moves ~ newSq);
}
void main(in string[] args) {
enum toSq = (in string xy) => Square(xy[0] - '`', xy[1] - '0');
immutable toAlg = (in Square s) => [dchar(s.x + '`'), dchar(s.y + '0')];
immutable sq = toSq((args.length == 2) ? args[1] : "e5");
const board = iota(1, 9).cartesianProduct(iota(1, 9)).map!Square.array;
writefln("%(%-(%s -> %)\n%)", board.knightTour([sq]).map!toAlg.chunks(8));
}</lang>
- Output:
e5 -> d7 -> b8 -> a6 -> b4 -> a2 -> c1 -> b3 a1 -> c2 -> a3 -> b1 -> d2 -> f1 -> h2 -> g4 h6 -> g8 -> e7 -> c8 -> a7 -> c6 -> a5 -> b7 d8 -> f7 -> h8 -> g6 -> f8 -> h7 -> f6 -> e8 g7 -> h5 -> g3 -> h1 -> f2 -> d1 -> b2 -> a4 b6 -> a8 -> c7 -> b5 -> c3 -> d5 -> e3 -> c4 d6 -> e4 -> c5 -> d3 -> e1 -> g2 -> h4 -> f5 d4 -> e2 -> f4 -> e6 -> g5 -> f3 -> g1 -> h3
EchoLisp
The algorithm uses iterative backtracking and Warnsdorff's heuristic. It can output closed or non-closed tours. <lang lisp> (require 'plot) (define *knight-moves* '((2 . 1)(2 . -1 ) (1 . -2) (-1 . -2 )(-2 . -1) (-2 . 1) (-1 . 2) (1 . 2))) (define *hit-squares* null) (define *legal-moves* null) (define *tries* 0)
(define (square x y n ) (+ y (* x n))) (define (dim n) (1- (* n n))) ; n^2 - 1
- check legal knight move from sq
- return null or (list destination-square)
(define (legal-disp n sq k-move)
(let ((x (+ (quotient sq n) (first k-move)))
(y (+ (modulo sq n) (rest k-move))))
(if (and (>= x 0) (< x n) (>= y 0) (< y n))
(list (square x y n)) null)))
;; list of legal destination squares from sq
(define (legal-moves sq k-moves n )
(if (null? k-moves) null
(append (legal-moves sq (rest k-moves) n) (legal-disp n sq (first k-moves)))))
- square freedom = number of destination squares not already reached
(define (freedom sq) (for/sum ((dest (vector-ref *legal-moves* sq))) (if (vector-ref *hit-squares* dest) 0 1)))
- The chess adage" A knight on the rim is dim" is false here
- choose to move to square with smallest freedom
- Warnsdorf's rule
(define (square-sort a b) (< (freedom a) (freedom b)))
- knight tour engine
(define (play sq step starter last-one wants-open) (set! *tries* (1+ *tries*)) (vector-set! *hit-squares* sq step) ;; flag used square (if (= step last-one) (throw 'HIT last-one)) ;; stop on first path found
(when (or wants-open ;; cut search iff closed path (and (< step last-one) (> (freedom starter) 0))) ;; this ensures a closed path
(for ((target (list-sort square-sort (vector-ref *legal-moves* sq)))) (unless (vector-ref *hit-squares* target) (play target (1+ step) starter last-one wants-open)))) (vector-set! *hit-squares* sq #f)) ;; unflag used square
(define (show-steps n wants-open) (string-delimiter "") (if wants-open (printf "♘-tour: %d tries." *tries*) (printf "♞-closed-tour: %d tries." *tries*)) (for ((x n)) (writeln) (for((y n))
(write (string-pad-right (vector-ref *hit-squares* (square x y n)) 4)))))
(define (k-tour (n 8) (starter 0) (wants-open #t))
(set! *hit-squares* (make-vector (* n n) #f))
- build vector of legal moves for squares 0..n^2-1
(set! *legal-moves* (build-vector (* n n) (lambda(sq) (legal-moves sq *knight-moves* n)))) (set! *tries* 0) ; counter (try (play starter 0 starter (dim n) wants-open) (catch (hit mess) (show-steps n wants-open)))) </lang>
- Output:
<lang lisp> (k-tour 8 0 #f) ♞-closed-tour: 66 tries. 0 47 14 31 62 27 12 29 15 32 63 54 13 30 57 26 48 1 46 61 56 59 28 11 33 16 55 50 53 44 25 58 2 49 42 45 60 51 10 39 17 34 19 52 43 40 7 24 20 3 36 41 22 5 38 9 35 18 21 4 37 8 23 6
(k-tour 20 57) ♘-tour: 400 tries. 31 34 29 104 209 36 215 300 211 38 213 354 343 40 345 386 383 42 1 388 28 103 32 35 216 299 210 37 214 335 342 39 346 385 382 41 390 387 396 43 33 30 105 208 201 308 301 336 323 212 353 340 355 344 391 384 395 0 389 2 102 27 202 219 298 217 322 309 334 341 356 347 358 351 376 381 378 399 44 397 203 106 207 200 307 228 311 302 337 324 339 352 373 364 379 392 375 394 3 368 26 101 220 229 218 297 304 321 310 333 348 357 350 359 374 377 380 367 398 45 107 204 199 206 227 306 231 312 303 338 325 330 363 372 365 328 393 254 369 4 100 25 122 221 230 233 296 305 320 313 332 349 326 329 360 371 366 251 46 253 121 108 205 198 145 226 237 232 295 286 319 314 331 362 327 316 255 370 5 178 24 99 144 123 222 129 234 279 236 281 294 289 318 315 256 361 250 179 252 47 109 120 111 130 197 146 225 238 285 278 287 272 293 290 317 180 257 162 177 6 98 23 124 143 128 223 276 235 280 239 282 291 288 265 270 249 176 181 48 161 115 110 119 112 131 196 147 224 277 284 273 266 271 292 245 258 163 174 7 58 22 97 114 125 142 127 140 275 194 267 240 283 264 269 248 175 182 59 160 49 87 116 95 118 113 132 195 148 187 274 263 268 191 244 259 246 173 164 57 8 96 21 88 133 126 141 150 139 262 193 190 241 260 247 172 183 60 159 50 65 77 86 117 94 89 138 135 188 149 186 261 192 171 184 243 156 165 64 9 56 20 81 78 85 134 93 90 151 136 189 170 185 242 155 166 61 158 53 66 51 79 76 83 18 91 74 137 16 169 72 153 14 167 70 157 12 63 68 55 10 82 19 80 75 84 17 92 73 152 15 168 71 154 13 62 69 54 11 52 67 </lang>
- Plotting
64 shades of gray. We plot the move sequence in shades of gray, from black to white. The starting square is red. The ending square is green. One can observe that the squares near the border are played first (dark squares). <lang lisp> (define (step-color x y n last-one) (letrec ((sq (square (floor x) (floor y) n)) (step (vector-ref *hit-squares* sq) n n)) (cond ((= 0 step) (rgb 1 0 0)) ;; red starter ((= last-one step) (rgb 0 1 0)) ;; green end (else (gray (// step n n))))))
(define ( k-plot n) (plot-rgb (lambda (x y) (step-color x y n (dim n))) (- n epsilon) (- n epsilon))) </lang>
Closed path on a 12x12 board: [1]
![[1]](http://www.echolalie.org/echolisp/images/k-tour-12.png){kind=link}
Open path on a 24x24 board: [2]
![[2]](http://www.echolalie.org/echolisp/images/k-tour-24.png){kind=link}
Elixir
<lang elixir>defmodule Board do
import Integer, only: [is_odd: 1]
defmodule Cell do
defstruct [:value, :adj]
end
@adjacent [[-1,-2],[-2,-1],[-2,1],[-1,2],[1,2],[2,1],[2,-1],[1,-2]]
defp initialize(rows, cols) do
board = for i <- 1..rows, j <- 1..cols, into: %{}, do: {{i,j}, true}
for i <- 1..rows, j <- 1..cols, into: %{} do
adj = for [di,dj] <- @adjacent, board[{i+di, j+dj}], do: {i+di, j+dj}
{{i,j}, %Cell{value: 0, adj: adj}}
end
end
defp solve(board, ij, num, goal) do
board = Map.update!(board, ij, fn cell -> %{cell | value: num} end)
if num == goal do
throw({:ok, board})
else
wdof(board, ij)
|> Enum.each(fn k -> solve(board, k, num+1, goal) end)
end
end
defp wdof(board, ij) do # Warnsdorf's rule
board[ij].adj
|> Enum.filter(fn k -> board[k].value == 0 end)
|> Enum.sort_by(fn k ->
Enum.count(board[k].adj, fn x -> board[x].value == 0 end)
end)
end
defp to_string(board, rows, cols) do
width = to_string(rows * cols) |> String.length
format = String.duplicate("~#{width}w ", cols)
Enum.map_join(1..rows, "\n", fn i ->
:io_lib.fwrite format, (for j <- 1..cols, do: board[{i,j}].value)
end)
end
def knight_tour(rows, cols, sx, sy) do
IO.puts "\nBoard (#{rows} x #{cols}), Start: [#{sx}, #{sy}]"
if is_odd(rows*cols) and is_odd(sx+sy) do
IO.puts "No solution"
else
try do
initialize(rows, cols)
|> solve({sx,sy}, 1, rows*cols)
IO.puts "No solution"
catch
{:ok, board} -> IO.puts to_string(board, rows, cols)
end
end
end
end
Board.knight_tour(8,8,4,2) Board.knight_tour(5,5,3,3) Board.knight_tour(4,9,1,1) Board.knight_tour(5,5,1,2) Board.knight_tour(12,12,2,2)</lang>
- Output:
Board (8 x 8), Start: [4, 2] 23 20 3 32 25 10 5 8 2 35 24 21 4 7 26 11 19 22 33 36 31 28 9 6 34 1 50 29 48 37 12 27 51 18 53 44 61 30 47 38 54 43 56 49 58 45 62 13 17 52 41 60 15 64 39 46 42 55 16 57 40 59 14 63 Board (5 x 5), Start: [3, 3] 19 8 3 14 25 2 13 18 9 4 7 20 1 24 15 12 17 22 5 10 21 6 11 16 23 Board (4 x 9), Start: [1, 1] 1 34 3 28 13 24 9 20 17 4 29 6 33 8 27 18 23 10 35 2 31 14 25 12 21 16 19 30 5 36 7 32 15 26 11 22 Board (5 x 5), Start: [1, 2] No solution Board (12 x 12), Start: [2, 2] 87 24 59 2 89 26 61 4 39 8 31 6 58 1 88 25 60 3 92 27 30 5 38 9 23 86 83 90 103 98 29 62 93 40 7 32 82 57 102 99 84 91 104 97 28 37 10 41 101 22 85 114 105 116 111 94 63 96 33 36 56 81 100 123 128 113 106 117 110 35 42 11 21 122 141 80 115 124 127 112 95 64 109 34 144 55 78 121 142 129 118 107 126 133 12 43 51 20 143 140 79 120 125 138 69 108 65 134 54 73 52 77 130 139 70 119 132 137 44 13 19 50 75 72 17 48 131 68 15 46 135 66 74 53 18 49 76 71 16 47 136 67 14 45
Elm
<lang elm>import List exposing (concatMap, foldl, head,member,filter,length,minimum,concat,map,map2,tail) import List.Extra exposing (minimumBy, andThen) import String exposing (join) import Html as H import Html.Attributes as HA import Html.App exposing (program) import Time exposing (Time,every, second) import Svg exposing (rect, line, svg, g) import Svg.Events exposing (onClick) import Svg.Attributes exposing (version, viewBox, x, y, x1, y1, x2, y2, fill, style, width, height)
w = 450 h = 450 rowCount=20 colCount=20 dt = 0.03
type alias Cell = (Int, Int)
type alias Model =
{ path : List Cell
, board : List Cell
}
type Msg = NoOp | Tick Time | SetStart Cell
init : (Model,Cmd Msg) init =
let board = [0..rowCount-1] `andThen` \r ->
[0..colCount-1] `andThen` \c ->
[(r, c)]
path = []
in (Model path board, Cmd.none)
view : Model -> H.Html Msg view model =
let
showChecker row col =
rect [ x <| toString col
, y <| toString row
, width "1"
, height "1"
, fill <| if (row + col) % 2 == 0 then "blue" else "grey"
, onClick <| SetStart (row, col)
]
[]
showMove (row0,col0) (row1,col1) =
line [ x1 <| toString ((toFloat col0) + 0.5)
, y1 <| toString ((toFloat row0) + 0.5)
, x2 <| toString ((toFloat col1) + 0.5)
, y2 <| toString ((toFloat row1) + 0.5)
, style "stroke:yellow;stroke-width:0.05"
]
[]
render model =
let checkers = model.board `andThen` \(r,c) ->
[showChecker r c]
moves = case List.tail model.path of
Nothing -> []
Just tl -> List.map2 showMove model.path tl
in checkers ++ moves
unvisited = length model.board - length model.path
center = HA.style [ ( "text-align", "center") ]
in
H.div
[]
[ H.h2 [center] [H.text "Knight's Tour"]
, H.h2 [center] [H.text <| "Unvisited count : " ++ toString unvisited ]
, H.h2 [center] [H.text "(pick a square)"]
, H.div
[center]
[ svg
[ version "1.1"
, width (toString w)
, height (toString h)
, viewBox (join " "
[ toString 0
, toString 0
, toString colCount
, toString rowCount ])
]
[ g [] <| render model]
]
]
nextMoves : Model -> Cell -> List Cell nextMoves model (stRow,stCol) =
let c = [ 1, 2, -1, -2]
km = c `andThen` \cRow ->
c `andThen` \cCol ->
if abs(cRow) == abs(cCol) then [] else [(cRow,cCol)]
jumps = List.map (\(kmRow,kmCol) -> (kmRow + stRow, kmCol + stCol)) km
in List.filter (\j -> List.member j model.board && not (List.member j model.path) ) jumps
bestMove : Model -> Maybe Cell bestMove model =
case List.head (model.path) of
Nothing -> Nothing
Just mph -> minimumBy (List.length << nextMoves model) (nextMoves model mph)
update : Msg -> Model -> (Model, Cmd Msg) update msg model =
let mo = case msg of
SetStart start ->
{model | path = [start]}
Tick t ->
case model.path of
[] -> model
_ -> case bestMove model of
Nothing -> model
Just best -> {model | path = best::model.path }
NoOp -> model
in (mo, Cmd.none)
subscriptions : Model -> Sub Msg subscriptions _ =
Time.every (dt * second) Tick
main =
program
{ init = init
, view = view
, update = update
, subscriptions = subscriptions
}</lang>
Link to live demo: http://dc25.github.io/knightsTourElm/
Erlang
Again I use backtracking. It seemed easier this time. <lang Erlang> -module( knights_tour ).
-export( [display/1, solve/1, task/0] ).
display( Moves ) -> %% The knigh walks the moves {Position, Step_nr} order. %% Top left corner is {$a, 8}, Bottom right is {$h, 1}. io:fwrite( "Moves:" ), lists:foldl( fun display_moves/2, erlang:length(Moves), lists:keysort(2, Moves) ), io:nl(), [display_row(Y, Moves) || Y <- lists:seq(8, 1, -1)].
solve( First_square ) ->
try
bt_loop( 1, next_moves(First_square), [{First_square, 1}] )
catch
_:{ok, Moves} -> Moves
end.
task() -> io:fwrite( "Starting {a, 1}~n" ), Moves = solve( {$a, 1} ), display( Moves ).
bt( N, Move, Moves ) -> bt_reject( is_not_allowed_knight_move(Move, Moves), N, Move, [{Move, N} | Moves] ).
bt_accept( true, _N, _Move, Moves ) -> erlang:throw( {ok, Moves} ); bt_accept( false, N, Move, Moves ) -> bt_loop( N, next_moves(Move), Moves ).
bt_loop( N, New_moves, Moves ) -> [bt( N+1, X, Moves ) || X <- New_moves].
bt_reject( true, _N, _Move, _Moves ) -> backtrack; bt_reject( false, N, Move, Moves ) -> bt_accept( is_all_knights(Moves), N, Move, Moves ).
display_moves( {{X, Y}, 1}, Max ) -> io:fwrite(" ~p. N~c~p", [1, X, Y]), Max; display_moves( {{X, Y}, Max}, Max ) -> io:fwrite(" N~c~p~n", [X, Y]), Max; display_moves( {{X, Y}, Step_nr}, Max ) when Step_nr rem 8 =:= 0 -> io:fwrite(" N~c~p~n~p. N~c~p", [X, Y, Step_nr, X, Y]), Max; display_moves( {{X, Y}, Step_nr}, Max ) -> io:fwrite(" N~c~p ~p. N~c~p", [X, Y, Step_nr, X, Y]), Max.
display_row( Row, Moves ) -> [io:fwrite(" ~2b", [proplists:get_value({X, Row}, Moves)]) || X <- [$a, $b, $c, $d, $e, $f, $g, $h]], io:nl().
is_all_knights( Moves ) when erlang:length(Moves) =:= 64 -> true; is_all_knights( _Moves ) -> false.
is_asymetric( Start_column, Start_row, Stop_column, Stop_row ) -> erlang:abs( Start_column - Stop_column ) =/= erlang:abs( Start_row - Stop_row ).
is_not_allowed_knight_move( Move, Moves ) -> no_such_move =/= proplists:get_value( Move, Moves, no_such_move ).
next_moves( {Column, Row} ) -> [{X, Y} || X <- next_moves_column(Column), Y <- next_moves_row(Row), is_asymetric(Column, Row, X, Y)].
next_moves_column( $a ) -> [$b, $c]; next_moves_column( $b ) -> [$a, $c, $d]; next_moves_column( $g ) -> [$e, $f, $h]; next_moves_column( $h ) -> [$g, $f]; next_moves_column( C ) -> [C - 2, C - 1, C + 1, C + 2].
next_moves_row( 1 ) -> [2, 3]; next_moves_row( 2 ) -> [1, 3, 4]; next_moves_row( 7 ) -> [5, 6, 8]; next_moves_row( 8 ) -> [6, 7]; next_moves_row( N ) -> [N - 2, N - 1, N + 1, N + 2]. </lang>
- Output:
17> knights_tour:task().
Starting {a, 1}
Moves: 1. Na1 Nb3 2. Nb3 Na5 3. Na5 Nb7 4. Nb7 Nc5 5. Nc5 Na4 6. Na4 Nb2 7. Nb2 Nc4
8. Nc4 Na3 9. Na3 Nb1 10. Nb1 Nc3 11. Nc3 Na2 12. Na2 Nb4 13. Nb4 Na6 14. Na6 Nb8 15. Nb8 Nc6
16. Nc6 Na7 17. Na7 Nb5 18. Nb5 Nc7 19. Nc7 Na8 20. Na8 Nb6 21. Nb6 Nc8 22. Nc8 Nd6 23. Nd6 Ne4
24. Ne4 Nd2 25. Nd2 Nf1 26. Nf1 Ne3 27. Ne3 Nc2 28. Nc2 Nd4 29. Nd4 Ne2 30. Ne2 Nc1 31. Nc1 Nd3
32. Nd3 Ne1 33. Ne1 Ng2 34. Ng2 Nf4 35. Nf4 Nd5 36. Nd5 Ne7 37. Ne7 Ng8 38. Ng8 Nh6 39. Nh6 Nf5
40. Nf5 Nh4 41. Nh4 Ng6 42. Ng6 Nh8 43. Nh8 Nf7 44. Nf7 Nd8 45. Nd8 Ne6 46. Ne6 Nf8 47. Nf8 Nd7
48. Nd7 Ne5 49. Ne5 Ng4 50. Ng4 Nh2 51. Nh2 Nf3 52. Nf3 Ng1 53. Ng1 Nh3 54. Nh3 Ng5 55. Ng5 Nh7
56. Nh7 Nf6 57. Nf6 Ne8 58. Ne8 Ng7 59. Ng7 Nh5 60. Nh5 Ng3 61. Ng3 Nh1 62. Nh1 Nf2 63. Nf2 Nd1
20 15 22 45 58 47 38 43
17 4 19 48 37 44 59 56
14 21 16 23 46 57 42 39
3 18 5 36 49 40 55 60
6 13 8 29 24 35 50 41
9 2 11 32 27 52 61 54
12 7 28 25 30 63 34 51
1 10 31 64 33 26 53 62
ERRE
Taken from ERRE distribution disk. Comments are in Italian. <lang ERRE> ! ********************************************************************** ! * * ! * IL GIRO DEL CAVALLO - come collocare un cavallo su di una * ! * scacchiera n*n passando una sola volta * ! * per ogni casella. * ! * * ! ********************************************************************** ! ---------------------------------------------------------------------- ! Inizializzazione dei parametri ! ----------------------------------------------------------------------
PROGRAM KNIGHT
!$INTEGER !$KEY
DIM H[25,25],A[8],B[8],P0[8],P1[8]
!$INCLUDE="PC.LIB"
PROCEDURE INIT_SCACCHIERA ! ********************************************************************** ! * Routine di inizializzazione scacchiera * ! **********************************************************************
FOR I1=1 TO 8 DO
U=X+A[I1] V=Y+B[I1]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
H[U,V]=H[U,V]-1
END IF
END FOR
END PROCEDURE
PROCEDURE MOSTRA_SCACCHIERA ! ********************************************************************* ! * Routine di visualizzazione della scacchiera * ! *********************************************************************
LOCATE(5,1) COLOR(0,7) PRINT(" Mossa num.";NMOS) COLOR(7,0)
L2=N
FOR I2=1 TO N DO
PRINT
FOR L1=1 TO N DO
IF H[L1,L2]>0 THEN COLOR(15,0) END IF
WRITE("####";H[L1,L2];)
COLOR(7,0)
END FOR
L2=L2-1
END FOR
END PROCEDURE
PROCEDURE AGGIORNA_SCACCHIERA ! ********************************************************************* ! * Routine di Aggiornamento Scacchiera * ! *********************************************************************
B=1
FOR I1=1 TO 8 DO
U=X+A[I1] V=Y+B[I1]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 THEN
H[U,V]=H[U,V]+1 B=0
END IF
END IF
END FOR
IF B=1 THEN Q1=0 END IF
END PROCEDURE
PROCEDURE MOSSA_MAX_PESO ! ********************************************************************* ! * Cerca la prossima mossa con il massimo peso * ! *********************************************************************
M1=0 RO=1
FOR W=1 TO 8 DO
U=Z1+A[W] V=Z2+B[W]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 AND H[U,V]<=M1 THEN
IF H[U,V]=M1 THEN
RO=RO+1 P0[RO]=W
ELSE
M1=H[U,V] Q1=1 T1=U T2=V RO=1 P0[1]=W
END IF
END IF
END IF
END FOR
END PROCEDURE
PROCEDURE MOSSA_MIN_PESO ! ********************************************************************* ! * Cerca la prossima mossa con il minimo peso * ! *********************************************************************
M1=-9 RO=1
FOR W=1 TO 8 DO
U=Z1+A[W] V=Z2+B[W]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 AND H[U,V]>=M1 THEN
IF H[U,V]=M1 THEN
RO=RO+1 P0[RO]=W
ELSE
M1=H[U,V] Q1=1 T1=U T2=V RO=1 P0[1]=W
END IF
END IF
END IF
END FOR
END PROCEDURE
BEGIN
A[1]=1 A[2]=2 A[3]=2 A[4]=1
A[5]=-1 A[6]=-2 A[7]=-2 A[8]=-1
B[1]=2 B[2]=1 B[3]=-1 B[4]=-2
B[5]=-2 B[6]=-1 B[7]=1 B[8]=2
CLS
PRINT(" *** LA GALOPPATA DEL CAVALIERE ***")
PRINT
PRINT("Inserire la dimensione della scacchiera (max. 25)";)
INPUT(N)
PRINT("Inserire la caselle di partenza (x,y) ";)
INPUT(X1,Y1)
NMOS=1 A1=1 N1=N*N ESCAPE=FALSE
! ---------------------------------------------------------------------- ! Set della scacchiera ! ----------------------------------------------------------------------
WHILE NOT ESCAPE DO
FOR I=1 TO N DO
FOR J=1 TO N DO
H[I,J]=0
END FOR
END FOR
FOR I=1 TO N DO
FOR J=1 TO N DO
X=I Y=J
INIT_SCACCHIERA
END FOR
END FOR
! ---------------------------------------------------------------------- ! Effettua la prima mossa ! ----------------------------------------------------------------------
X=X1 Y=Y1 H[X,Y]=1 L=2
AGGIORNA_SCACCHIERA
Q1=1 Q2=1
! ----------------------------------------------------------------------- ! Trova la prossima mossa ! -----------------------------------------------------------------------
WHILE Q1<>0 AND Q2<>0 DO
Q1=0 Z1=X Z2=Y
MOSSA_MIN_PESO
IF RO<=1 THEN
C1=T1 C2=T2
ELSE
! ------------------------------------------------------------------------ ! Esamina tutti i vincoli ! ------------------------------------------------------------------------
FOR K=1 TO RO DO
P1[K]=P0[K]
END FOR
R1=RO
IF A1=1 THEN M2=-9 ELSE M2=0 END IF
FOR K=1 TO R1 DO
F1=P1[K] Z1=X+A[F1] Z2=Y+B[F1]
IF A1=1 THEN
MOSSA_MAX_PESO
IF M1<=M2 THEN
!$NULL
ELSE
M2=M1 C1=Z1 C2=Z2
END IF
ELSE
MOSSA_MIN_PESO
IF M1>=M2 THEN
!$NULL
ELSE
M2=M1 C1=Z1 C2=Z2
END IF
END IF
END FOR
! ------------------------------------------------------------------------ ! Prossima mossa trovata:aggiorna la scacchiera ! ------------------------------------------------------------------------
END IF
IF Q1<>0 THEN
X=C1 Y=C2 H[X,Y]=L
AGGIORNA_SCACCHIERA
IF L=N1 THEN Q2=0 END IF
END IF
L=L+1
MOSTRA_SCACCHIERA
NMOS=NMOS+1
END WHILE
! ------------------------------------------------------------------------ ! La ricerca è terminata: visualizza i risultati ! ------------------------------------------------------------------------
PRINT PRINT
IF Q2<>1 THEN
PRINT("*** Trovata la soluzione! ***")
MOSTRA_SCACCHIERA
ESCAPE=TRUE
ELSE
IF A1=0 THEN
PRINT("Nessuna soluzione.")
ESCAPE=TRUE
ELSE
BEEP
A1=0
END IF
END IF
END WHILE
REPEAT
GET(A$)
UNTIL A$<>""
END PROGRAM </lang>
- Output:
*** LA GALOPPATA DEL CAVALIERE *** Inserire la dimensione della scacchiera (max. 25)? 8 Inserire la caselle di partenza (x,y) ? 1,1 Mossa num. 64 64 7 54 41 60 9 48 39 53 42 61 8 55 40 35 10 6 63 44 59 34 49 38 47 43 52 21 62 45 56 11 36 20 5 58 33 50 37 46 25 31 2 51 22 57 26 15 12 4 19 32 29 14 17 24 27 1 30 3 18 23 28 13 16 *** Trovata la soluzione! ***
Fōrmulæ
In this page you can see the solution of this task.
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.
The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.
Go
Warnsdorf's rule
<lang go>package main
import (
"fmt" "math/rand" "time"
)
// input, 0-based start position const startRow = 0 const startCol = 0
func main() {
rand.Seed(time.Now().Unix())
for !knightTour() {
}
}
var moves = []struct{ dr, dc int }{
{2, 1},
{2, -1},
{1, 2},
{1, -2},
{-1, 2},
{-1, -2},
{-2, 1},
{-2, -1},
}
// Attempt knight tour starting at startRow, startCol using Warnsdorff's rule // and random tie breaking. If a tour is found, print it and return true. // Otherwise no backtracking, just return false. func knightTour() bool {
// 8x8 board. squares hold 1-based visit order. 0 means unvisited.
board := make([][]int, 8)
for i := range board {
board[i] = make([]int, 8)
}
r := startRow
c := startCol
board[r][c] = 1 // first move
for move := 2; move <= 64; move++ {
minNext := 8
var mr, mc, nm int
candidateMoves:
for _, cm := range moves {
cr := r + cm.dr
if cr < 0 || cr >= 8 { // off board
continue
}
cc := c + cm.dc
if cc < 0 || cc >= 8 { // off board
continue
}
if board[cr][cc] > 0 { // already visited
continue
}
// cr, cc candidate legal move.
p := 0 // count possible next moves.
for _, m2 := range moves {
r2 := cr + m2.dr
if r2 < 0 || r2 >= 8 {
continue
}
c2 := cc + m2.dc
if c2 < 0 || c2 >= 8 {
continue
}
if board[r2][c2] > 0 {
continue
}
p++
if p > minNext { // bail out as soon as it's eliminated
continue candidateMoves
}
}
if p < minNext { // it's better. keep it.
minNext = p // new min possible next moves
nm = 1 // number of candidates with this p
mr = cr // best candidate move
mc = cc
continue
}
// it ties for best so far.
// keep it with probability 1/(number of tying moves)
nm++ // number of tying moves
if rand.Intn(nm) == 0 { // one chance to keep it
mr = cr
mc = cc
}
}
if nm == 0 { // no legal move
return false
}
// make selected move
r = mr
c = mc
board[r][c] = move
}
// tour complete. print board.
for _, r := range board {
for _, m := range r {
fmt.Printf("%3d", m)
}
fmt.Println()
}
return true
}</lang>
- Output:
1 4 39 20 23 6 63 58 40 19 2 5 62 57 22 7 3 38 41 48 21 24 59 64 18 43 32 37 56 61 8 25 31 14 47 42 49 36 53 60 46 17 44 33 52 55 26 9 13 30 15 50 11 28 35 54 16 45 12 29 34 51 10 27
Ant colony
<lang go>/* Adapted from "Enumerating Knight's Tours using an Ant Colony Algorithm" by Philip Hingston and Graham Kendal, PDF at http://www.cs.nott.ac.uk/~gxk/papers/cec05knights.pdf. */
package main
import (
"fmt" "math/rand" "sync" "time"
)
const boardSize = 8 const nSquares = boardSize * boardSize const completeTour = nSquares - 1
// task input: starting square. These are 1 based, but otherwise 0 based // row and column numbers are used througout the program. const rStart = 2 const cStart = 3
// pheromone representation read by ants var tNet = make([]float64, nSquares*8)
// row, col deltas of legal moves var drc = [][]int{{1, 2}, {2, 1}, {2, -1}, {1, -2},
{-1, -2}, {-2, -1}, {-2, 1}, {-1, 2}}
// get square reached by following edge k from square (r, c) func dest(r, c, k int) (int, int, bool) {
r += drc[k][0] c += drc[k][1] return r, c, r >= 0 && r < boardSize && c >= 0 && c < boardSize
}
// struct represents a pheromone amount associated with a move type rckt struct {
r, c, k int t float64
}
func main() {
fmt.Println("Starting square: row", rStart, "column", cStart)
// initialize board
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
for k := 0; k < 8; k++ {
if _, _, ok := dest(r, c, k); ok {
tNet[(r*boardSize+c)*8+k] = 1e-6
}
}
}
}
// waitGroups for ant release clockwork var start, reset sync.WaitGroup start.Add(1) // channel for ants to return tours with pheremone updates tch := make(chan []rckt)
// create an ant for each square
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
go ant(r, c, &start, &reset, tch)
}
}
// accumulator for new pheromone amounts tNew := make([]float64, nSquares*8)
// each iteration is a "cycle" as described in the paper
for {
// evaporate pheromones
for i := range tNet {
tNet[i] *= .75
}
reset.Add(nSquares) // number of ants to release
start.Done() // release them
reset.Wait() // wait for them to begin searching
start.Add(1) // reset start signal for next cycle
// gather tours from ants
for i := 0; i < nSquares; i++ {
tour := <-tch
// watch for a complete tour from the specified starting square
if len(tour) == completeTour &&
tour[0].r == rStart-1 && tour[0].c == cStart-1 {
// task output: move sequence in a grid.
seq := make([]int, nSquares)
for i, sq := range tour {
seq[sq.r*boardSize+sq.c] = i + 1
}
last := tour[len(tour)-1]
r, c, _ := dest(last.r, last.c, last.k)
seq[r*boardSize+c] = nSquares
fmt.Println("Move sequence:")
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
fmt.Printf(" %3d", seq[r*boardSize+c])
}
fmt.Println()
}
return // task only requires finding a single tour
}
// accumulate pheromone amounts from all ants
for _, move := range tour {
tNew[(move.r*boardSize+move.c)*8+move.k] += move.t
}
}
// update pheromone amounts on network, reset accumulator
for i, tn := range tNew {
tNet[i] += tn
tNew[i] = 0
}
}
}
type square struct {
r, c int
}
func ant(r, c int, start, reset *sync.WaitGroup, tourCh chan []rckt) {
rnd := rand.New(rand.NewSource(time.Now().UnixNano())) tabu := make([]square, nSquares) moves := make([]rckt, nSquares) unexp := make([]rckt, 8) tabu[0].r = r tabu[0].c = c
for {
// cycle initialization
moves = moves[:0]
tabu = tabu[:1]
r := tabu[0].r
c := tabu[0].c
// wait for start signal
start.Wait()
reset.Done()
for {
// choose next move
unexp = unexp[:0]
var tSum float64
findU:
for k := 0; k < 8; k++ {
dr, dc, ok := dest(r, c, k)
if !ok {
continue
}
for _, t := range tabu {
if t.r == dr && t.c == dc {
continue findU
}
}
tk := tNet[(r*boardSize+c)*8+k]
tSum += tk
// note: dest r, c stored here
unexp = append(unexp, rckt{dr, dc, k, tk})
}
if len(unexp) == 0 {
break // no moves
}
rn := rnd.Float64() * tSum
var move rckt
for _, move = range unexp {
if rn <= move.t {
break
}
rn -= move.t
}
// move to new square
move.r, r = r, move.r
move.c, c = c, move.c
tabu = append(tabu, square{r, c})
moves = append(moves, move)
}
// compute pheromone amount to leave
for i := range moves {
moves[i].t = float64(len(moves)-i) / float64(completeTour-i)
}
// return tour found for this cycle
tourCh <- moves
}
}</lang> Output:
Starting square: row 2 column 3 Move sequence: 64 33 36 3 54 49 38 51 35 4 1 30 37 52 55 48 32 63 34 53 2 47 50 39 5 18 31 46 29 20 13 56 62 27 44 19 14 11 40 21 17 6 15 28 45 22 57 12 26 61 8 43 24 59 10 41 7 16 25 60 9 42 23 58
Haskell
<lang Haskell>import Data.Char (ord, chr) import Data.Ord (comparing) import Data.List (minimumBy, (\\), intercalate, sort)
type Square = (Int, Int)
board :: [Square] board =
[ (x, y) | x <- [1 .. 8] , y <- [1 .. 8] ]
knightMoves :: Square -> [Square] knightMoves (x, y) = filter (`elem` board) jumps
where
jumps =
[ (x + i, y + j)
| i <- jv
, j <- jv
, abs i /= abs j ]
jv = [1, -1, 2, -2]
knightTour :: [Square] -> [Square] knightTour moves
| null candMoves = reverse moves | otherwise = knightTour $ newSquare : moves where newSquare = minimumBy (comparing (length . findMoves)) candMoves candMoves = findMoves $ head moves findMoves = (\\ moves) . knightMoves
toSq :: String -> (Int, Int) toSq [x, y] = (ord x - 96, ord y - 48)
toAlg :: (Int, Int) -> String toAlg (x, y) = [chr (x + 96), chr (y + 48)]
-- TEST ----------------------------------------------------------------------- sq :: (Int, Int) sq = toSq "e5" -- Input: starting position on the board, e.g. (5, 5) as "e5"
main :: IO () main = printTour $ toAlg <$> knightTour [sq]
where
printTour [] = return ()
printTour tour = do
putStrLn $ intercalate " -> " $ take 8 tour
printTour $ drop 8 tour</lang>
- Output:
e5 -> f7 -> h8 -> g6 -> h4 -> g2 -> e1 -> f3 g1 -> h3 -> g5 -> h7 -> f8 -> d7 -> b8 -> a6 b4 -> a2 -> c1 -> d3 -> b2 -> a4 -> b6 -> a8 c7 -> e8 -> g7 -> h5 -> f4 -> e6 -> d8 -> b7 c5 -> b3 -> a1 -> c2 -> a3 -> b1 -> d2 -> c4 a5 -> c6 -> a7 -> c8 -> d6 -> b5 -> d4 -> e2 c3 -> d1 -> f2 -> h1 -> g3 -> e4 -> f6 -> g8 h6 -> g4 -> h2 -> f1 -> e3 -> f5 -> e7 -> d5
Icon and Unicon
This implements Warnsdorff's algorithm using unordered sets.
- The board must be square (it has only been tested on 8x8 and 7x7). Currently the maximum size board (due to square notation) is 26x26.
- Tie breaking is selectable with 3 variants supplied (first in list, random, and Roth's distance heuristic).
- A debug log can be generated showing the moves and choices considered for tie breaking.
The algorithm doesn't always generate a complete tour. <lang Icon>link printf
procedure main(A) ShowTour(KnightsTour(Board(8))) end
procedure KnightsTour(B,sq,tbrk,debug) #: Warnsdorff’s algorithm
/B := Board(8) # create 8x8 board if none given /sq := ?B.files || ?B.ranks # random initial position (default) sq2fr(sq,B) # validate initial sq if type(tbrk) == "procedure" then
B.tiebreak := tbrk # override tie-breaker
if \debug then write("Debug log : move#, move : (accessibility) choices")
choices := [] # setup to track moves and choices every (movesto := table())[k := key(B.movesto)] := copy(B.movesto[k])
B.tour := [] # new tour repeat {
put(B.tour,sq) # record move
ac := 9 # accessibility counter > maximum
while get(choices) # empty choices for tiebreak
every delete(movesto[nextsq := !movesto[sq]],sq) do { # make sq unavailable
if ac >:= *movesto[nextsq] then # reset to lower accessibility count
while get(choices) # . re-empty choices
if ac = *movesto[nextsq] then
put(choices,nextsq) # keep least accessible sq and any ties
}
if \debug then { # move#, move, (accessibility), choices
writes(sprintf("%d. %s : (%d) ",*B.tour,sq,ac))
every writes(" ",!choices|"\n")
}
sq := B.tiebreak(choices,B) | break # choose next sq until out of choices
}
return B end
procedure RandomTieBreaker(S,B) # random choice return ?S end
procedure FirstTieBreaker(S,B) # first one in the list return !S end
procedure RothTieBreaker(S,B) # furthest from the center if *S = 0 then fail # must fail if [] every fr := sq2fr(s := !S,B) do {
d := sqrt(abs(fr[1]-1 - (B.N-1)*0.5)^2 + abs(fr[2]-1 - (B.N-1)*0.5)^2) if (/md := d) | ( md >:= d) then msq := s # save sq }
return msq end
record board(N,ranks,files,movesto,tiebreak,tour) # structure for board
procedure Board(N) #: create board N := *&lcase >=( 0 < integer(N)) | stop("N=",image(N)," is out of range.") B := board(N,[],&lcase[1+:N],table(),RandomTieBreaker) # setup every put(B.ranks,N to 1 by -1) # add rank #s every sq := !B.files || !B.ranks do # for each sq add
every insert(B.movesto[sq] := set(), KnightMoves(sq,B)) # moves to next sq
return B end
procedure sq2fr(sq,B) #: return numeric file & rank f := find(sq[1],B.files) | runerr(205,sq) r := integer(B.ranks[sq[2:0]]) | runerr(205,sq) return [f,r] end
procedure KnightMoves(sq,B) #: generate all Kn accessible moves from sq fr := sq2fr(sq,B) every ( i := -2|-1|1|2 ) & ( j := -2|-1|1|2 ) do
if (abs(i)~=abs(j)) & (0<(ri:=fr[2]+i)<=B.N) & (0<(fj:=fr[1]+j)<=B.N) then
suspend B.files[fj]||B.ranks[ri]
end
procedure ShowTour(B) #: show the tour write("Board size = ",B.N) write("Tour length = ",*B.tour) write("Tie Breaker = ",image(B.tiebreak))
every !(squares := list(B.N)) := list(B.N,"-") every fr := sq2fr(B.tour[m := 1 to *B.tour],B) do
squares[fr[2],fr[1]] := m
every (hdr1 := " ") ||:= right(!B.files,3) every (hdr2 := " +") ||:= repl((1 to B.N,"-"),3) | "-+"
every write(hdr1|hdr2) every r := 1 to B.N do {
writes(right(B.ranks[r],3)," |")
every writes(right(squares[r,f := 1 to B.N],3))
write(" |",right(B.ranks[r],3))
}
every write(hdr2|hdr1|&null) end</lang>
The following can be used when debugging to validate the board structure and to image the available moves on the board. <lang Icon>procedure DumpBoard(B) #: Dump Board internals write("Board size=",B.N) write("Available Moves at start of tour:", ImageMovesTo(B.movesto)) end
procedure ImageMovesTo(movesto) #: image of available moves every put(K := [],key(movesto)) every (s := "\n") ||:= (k := !sort(K)) || " : " do
every s ||:= " " || (!sort(movesto[k])|"\n")
return s end</lang>
Sample output:
Board size = 8
Tour length = 64
Tie Breaker = procedure RandomTieBreaker
a b c d e f g h
+-------------------------+
8 | 53 10 29 26 55 12 31 16 | 8
7 | 28 25 54 11 30 15 48 13 | 7
6 | 9 52 27 62 47 56 17 32 | 6
5 | 24 61 38 51 36 45 14 49 | 5
4 | 39 8 63 46 57 50 33 18 | 4
3 | 64 23 60 37 42 35 44 3 | 3
2 | 7 40 21 58 5 2 19 34 | 2
1 | 22 59 6 41 20 43 4 1 | 1
+-------------------------+
a b c d e f g hTwo 7x7 boards:
Board size = 7
Tour length = 33
Tie Breaker = procedure RandomTieBreaker
a b c d e f g
+----------------------+
7 | 33 4 15 - 29 6 17 | 7
6 | 14 - 30 5 16 - 28 | 6
5 | 3 32 - - - 18 7 | 5
4 | - 13 - 31 - 27 - | 4
3 | 23 2 - - - 8 19 | 3
2 | 12 - 24 21 10 - 26 | 2
1 | 1 22 11 - 25 20 9 | 1
+----------------------+
a b c d e f g
Board size = 7
Tour length = 49
Tie Breaker = procedure RothTieBreaker
a b c d e f g
+----------------------+
7 | 35 14 21 46 7 12 9 | 7
6 | 20 49 34 13 10 23 6 | 6
5 | 15 36 45 22 47 8 11 | 5
4 | 42 19 48 33 40 5 24 | 4
3 | 37 16 41 44 27 32 29 | 3
2 | 18 43 2 39 30 25 4 | 2
1 | 1 38 17 26 3 28 31 | 1
+----------------------+
a b c d e f g
J
Solution:
The Knight's tour essay on the Jwiki shows a couple of solutions including one using Warnsdorffs algorithm.
<lang j>NB. knight moves for each square of a (y,y) board
kmoves=: monad define
t=. (>,{;~i.y) +"1/ _2]\2 1 2 _1 1 2 1 _2 _1 2 _1 _2 _2 1 _2 _1
(*./"1 t e. i.y) <@#"1 y#.t
)
ktourw=: monad define
M=. >kmoves y
p=. k=. 0
b=. 1 $~ *:y
for. i.<:*:y do.
b=. 0 k}b
p=. p,k=. ((i.<./) +/"1 b{~j{M){j=. ({&b # ]) k{M
end.
assert. ~:p
(,~y)$/:p
)</lang>
Example Use: <lang j> ktourw 8 NB. solution for an 8 x 8 board
0 25 14 23 28 49 12 31
15 22 27 50 13 30 63 48 26 1 24 29 62 59 32 11 21 16 51 58 43 56 47 60
2 41 20 55 52 61 10 33
17 38 53 42 57 44 7 46 40 3 36 19 54 5 34 9 37 18 39 4 35 8 45 6
9!:37]0 64 4 4 NB. truncate lines longer than 64 characters and only show first and last four lines
ktourw 202 NB. 202x202 board -- this implementation failed for 200 and 201 0 401 414 405 398 403 424 417 396 419 43... 413 406 399 402 425 416 397 420 439 430 39... 400 1 426 415 404 423 448 429 418 437 4075... 409 412 407 446 449 428 421 440 40739 40716 43...
...
550 99 560 569 9992 779 786 773 10002 9989 78... 555 558 553 778 563 570 775 780 785 772 1000... 100 551 556 561 102 777 572 771 104 781 57... 557 554 101 552 571 562 103 776 573 770 10...</lang>
Java
<lang java>import java.util.*;
public class KnightsTour {
private final static int base = 12;
private final static int[][] moves = {{1,-2},{2,-1},{2,1},{1,2},{-1,2},
{-2,1},{-2,-1},{-1,-2}};
private static int[][] grid;
private static int total;
public static void main(String[] args) {
grid = new int[base][base];
total = (base - 4) * (base - 4);
for (int r = 0; r < base; r++)
for (int c = 0; c < base; c++)
if (r < 2 || r > base - 3 || c < 2 || c > base - 3)
grid[r][c] = -1;
int row = 2 + (int) (Math.random() * (base - 4));
int col = 2 + (int) (Math.random() * (base - 4));
grid[row][col] = 1;
if (solve(row, col, 2))
printResult();
else System.out.println("no result");
}
private static boolean solve(int r, int c, int count) {
if (count > total)
return true;
List<int[]> nbrs = neighbors(r, c);
if (nbrs.isEmpty() && count != total)
return false;
Collections.sort(nbrs, new Comparator<int[]>() {
public int compare(int[] a, int[] b) {
return a[2] - b[2];
}
});
for (int[] nb : nbrs) {
r = nb[0];
c = nb[1];
grid[r][c] = count;
if (!orphanDetected(count, r, c) && solve(r, c, count + 1))
return true;
grid[r][c] = 0;
}
return false; }
private static List<int[]> neighbors(int r, int c) {
List<int[]> nbrs = new ArrayList<>();
for (int[] m : moves) {
int x = m[0];
int y = m[1];
if (grid[r + y][c + x] == 0) {
int num = countNeighbors(r + y, c + x);
nbrs.add(new int[]{r + y, c + x, num});
}
}
return nbrs;
}
private static int countNeighbors(int r, int c) {
int num = 0;
for (int[] m : moves)
if (grid[r + m[1]][c + m[0]] == 0)
num++;
return num;
}
private static boolean orphanDetected(int cnt, int r, int c) {
if (cnt < total - 1) {
List<int[]> nbrs = neighbors(r, c);
for (int[] nb : nbrs)
if (countNeighbors(nb[0], nb[1]) == 0)
return true;
}
return false;
}
private static void printResult() {
for (int[] row : grid) {
for (int i : row) {
if (i == -1) continue;
System.out.printf("%2d ", i);
}
System.out.println();
}
}
}</lang>
34 17 20 3 36 7 22 5 19 2 35 40 21 4 37 8 16 33 18 51 44 39 6 23 1 50 43 46 41 56 9 38 32 15 54 61 52 45 24 57 49 62 47 42 55 60 27 10 14 31 64 53 12 29 58 25 63 48 13 30 59 26 11 28
JavaScript
Procedural
Using Warnsdorff rule and Backtracking.
You can test it here.
<lang javascript> class KnightTour {
constructor() {
this.width = 856;
this.height = 856;
this.cellCount = 8;
this.size = 0;
this.knightPiece = "\u2658";
this.knightPos = {
x: 0,
y: 0
};
this.ctx = null;
this.step = this.width / this.cellCount;
this.lastTime = 0;
this.wait;
this.delay;
this.success;
this.jumps;
this.directions = [];
this.visited = [];
this.path = [];
document.getElementById("start").addEventListener("click", () => {
this.startHtml();
});
this.init();
this.drawBoard();
}
drawBoard() {
let a = false, xx, yy;
for (let y = 0; y < this.cellCount; y++) {
for (let x = 0; x < this.cellCount; x++) {
if (a) {
this.ctx.fillStyle = "#607db8";
} else {
this.ctx.fillStyle = "#aecaf0";
}
a = !a;
xx = x * this.step;
yy = y * this.step;
this.ctx.fillRect(xx, yy, xx + this.step, yy + this.step);
}
if (!(this.cellCount & 1)) a = !a;
}
if (this.path.length) {
const s = this.step >> 1;
this.ctx.lineWidth = 3;
this.ctx.fillStyle = "black";
this.ctx.beginPath();
this.ctx.moveTo(this.step * this.knightPos.x + s, this.step * this.knightPos.y + s);
let a, b, v = this.path.length - 1;
for (; v > -1; v--) {
a = this.path[v].pos.x * this.step + s;
b = this.path[v].pos.y * this.step + s;
this.ctx.lineTo(a, b);
this.ctx.stroke();
this.ctx.fillRect(a - 5, b - 5, 10, 10);
}
}
}
createMoves(pos) {
const possibles = [];
let x = 0,
y = 0,
m = 0,
l = this.directions.length;
for (; m < l; m++) {
x = pos.x + this.directions[m].x;
y = pos.y + this.directions[m].y;
if (x > -1 && x < this.cellCount && y > -1 && y < this.cellCount && !this.visited[x + y * this.cellCount]) {
possibles.push({
x,
y
})
}
}
return possibles;
}
warnsdorff(pos) {
const possibles = this.createMoves(pos);
if (possibles.length < 1) return [];
const moves = [];
for (let p = 0, l = possibles.length; p < l; p++) {
let ps = this.createMoves(possibles[p]);
moves.push({
len: ps.length,
pos: possibles[p]
});
}
moves.sort((a, b) => {
return b.len - a.len;
});
return moves;
}
startHtml() {
this.cellCount = parseInt(document.getElementById("cellCount").value);
this.size = Math.floor(this.width / this.cellCount)
this.wait = this.delay = parseInt(document.getElementById("delay").value);
this.step = this.width / this.cellCount;
this.ctx.font = this.size + "px Arial";
document.getElementById("log").innerText = "";
document.getElementById("path").innerText = "";
this.path = [];
this.jumps = 1;
this.success = true;
this.visited = [];
const cnt = this.cellCount * this.cellCount;
for (let a = 0; a < cnt; a++) {
this.visited.push(false);
}
const kx = parseInt(document.getElementById("knightx").value),
ky = parseInt(document.getElementById("knighty").value);
this.knightPos = {
x: (kx > this.cellCount || kx < 0) ? Math.floor(Math.random() * this.cellCount) : kx,
y: (ky > this.cellCount || ky < 0) ? Math.floor(Math.random() * this.cellCount) : ky
};
this.mainLoop = (time = 0) => {
const dif = time - this.lastTime;
this.lastTime = time;
this.wait -= dif;
if (this.wait > 0) {
requestAnimationFrame(this.mainLoop);
return;
}
this.wait = this.delay;
let moves;
if (this.success) {
moves = this.warnsdorff(this.knightPos);
} else {
if (this.path.length > 0) {
const path = this.path[this.path.length - 1];
moves = path.m;
if (moves.length < 1) this.path.pop();
this.knightPos = path.pos
this.visited[this.knightPos.x + this.knightPos.y * this.cellCount] = false;
this.jumps--;
this.wait = this.delay;
} else {
document.getElementById("log").innerText = "Can't find a solution!";
return;
}
}
this.drawBoard();
const ft = this.step - (this.step >> 3);
this.ctx.fillStyle = "#000";
this.ctx.fillText(this.knightPiece, this.knightPos.x * this.step, this.knightPos.y * this.step + ft);
if (moves.length < 1) {
if (this.jumps === this.cellCount * this.cellCount) {
document.getElementById("log").innerText = "Tour finished!";
let str = "";
for (let z of this.path) {
str += `${1 + z.pos.x + z.pos.y * this.cellCount}, `;
}
str += `${1 + this.knightPos.x + this.knightPos.y * this.cellCount}`;
document.getElementById("path").innerText = str;
return;
} else {
this.success = false;
}
} else {
this.visited[this.knightPos.x + this.knightPos.y * this.cellCount] = true;
const move = moves.pop();
this.path.push({
pos: this.knightPos,
m: moves
});
this.knightPos = move.pos
this.success = true;
this.jumps++;
}
requestAnimationFrame(this.mainLoop);
};
this.mainLoop();
}
init() {
const canvas = document.createElement("canvas");
canvas.id = "cv";
canvas.width = this.width;
canvas.height = this.height;
this.ctx = canvas.getContext("2d");
document.getElementById("out").appendChild(canvas);
this.directions = [{
x: -1,
y: -2
}, {
x: -2,
y: -1
}, {
x: 1,
y: -2
}, {
x: 2,
y: -1
},
{
x: -1,
y: 2
}, {
x: -2,
y: 1
}, {
x: 1,
y: 2
}, {
x: 2,
y: 1
}
];
}
} new KnightTour(); </lang> To test it, you'll need an index.html
<!DOCTYPE html> <html> <head> <meta charset="UTF-8"> <title>Knight's Tour</title> <link rel="stylesheet" type="text/css" media="screen" href="style.css" /> </head> <body> <div id='out'></div> <div id='ctrls'> <span>Cells: </span><input id="cellCount" value="8" type="number" max="250" min="5"><br /> <span>Delay: </span><input id="delay" value="500" type="number" max="2000" min="0"><br /> <span>Knight X: </span><input id="knightx" value="-1" type="number" max="250" min="-1"><br /> <span>Knight Y: </span><input id="knighty" value="-1" type="number" max="250" min="-1"><br /> <button id="start">Start</button> <div id='log'></div> <div id="path"></div> </div> <script src="tour_bt.js" type="module"></script> </body> </html>
And a style.css
body {
font-family: verdana;
color: white;
font-size: 36px;
background-color: #001f33
}
button {
width: 100%;
height: 40px;
margin: 20px 0px 20px 0px;
font-size: 28px
}
canvas {
border: 4px solid #000;
margin: 40px;
}
#out {
float: left;
}
#ctrls {
margin-top: 40px;
text-align: left;
width: 280px;
line-height: 40px;
float: left;
}
#ctrls input {
float: right;
width: 80px;
height: 24px;
margin-top: 6px;
font-size: 22px;
}
#path {
margin-top: 10px;
font-size: 12px;
line-height: 16px;
}
Functional
A composition of values, drawing on generic abstractions:
<lang javascript>(() => {
'use strict';
// knightsTour :: Int -> [(Int, Int)] -> [(Int, Int)]
const knightsTour = rowLength => moves => {
const go = path => {
const
findMoves = xy => difference(knightMoves(xy), path),
warnsdorff = minimumBy(
comparing(compose(length, findMoves))
),
options = findMoves(path[0]);
return 0 < options.length ? (
go([warnsdorff(options)].concat(path))
) : reverse(path);
};
// board :: (Int, Int) const board = concatMap( col => concatMap( row => [ [col, row] ], enumFromTo(1, rowLength)), enumFromTo(1, rowLength) );
// knightMoves :: (Int, Int) -> [(Int, Int)]
const knightMoves = ([x, y]) =>
concatMap(
([dx, dy]) => {
const ab = [x + dx, y + dy];
return elem(ab, board) ? (
[ab]
) : [];
}, [
[-2, -1],
[-2, 1],
[-1, -2],
[-1, 2],
[1, -2],
[1, 2],
[2, -1],
[2, 1]
]
);
return go(moves);
};
// TEST -----------------------------------------------
// main :: IO()
const main = () => {
// boardSize :: Int
const boardSize = 8;
// tour :: [(Int, Int)]
const tour = knightsTour(boardSize)(
[fromAlgebraic('e5')]
);
// report :: String
const report = '(Board size ' +
boardSize + '*' + boardSize + ')\n\n' +
'Route: \n\n' +
showRoute(boardSize)(tour) + '\n\n' +
'Coverage and order: \n\n' +
showCoverage(boardSize)(tour) + '\n\n';
return (
console.log(report),
report
);
}
// DISPLAY --------------------------------------------
// algebraic :: (Int, Int) -> String
const algebraic = ([x, y]) =>
chr(x + 96) + y.toString();
// fromAlgebraic :: String -> (Int, Int)
const fromAlgebraic = s =>
2 <= s.length ? (
[ord(s[0]) - 96, parseInt(s.slice(1))]
) : undefined;
// showCoverage :: Int -> [(Int, Int)] -> String
const showCoverage = rowLength => xys => {
const
intMax = xys.length,
w = 1 + intMax.toString().length
return unlines(map(concat,
chunksOf(
rowLength,
map(composeList([justifyRight(w, ' '), str, fst]),
sortBy(
mappendComparing([
compose(fst, snd),
compose(snd, snd)
]),
zip(enumFromTo(1, intMax), xys)
)
)
)
));
};
// showRoute :: Int -> [(Int, Int)] -> String
const showRoute = rowLength => xys => {
const w = 1 + rowLength.toString().length;
return unlines(map(
xs => xs.join(' -> '),
chunksOf(
rowLength,
map(compose(justifyRight(w, ' '), algebraic), xys)
)
));
};
// GENERIC FUNCTIONS ----------------------------------
// Tuple (,) :: a -> b -> (a, b)
const Tuple = (a, b) => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});
// chr :: Int -> Char const chr = x => String.fromCodePoint(x);
// chunksOf :: Int -> [a] -> a const chunksOf = (n, xs) => enumFromThenTo(0, n, xs.length - 1) .reduce( (a, i) => a.concat([xs.slice(i, (n + i))]), [] );
// compare :: a -> a -> Ordering
const compare = (a, b) =>
a < b ? -1 : (a > b ? 1 : 0);
// comparing :: (a -> b) -> (a -> a -> Ordering)
const comparing = f =>
(x, y) => {
const
a = f(x),
b = f(y);
return a < b ? -1 : (a > b ? 1 : 0);
};
// compose (<<<) :: (b -> c) -> (a -> b) -> a -> c const compose = (f, g) => x => f(g(x));
// composeList :: [(a -> a)] -> (a -> a)
const composeList = fs =>
x => fs.reduceRight((a, f) => f(a), x, fs);
// concat :: a -> [a] // concat :: [String] -> String const concat = xs => 0 < xs.length ? (() => { const unit = 'string' !== typeof xs[0] ? ( [] ) : ; return unit.concat.apply(unit, xs); })() : [];
// concatMap :: (a -> [b]) -> [a] -> [b]
const concatMap = (f, xs) =>
xs.reduce((a, x) => a.concat(f(x)), []);
// difference :: Eq a => [a] -> [a] -> [a]
const difference = (xs, ys) => {
const s = new Set(ys.map(str));
return xs.filter(x => !s.has(str(x)));
};
// elem :: Eq a => a -> [a] -> Bool const elem = (x, xs) => xs.some(eq(x))
// enumFromThenTo :: Int -> Int -> Int -> [Int]
const enumFromThenTo = (x1, x2, y) => {
const d = x2 - x1;
return Array.from({
length: Math.floor(y - x2) / d + 2
}, (_, i) => x1 + (d * i));
};
// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = (m, n) =>
Array.from({
length: 1 + n - m
}, (_, i) => m + i);
// eq (==) :: Eq a => a -> a -> Bool
const eq = a => b => {
const t = typeof a;
return t !== typeof b ? (
false
) : 'object' !== t ? (
'function' !== t ? (
a === b
) : a.toString() === b.toString()
) : (() => {
const kvs = Object.entries(a);
return kvs.length !== Object.keys(b).length ? (
false
) : kvs.every(([k, v]) => eq(v)(b[k]));
})();
};
// fst :: (a, b) -> a const fst = tpl => tpl[0];
// justifyRight :: Int -> Char -> String -> String
const justifyRight = (n, cFiller) => s =>
n > s.length ? (
s.padStart(n, cFiller)
) : s;
// length :: [a] -> Int
const length = xs =>
(Array.isArray(xs) || 'string' === typeof xs) ? (
xs.length
) : Infinity;
// map :: (a -> b) -> [a] -> [b]
const map = (f, xs) =>
(Array.isArray(xs) ? (
xs
) : xs.split()).map(f);
// mappendComparing :: [(a -> b)] -> (a -> a -> Ordering)
const mappendComparing = fs =>
(x, y) => fs.reduce(
(ordr, f) => (ordr || compare(f(x), f(y))),
0
);
// minimumBy :: (a -> a -> Ordering) -> [a] -> a
const minimumBy = f => xs =>
xs.reduce((a, x) => undefined === a ? x : (
0 > f(x, a) ? x : a
), undefined);
// ord :: Char -> Int const ord = c => c.codePointAt(0);
// reverse :: [a] -> [a]
const reverse = xs =>
'string' !== typeof xs ? (
xs.slice(0).reverse()
) : xs.split().reverse().join();
// snd :: (a, b) -> b const snd = tpl => tpl[1];
// sortBy :: (a -> a -> Ordering) -> [a] -> [a]
const sortBy = (f, xs) =>
xs.slice()
.sort(f);
// str :: a -> String const str = x => x.toString();
// take :: Int -> [a] -> [a]
// take :: Int -> String -> String
const take = (n, xs) =>
xs.slice(0, n);
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// Use of `take` and `length` here allows for zipping with non-finite // lists - i.e. generators like cycle, repeat, iterate.
// zip :: [a] -> [b] -> [(a, b)]
const zip = (xs, ys) => {
const lng = Math.min(length(xs), length(ys));
const bs = take(lng, ys);
return take(lng, xs).map((x, i) => Tuple(x, bs[i]));
};
// MAIN --- return main();
})();</lang>
- Output:
(Board size 8*8) Route: e5 -> d7 -> b8 -> a6 -> b4 -> a2 -> c1 -> b3 a1 -> c2 -> a3 -> b1 -> d2 -> f1 -> h2 -> g4 h6 -> g8 -> e7 -> c8 -> a7 -> c6 -> a5 -> b7 d8 -> f7 -> h8 -> g6 -> f8 -> h7 -> f6 -> e8 g7 -> h5 -> g3 -> h1 -> f2 -> d1 -> b2 -> a4 b6 -> a8 -> c7 -> b5 -> c3 -> d5 -> e3 -> c4 d6 -> e4 -> c5 -> d3 -> e1 -> g2 -> h4 -> f5 d4 -> e2 -> f4 -> e6 -> g5 -> f3 -> g1 -> h3 Coverage and order: 9 6 11 40 23 4 21 42 12 39 8 5 44 41 24 3 7 10 45 48 51 22 43 20 38 13 52 57 46 49 2 25 53 58 47 50 1 60 19 32 14 37 62 59 56 31 26 29 63 54 35 16 61 28 33 18 36 15 64 55 34 17 30 27
Julia
Uses the Hidato puzzle solver module, which has its source code listed here in the Hadato task. <lang julia>using .Hidato # Note that the . here means to look locally for the module rather than in the libraries
const chessboard = """
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 """
const knightmoves = [[-2, -1], [-2, 1], [-1, -2], [-1, 2], [1, -2], [1, 2], [2, -1], [2, 1]]
board, maxmoves, fixed, starts = hidatoconfigure(chessboard) printboard(board, " 0", " ") hidatosolve(board, maxmoves, knightmoves, fixed, starts[1][1], starts[1][2], 1) printboard(board)
</lang>
- Output:
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 12 9 6 3 14 17 20 10 7 2 13 18 21 4 15 31 28 11 8 5 16 19 22 64 25 32 29 36 23 48 45 33 30 27 24 49 46 37 58 26 63 52 35 40 57 44 47 53 34 61 50 55 42 59 38 62 51 54 41 60 39 56 43
Kotlin
<lang scala>data class Square(val x : Int, val y : Int)
val board = Array(8 * 8, { Square(it / 8 + 1, it % 8 + 1) }) val axisMoves = arrayOf(1, 2, -1, -2)
fun <T> allPairs(a: Array<T>) = a.flatMap { i -> a.map { j -> Pair(i, j) } }
fun knightMoves(s : Square) : List<Square> {
val moves = allPairs(axisMoves).filter{ Math.abs(it.first) != Math.abs(it.second) }
fun onBoard(s : Square) = board.any {it == s}
return moves.map { Square(s.x + it.first, s.y + it.second) }.filter(::onBoard)
}
fun knightTour(moves : List<Square>) : List<Square> {
fun findMoves(s: Square) = knightMoves(s).filterNot { m -> moves.any { it == m } }
val newSquare = findMoves(moves.last()).minBy { findMoves(it).size }
return if (newSquare == null) moves else knightTour(moves + newSquare)
}
fun knightTourFrom(start : Square) = knightTour(listOf(start))
fun main(args : Array<String>) {
var col = 0
for ((x, y) in knightTourFrom(Square(1, 1))) {
System.out.print("$x,$y")
System.out.print(if (col == 7) "\n" else " ")
col = (col + 1) % 8
}
}</lang>
- Output:
1,1 2,3 3,1 1,2 2,4 1,6 2,8 4,7 6,8 8,7 7,5 8,3 7,1 5,2 7,3 8,1 6,2 4,1 2,2 1,4 2,6 1,8 3,7 5,8 7,7 8,5 6,6 7,8 8,6 7,4 8,2 6,1 4,2 2,1 3,3 5,4 3,5 4,3 5,1 6,3 8,4 7,2 6,4 5,6 4,8 2,7 1,5 3,6 1,7 3,8 5,7 4,5 5,3 6,5 4,4 3,2 1,3 2,5 4,6 3,4 5,5 6,7 8,8 7,6
Locomotive Basic
Influenced by the Python version, although computed tours are different.
<lang locobasic>10 mode 1:defint a-z 20 input "Board size: ",size 30 input "Start position: ",a$ 40 x=asc(mid$(a$,1,1))-96 50 y=val(mid$(a$,2,1)) 60 dim play(size,size) 70 for q=1 to 8 80 read dx(q),dy(q) 90 next 100 data 2,1,1,2,-1,2,-2,1,-2,-1,-1,-2,1,-2,2,-1 110 pen 0:paper 1 120 for q=1 to size 130 locate 3*q+1,24-size 140 print chr$(96+q); 150 locate 3*(size+1)+1,26-q 160 print using "#"; q; 170 next 180 pen 1:paper 0 190 ' main loop 200 n=n+1 210 play(x,y)=n 220 locate 3*x,26-y 230 print using "##"; n; 240 if n=size*size then call &bb06:end 250 nmov=100 260 for q=1 to 8 270 xc=x+dx(q) 280 yc=y+dy(q) 290 gosub 360 300 if nm<nmov then nmov=nm:qm=q 310 next 320 x=x+dx(qm) 330 y=y+dy(qm) 340 goto 200 350 ' find moves 360 if xc<1 or yc<1 or xc>size or yc>size then nm=1000:return 370 if play(xc,yc) then nm=2000:return 380 nm=0 390 for q2=1 to 8 400 xt=xc+dx(q2) 410 yt=yc+dy(q2) 420 if xt<1 or yt<1 or xt>size or yt>size then 460 430 if play(xt,yt) then 460 440 nm=nm+1 450 ' skip this move 460 next 470 return</lang>
Lua
<lang lua>N = 8
moves = { {1,-2},{2,-1},{2,1},{1,2},{-1,2},{-2,1},{-2,-1},{-1,-2} }
function Move_Allowed( board, x, y )
if board[x][y] >= 8 then return false end
local new_x, new_y = x + moves[board[x][y]+1][1], y + moves[board[x][y]+1][2] if new_x >= 1 and new_x <= N and new_y >= 1 and new_y <= N and board[new_x][new_y] == 0 then return true end return false
end
board = {}
for i = 1, N do
board[i] = {}
for j = 1, N do
board[i][j] = 0
end
end
x, y = 1, 1
lst = {} lst[1] = { x, y }
repeat
if Move_Allowed( board, x, y ) then
board[x][y] = board[x][y] + 1
x, y = x+moves[board[x][y]][1], y+moves[board[x][y]][2]
lst[#lst+1] = { x, y }
else
if board[x][y] >= 8 then
board[x][y] = 0
lst[#lst] = nil
if #lst == 0 then
print "No solution found."
os.exit(1)
end
x, y = lst[#lst][1], lst[#lst][2]
end
board[x][y] = board[x][y] + 1
end
until #lst == N^2
last = lst[1] for i = 2, #lst do
print( string.format( "%s%d - %s%d", string.sub("ABCDEFGH",last[1],last[1]), last[2], string.sub("ABCDEFGH",lst[i][1],lst[i][1]), lst[i][2] ) )
last = lst[i]
end</lang>
Mathematica
Solution <lang Mathematica> knightsTourMoves[start_] :=
Module[{
vertexLabels = (# -> ToString@c[[Quotient[# - 1, 8] + 1]] <> ToString[Mod[# - 1, 8] + 1]) & /@ Range[64], knightsGraph,
hamiltonianCycle, end},
knightsGraph = KnightTourGraph[i, i, VertexLabels -> vertexLabels, ImagePadding -> 15];
hamiltonianCycle = ((FindHamiltonianCycle[knightsGraph] /. UndirectedEdge -> DirectedEdge) /. labels)1;
end = Cases[hamiltonianCycle, (x_ \[DirectedEdge] start) :> x]1;
FindShortestPath[g, start, end]]
</lang>
Usage <lang Mathematica>
knightsTourMoves["d8"]
(* out *)
{"d8", "e6", "d4", "c2", "a1", "b3", "a5", "b7", "c5", "a4", "b2", "c4", "a3", "b1", "c3", "a2", "b4", "a6", "b8", "c6", "a7", "b5", \
"c7", "a8", "b6", "c8", "d6", "e4", "d2", "f1", "e3", "d1", "f2", "h1", "g3", "e2", "c1", "d3", "e1", "g2", "h4", "f5", "e7", "d5", \ "f4", "h5", "g7", "e8", "f6", "g8", "h6", "g4", "h2", "f3", "g1", "h3", "g5", "h7", "f8", "d7", "e5", "g6", "h8", "f7"} </lang>
Analysis
vertexLabels replaces the default vertex (i.e. square) names of the chessboard with the standard algebraic names "a1", "a2",...,"h8". <lang Mathematica>
vertexLabels = (# -> ToString@c[[Quotient[# - 1, 8] + 1]] <> ToString[Mod[# - 1, 8] + 1]) & /@ Range[64]
(* out *)
{1 -> "a1", 2 -> "a2", 3 -> "a3", 4 -> "a4", 5 -> "a5", 6 -> "a6", 7 -> "a7", 8 -> "a8",
9 -> "b1", 10 -> "b2", 11 -> "b3", 12 -> "b4", 13 -> "b5", 14 -> "b6", 15 -> "b7", 16 -> "b8",
17 -> "c1", 18 -> "c2", 19 -> "c3", 20 -> "c4", 21 -> "c5", 22 -> "c6", 23 -> "c7", 24 -> "c8",
25 -> "d1", 26 -> "d2", 27 -> "d3", 28 -> "d4", 29 -> "d5", 30 -> "d6", 31 -> "d7", 32 -> "d8",
33 -> "e1", 34 -> "e2", 35 -> "e3", 36 -> "e4", 37 -> "e5", 38 -> "e6", 39 -> "e7", 40 -> "e8",
41 -> "f1", 42 -> "f2", 43 -> "f3", 44 -> "f4", 45 -> "f5", 46 -> "f6", 47 -> "f7", 48 -> "f8",
49 -> "g1", 50 -> "g2", 51 -> "g3", 52 -> "g4", 53 -> "g5", 54 -> "g6",55 -> "g7", 56 -> "g8",
57 -> "h1", 58 -> "h2", 59 -> "h3", 60 -> "h4", 61 -> "h5", 62 -> "h6", 63 -> "h7", 64 -> "h8"}
</lang>
knightsGraph creates a graph of the solution space.
<lang Mathematica>
knightsGraph = KnightTourGraph[i, i, VertexLabels -> vertexLabels, ImagePadding -> 15];
</lang>
Find a Hamiltonian cycle (a path that visits each square exactly one time.)
<lang Mathematica>
hamiltonianCycle = ((FindHamiltonianCycle[knightsGraph] /. UndirectedEdge -> DirectedEdge) /. labels)1;
</lang>
Find the end square:
<lang Mathematica>
end = Cases[hamiltonianCycle, (x_ \[DirectedEdge] start) :> x]1;
</lang>
Find shortest path from the start square to the end square.
<lang Mathematica>
FindShortestPath[g, start, end]]
</lang>
Mathprog
While a little slower than using Warnsdorff this solution is interesting:
1. It shows that Hidato and Knights Tour are essentially the same problem.
2. It is possible to specify which square is used for any Knights Move.
<lang> /*Knights.mathprog
Find a Knights Tour
Nigel_Galloway January 11th., 2012
- /
param ZBLS; param ROWS; param COLS; param D := 2; set ROWSR := 1..ROWS; set COLSR := 1..COLS; set ROWSV := (1-D)..(ROWS+D); set COLSV := (1-D)..(COLS+D); param Iz{ROWSR,COLSR}, integer, default 0; set ZBLSV := 1..(ZBLS+1); set ZBLSR := 1..ZBLS;
var BR{ROWSV,COLSV,ZBLSV}, binary;
void0{r in ROWSV, z in ZBLSR,c in (1-D)..0}: BR[r,c,z] = 0; void1{r in ROWSV, z in ZBLSR,c in (COLS+1)..(COLS+D)}: BR[r,c,z] = 0; void2{c in COLSV, z in ZBLSR,r in (1-D)..0}: BR[r,c,z] = 0; void3{c in COLSV, z in ZBLSR,r in (ROWS+1)..(ROWS+D)}: BR[r,c,z] = 0; void4{r in ROWSV,c in (1-D)..0}: BR[r,c,ZBLS+1] = 1; void5{r in ROWSV,c in (COLS+1)..(COLS+D)}: BR[r,c,ZBLS+1] = 1; void6{c in COLSV,r in (1-D)..0}: BR[r,c,ZBLS+1] = 1; void7{c in COLSV,r in (ROWS+1)..(ROWS+D)}: BR[r,c,ZBLS+1] = 1;
Izfree{r in ROWSR, c in COLSR, z in ZBLSR : Iz[r,c] = -1}: BR[r,c,z] = 0; Iz1{Izr in ROWSR, Izc in COLSR, r in ROWSR, c in COLSR, z in ZBLSR : Izr=r and Izc=c and Iz[Izr,Izc]=z}: BR[r,c,z] = 1;
rule1{z in ZBLSR}: sum{r in ROWSR, c in COLSR} BR[r,c,z] = 1; rule2{r in ROWSR, c in COLSR}: sum{z in ZBLSV} BR[r,c,z] = 1; rule3{r in ROWSR, c in COLSR, z in ZBLSR}: BR[0,0,z+1] + BR[r-1,c-2,z+1] + BR[r-1,c+2,z+1] + BR[r-2,c-1,z+1] + BR[r-2,c+1,z+1] + BR[r+1,c+2,z+1] + BR[r+1,c-2,z+1] + BR[r+2,c-1,z+1] + BR[r+2,c+1,z+1] - BR[r,c,z] >= 0;
solve;
for {r in ROWSR} {
for {c in COLSR} {
printf " %2d", sum{z in ZBLSR} BR[r,c,z]*z;
}
printf "\n";
} data;
param ROWS := 5; param COLS := 5; param ZBLS := 25; param Iz: 1 2 3 4 5 :=
1 . . . . . 2 . 19 2 . . 3 . . . . . 4 . . . . . 5 . . . . . ; end;
</lang>
Produces:
<lang> GLPSOL: GLPK LP/MIP Solver, v4.47 Parameter(s) specified in the command line:
--minisat --math Knights.mathprog
Reading model section from Knights.mathprog... Reading data section from Knights.mathprog... 62 lines were read Generating void0... Generating void1... Generating void2... Generating void3... Generating void4... Generating void5... Generating void6... Generating void7... Generating Izfree... Generating Iz1... Generating rule1... Generating rule2... Generating rule3... Model has been successfully generated Will search for ANY feasible solution Translating to CNF-SAT... Original problem has 2549 rows, 2106 columns, and 9349 non-zeros 575 covering inequalities 1924 partitioning equalities Solving CNF-SAT problem... Instance has 3356 variables, 10874 clauses, and 34549 literals
============================[MINISAT]=============================
| Conflicts | ORIGINAL | LEARNT | Progress | | | Clauses Literals | Limit Clauses Literals Lit/Cl | |
==================================================================
| 0 | 9000 32675 | 3000 0 0 0.0 | 0.000 % | | 101 | 6025 21551 | 3300 93 1620 17.4 | 57.688 % | | 251 | 6025 21551 | 3630 243 4961 20.4 | 57.688 % |
==================================================================
SATISFIABLE Objective value = 0.000000000e+000 Time used: 0.0 secs Memory used: 6.5 Mb (6775701 bytes)
1 12 7 18 3 6 19 2 13 8 11 22 15 4 17 20 5 24 9 14 23 10 21 16 25
Model has been successfully processed </lang>
and
<lang> /*Knights.mathprog
Find a Knights Tour
Nigel_Galloway January 11th., 2012
- /
param ZBLS; param ROWS; param COLS; param D := 2; set ROWSR := 1..ROWS; set COLSR := 1..COLS; set ROWSV := (1-D)..(ROWS+D); set COLSV := (1-D)..(COLS+D); param Iz{ROWSR,COLSR}, integer, default 0; set ZBLSV := 1..(ZBLS+1); set ZBLSR := 1..ZBLS;
var BR{ROWSV,COLSV,ZBLSV}, binary;
void0{r in ROWSV, z in ZBLSR,c in (1-D)..0}: BR[r,c,z] = 0; void1{r in ROWSV, z in ZBLSR,c in (COLS+1)..(COLS+D)}: BR[r,c,z] = 0; void2{c in COLSV, z in ZBLSR,r in (1-D)..0}: BR[r,c,z] = 0; void3{c in COLSV, z in ZBLSR,r in (ROWS+1)..(ROWS+D)}: BR[r,c,z] = 0; void4{r in ROWSV,c in (1-D)..0}: BR[r,c,ZBLS+1] = 1; void5{r in ROWSV,c in (COLS+1)..(COLS+D)}: BR[r,c,ZBLS+1] = 1; void6{c in COLSV,r in (1-D)..0}: BR[r,c,ZBLS+1] = 1; void7{c in COLSV,r in (ROWS+1)..(ROWS+D)}: BR[r,c,ZBLS+1] = 1;
Izfree{r in ROWSR, c in COLSR, z in ZBLSR : Iz[r,c] = -1}: BR[r,c,z] = 0; Iz1{Izr in ROWSR, Izc in COLSR, r in ROWSR, c in COLSR, z in ZBLSR : Izr=r and Izc=c and Iz[Izr,Izc]=z}: BR[r,c,z] = 1;
rule1{z in ZBLSR}: sum{r in ROWSR, c in COLSR} BR[r,c,z] = 1; rule2{r in ROWSR, c in COLSR}: sum{z in ZBLSV} BR[r,c,z] = 1; rule3{r in ROWSR, c in COLSR, z in ZBLSR}: BR[0,0,z+1] + BR[r-1,c-2,z+1] + BR[r-1,c+2,z+1] + BR[r-2,c-1,z+1] + BR[r-2,c+1,z+1] + BR[r+1,c+2,z+1] + BR[r+1,c-2,z+1] + BR[r+2,c-1,z+1] + BR[r+2,c+1,z+1] - BR[r,c,z] >= 0;
solve;
for {r in ROWSR} {
for {c in COLSR} {
printf " %2d", sum{z in ZBLSR} BR[r,c,z]*z;
}
printf "\n";
} data;
param ROWS := 8; param COLS := 8; param ZBLS := 64; param Iz: 1 2 3 4 5 6 7 8 :=
1 . . . . . . . . 2 . . . . . . 48 . 3 . . . . . . . . 4 . . . . . . . . 5 . . . . . . . . 6 . . . . . . . . 7 . 58 . . . . . . 8 . . . . . . . . ; end;
</lang>
Produces:
<lang> GLPSOL: GLPK LP/MIP Solver, v4.47 Parameter(s) specified in the command line:
--minisat --math Knights.mathprog
Reading model section from Knights.mathprog... Reading data section from Knights.mathprog... 65 lines were read Generating void0... Generating void1... Generating void2... Generating void3... Generating void4... Generating void5... Generating void6... Generating void7... Generating Izfree... Generating Iz1... Generating rule1... Generating rule2... Generating rule3... Model has been successfully generated Will search for ANY feasible solution Translating to CNF-SAT... Original problem has 10466 rows, 9360 columns, and 55330 non-zeros 3968 covering inequalities 6370 partitioning equalities Solving CNF-SAT problem... Instance has 15056 variables, 46754 clauses, and 149794 literals
============================[MINISAT]=============================
| Conflicts | ORIGINAL | LEARNT | Progress | | | Clauses Literals | Limit Clauses Literals Lit/Cl | |
==================================================================
| 0 | 40512 143552 | 13504 0 0 0.0 | 0.000 % | | 100 | 32458 114610 | 14854 89 5138 57.7 | 46.633 % | | 250 | 32458 114610 | 16340 239 18544 77.6 | 46.633 % | | 475 | 27499 102956 | 17974 424 42212 99.6 | 46.892 % | | 813 | 27366 102490 | 19771 757 73184 96.7 | 51.541 % | | 1322 | 27366 102490 | 21748 1264 137991 109.2 | 52.245 % | | 2083 | 23226 92730 | 23923 2010 250286 124.5 | 53.620 % | | 3227 | 22239 90284 | 26315 3138 460582 146.8 | 53.620 % | | 4937 | 22239 90284 | 28947 4848 769486 158.7 | 53.620 % | | 7499 | 22206 90168 | 31842 7404 1258240 169.9 | 55.167 % | | 11346 | 21067 87284 | 35026 11248 2085553 185.4 | 55.167 % | | 17113 | 21067 87284 | 38528 17015 3625910 213.1 | 55.167 % | | 25763 | 21067 87284 | 42381 25665 5906283 230.1 | 55.167 % | | 38738 | 21051 87252 | 46619 38638 9316878 241.1 | 55.679 % | | 58199 | 21051 87252 | 51281 16434 3967196 241.4 | 55.685 % | | 87393 | 20707 86474 | 56410 45624 13013357 285.2 | 56.277 % | | 131184 | 20180 84834 | 62051 37252 8996727 241.5 | 56.542 % | | 196871 | 20180 84834 | 68256 49392 13807861 279.6 | 56.542 % | | 295399 | 20180 84834 | 75081 22688 5827696 256.9 | 56.542 % |
==================================================================
SATISFIABLE Objective value = 0.000000000e+000 Time used: 333.0 secs Memory used: 28.2 Mb (29609617 bytes)
51 24 31 6 49 26 33 64 30 5 50 25 32 63 48 43 23 52 7 4 27 44 15 34 8 29 60 45 62 47 42 17 59 22 53 28 3 16 35 14 54 9 56 61 46 39 18 41 21 58 11 38 19 2 13 36 10 55 20 57 12 37 40 1
Model has been successfully processed </lang>
Perl
Knight's tour using Warnsdorffs algorithm <lang perl>use strict; use warnings;
- Find a knight's tour
my @board;
- Choose starting position - may be passed in on command line; if
- not, choose random square.
my ($i, $j); if (my $sq = shift @ARGV) {
die "$0: illegal start square '$sq'\n" unless ($i, $j) = from_algebraic($sq);
} else {
($i, $j) = (int rand 8, int rand 8);
}
- Move sequence
my @moves = ();
foreach my $move (1..64) {
# Record current move push @moves, to_algebraic($i,$j); $board[$i][$j] = $move;
# Get list of possible next moves my @targets = possible_moves($i,$j);
# Find the one with the smallest degree
my @min = (9);
foreach my $target (@targets) {
my ($ni, $nj) = @$target;
my $next = possible_moves($ni,$nj);
@min = ($next, $ni, $nj) if $next < $min[0];
}
# And make it ($i, $j) = @min[1,2];
}
- Print the move list
for (my $i=0; $i<4; ++$i) {
for (my $j=0; $j<16; ++$j) {
my $n = $i*16+$j;
print $moves[$n];
print ', ' unless $n+1 >= @moves;
}
print "\n";
} print "\n";
- And the board, with move numbers
for (my $i=0; $i<8; ++$i) {
for (my $j=0; $j<8; ++$j) {
# Assumes (1) ANSI sequences work, and (2) output
# is light text on a dark background.
print "\e[7m" if ($i%2==$j%2);
printf " %2d", $board[$i][$j];
print "\e[0m";
}
print "\n";
}
- Find the list of positions the knight can move to from the given square
sub possible_moves {
my ($i, $j) = @_;
return grep { $_->[0] >= 0 && $_->[0] < 8
&& $_->[1] >= 0 && $_->[1] < 8
&& !$board[$_->[0]][$_->[1]] } (
[$i-2,$j-1], [$i-2,$j+1], [$i-1,$j-2], [$i-1,$j+2],
[$i+1,$j-2], [$i+1,$j+2], [$i+2,$j-1], [$i+2,$j+1]);
}
- Return the algebraic name of the square identified by the coordinates
- i=rank, 0=black's home row; j=file, 0=white's queen's rook
sub to_algebraic {
my ($i, $j) = @_;
chr(ord('a') + $j) . (8-$i);
}
- Return the coordinates matching the given algebraic name
sub from_algebraic {
my $square = shift;
return unless $square =~ /^([a-h])([1-8])$/;
return (8-$2, ord($1) - ord('a'));
}</lang>
Sample output (start square c3):
Perl 6
<lang perl6>my @board;
my $I = 8; my $J = 8; my $F = $I*$J > 99 ?? "%3d" !! "%2d";
- Choose starting position - may be passed in on command line; if
- not, choose random square.
my ($i, $j);
if my $sq = shift @*ARGS {
die "$*PROGRAM_NAME: illegal start square '$sq'\n" unless ($i, $j) = from_algebraic($sq);
} else {
($i, $j) = (^$I).pick, (^$J).pick;
}
- Move sequence
my @moves = ();
for 1 .. $I * $J -> $move {
# Record current move
push @moves, to_algebraic($i,$j);
# @board[$i] //= []; # (uncomment if autoviv is broken)
@board[$i][$j] = $move;
# Find move with the smallest degree
my @min = (9);
for possible_moves($i,$j) -> @target {
my ($ni, $nj) = @target;
my $next = possible_moves($ni,$nj);
@min = $next, $ni, $nj if $next < @min[0];
}
# And make it
($i, $j) = @min[1,2];
}
- Print the move list
for @moves.kv -> $i, $m {
print ',', $i %% 16 ?? "\n" !! " " if $i; print $m;
} say "\n";
- And the board, with move numbers
for ^$I -> $i {
for ^$J -> $j {
# Assumes (1) ANSI sequences work, and (2) output
# is light text on a dark background.
print "\e[7m" if $i % 2 == $j % 2;
printf $F, @board[$i][$j];
print "\e[0m";
}
print "\n";
}
- Find the list of positions the knight can move to from the given square
sub possible_moves($i,$j) {
grep -> [$ni, $nj] { $ni ~~ ^$I and $nj ~~ ^$J and !@board[$ni][$nj] },
[$i-2,$j-1], [$i-2,$j+1], [$i-1,$j-2], [$i-1,$j+2],
[$i+1,$j-2], [$i+1,$j+2], [$i+2,$j-1], [$i+2,$j+1];
}
- Return the algebraic name of the square identified by the coordinates
- i=rank, 0=black's home row; j=file, 0=white's queen's rook
sub to_algebraic($i,$j) {
chr(ord('a') + $j) ~ ($I - $i);
}
- Return the coordinates matching the given algebraic name
sub from_algebraic($square where /^ (<[a..z]>) (\d+) $/) {
$I - $1, ord(~$0) - ord('a');
}</lang> (Output identical to Perl's above.)
Phix
This is pretty fast (<<1s) up to size 48, before some sizes start to take quite some time to complete. It will even solve a 200x200 in 0.67s <lang Phix>constant size = 8 constant nchars = length(sprintf(" %d",size*size)) constant fmt = sprintf(" %%%dd",nchars-1) constant blank = repeat(' ',nchars)
-- to simplify output, each square is nchars sequence board = repeat(repeat(' ',size*nchars),size) -- keep current counts, immediately backtrack if any hit 0 -- (in line with the above, we only use every nth entry) sequence warnsdorffs = repeat(repeat(0,size*nchars),size)
constant ROW = 1, COL = 2 constant moves = {{-1,-2},{-2,-1},{-2,1},{-1,2},{1,2},{2,1},{2,-1},{1,-2}}
function onboard(integer row, integer col)
return row>=1 and row<=size and col>=nchars and col<=nchars*size
end function
procedure init_warnsdorffs() integer nrow,ncol
for row=1 to size do
for col=nchars to nchars*size by nchars do
for move=1 to length(moves) do
nrow = row+moves[move][ROW]
ncol = col+moves[move][COL]*nchars
if onboard(nrow,ncol) then
warnsdorffs[row][col] += 1
end if
end for
end for
end for
end procedure
atom t0 = time() integer tries = 0 atom t1 = time()+1 function solve(integer row, integer col, integer n) integer nrow, ncol if time()>t1 then
?{row,floor(col/nchars),n,tries}
puts(1,join(board,"\n"))
t1 = time()+1
-- if wait_key()='!' then ?9/0 end if end if
tries+= 1
if n>size*size then return 1 end if
sequence wmoves = {}
for move=1 to length(moves) do
nrow = row+moves[move][ROW]
ncol = col+moves[move][COL]*nchars
if onboard(nrow,ncol)
and board[nrow][ncol]=' ' then
wmoves = append(wmoves,{warnsdorffs[nrow][ncol],nrow,ncol})
end if
end for
wmoves = sort(wmoves)
-- avoid creating orphans
if length(wmoves)<2 or wmoves[2][1]>1 then
for m=1 to length(wmoves) do
{?,nrow,ncol} = wmoves[m]
warnsdorffs[nrow][ncol] -= 1
end for
for m=1 to length(wmoves) do
{?,nrow,ncol} = wmoves[m]
board[nrow][ncol-nchars+1..ncol] = sprintf(fmt,n)
if solve(nrow,ncol,n+1) then return 1 end if
board[nrow][ncol-nchars+1..ncol] = blank
end for
for m=1 to length(wmoves) do
{?,nrow,ncol} = wmoves[m]
warnsdorffs[nrow][ncol] += 1
end for
end if
return 0
end function
init_warnsdorffs() board[1][nchars] = '1' if solve(1,nchars,2) then
puts(1,join(board,"\n"))
printf(1,"\nsolution found in %d tries (%3.2fs)\n",{tries,time()-t0})
else
puts(1,"no solutions found\n")
end if
{} = wait_key()</lang>
- Output:
"started" 1 16 31 40 3 18 21 56 30 39 2 17 42 55 4 19 15 32 41 46 53 20 57 22 38 29 48 43 58 45 54 5 33 14 37 52 47 60 23 62 28 49 34 59 44 63 6 9 13 36 51 26 11 8 61 24 50 27 12 35 64 25 10 7 solution found in 64 tries (0.00s)
PicoLisp
<lang PicoLisp>(load "@lib/simul.l")
- Build board
(grid 8 8)
- Generate legal moves for a given position
(de moves (Tour)
(extract
'((Jump)
(let? Pos (Jump (car Tour))
(unless (memq Pos Tour)
Pos ) ) )
(quote # (taken from "games/chess.l")
((This) (: 0 1 1 0 -1 1 0 -1 1)) # South Southwest
((This) (: 0 1 1 0 -1 1 0 1 1)) # West Southwest
((This) (: 0 1 1 0 -1 -1 0 1 1)) # West Northwest
((This) (: 0 1 1 0 -1 -1 0 -1 -1)) # North Northwest
((This) (: 0 1 -1 0 -1 -1 0 -1 -1)) # North Northeast
((This) (: 0 1 -1 0 -1 -1 0 1 -1)) # East Northeast
((This) (: 0 1 -1 0 -1 1 0 1 -1)) # East Southeast
((This) (: 0 1 -1 0 -1 1 0 -1 1)) ) ) ) # South Southeast
- Build a list of moves, using Warnsdorff’s algorithm
(let Tour '(b1) # Start at b1
(while
(mini
'((P) (length (moves (cons P Tour))))
(moves Tour) )
(push 'Tour @) )
(flip Tour) )</lang>
Output:
-> (b1 a3 b5 a7 c8 b6 a8 c7 a6 b8 d7 f8 h7 g5 h3 g1 e2 c1 a2 b4 c2 a1 b3 a5 b7 d8 c6 d4 e6 c5 a4 c3 d1 b2 c4 d2 f1 h2 f3 e1 d3 e5 f7 h8 g6 h4 g2 f4 d5 e7 g8 h6 g4 e3 f5 d6 e8 g7 h5 f6 e4 g3 h1 f2)
PostScript
You probably shouldn't send this to a printer. Solution using Warnsdorffs algorithm. <lang postscript>%!PS-Adobe-3.0 %%BoundingBox: 0 0 300 300
/s { 300 n div } def /l { rlineto } def
% draws a square /bx { s mul exch s mul moveto s 0 l 0 s l s neg 0 l 0 s neg l } def
% draws checker board /xbd { 1 setgray
0 0 moveto 300 0 l 0 300 l -300 0 l fill
.7 1 .6 setrgbcolor
0 1 n1 { dup 2 mod 2 n1 { 1 index bx fill } for pop } for
0 setgray
} def
/ar1 { [ exch { 0 } repeat ] } def /ar2 { [ exch dup { dup ar1 exch } repeat pop ] } def
/neighbors {
-1 2 0
1 2 0
2 1 0
2 -1 0
1 -2 0
-1 -2 0
-2 -1 0
-2 1 0
%24 x y add 3 mul roll
} def
/func { 0 dict begin mark } def /var { counttomark -1 1 { 2 add -1 roll def } for cleartomark } def
% x y can_goto -> bool /can_goto {
func /x /y var
x 0 ge
x n lt
y 0 ge
y n lt
and and and {
occupied x get y get 0 eq
} { false } ifelse
end
} def
% x y num_access -> number of cells reachable from (x,y) /num_access {
func /x /y var
/count 0 def
x y can_goto {
neighbors
8 { pop y add exch x add exch can_goto {
/count count 1 add def
} if
} repeat
count 0 gt { count } { 9 } ifelse
} { 10 } ifelse
end
} def
% a circle /marker { x s mul y s mul s 20 div 0 360 arc fill } def
% n solve -> draws board of size n x n, calcs path and draws it /solve {
func /n var
/n1 n 1 sub def
/c false def
8 n div setlinewidth
gsave
0 1 n1 { /x exch def c not {
0 1 n1 {
/occupied n ar2 def
c not {
/c true def
/y exch def
grestore xbd gsave
s 2 div dup translate
n n mul 2 sub -1 0 { /iter exch def
c {
0 setgray marker x s mul y s mul moveto
occupied x get y 1 put
neighbors
8 { pop y add exch x add exch 2 copy num_access 24 3 roll } repeat
7 { dup 4 index lt { 6 3 roll } if pop pop pop } repeat
9 ge iter 0 gt and { /c false def } if
/y exch def
/x exch def
.2 setgray x s mul y s mul lineto stroke
} if } for
% to be nice, draw box at final position
.5 0 0 setrgbcolor marker
y .5 sub x .5 sub bx 1 setlinewidth stroke
stroke
} if
} for } if } for showpage
grestore
end
} def
3 1 100 { solve } for
%%EOF</lang>
Prolog
Knights tour using Warnsdorffs algorithm
<lang Prolog>% N is the number of lines of the chessboard knight(N) :- Max is N * N, length(L, Max), knight(N, 0, Max, 0, 0, L), display(N, 0, L).
% knight(NbCol, Coup, Max, Lig, Col, L), % NbCol : number of columns per line % Coup : number of the current move % Max : maximum number of moves % Lig/ Col : current position of the knight % L : the "chessboard"
% the game is over knight(_, Max, Max, _, _, _) :- !.
knight(NbCol, N, MaxN, Lg, Cl, L) :- % Is the move legal Lg >= 0, Cl >= 0, Lg < NbCol, Cl < NbCol,
Pos is Lg * NbCol + Cl, N1 is N+1, % is the place free nth0(Pos, L, N1),
LgM1 is Lg - 1, LgM2 is Lg - 2, LgP1 is Lg + 1, LgP2 is Lg + 2, ClM1 is Cl - 1, ClM2 is Cl - 2, ClP1 is Cl + 1, ClP2 is Cl + 2, maplist(best_move(NbCol, L), [(LgP1, ClM2), (LgP2, ClM1), (LgP2, ClP1),(LgP1, ClP2), (LgM1, ClM2), (LgM2, ClM1), (LgM2, ClP1),(LgM1, ClP2)], R), sort(R, RS), pairs_values(RS, Moves),
move(NbCol, N1, MaxN, Moves, L).
move(NbCol, N1, MaxN, [(Lg, Cl) | R], L) :- knight(NbCol, N1, MaxN, Lg, Cl, L); move(NbCol, N1, MaxN, R, L).
%% An illegal move is scored 1000 best_move(NbCol, _L, (Lg, Cl), 1000-(Lg, Cl)) :- ( Lg < 0 ; Cl < 0; Lg >= NbCol; Cl >= NbCol), !.
best_move(NbCol, L, (Lg, Cl), 1000-(Lg, Cl)) :- Pos is Lg*NbCol+Cl, nth0(Pos, L, V), \+var(V), !.
%% a legal move is scored with the number of moves a knight can make best_move(NbCol, L, (Lg, Cl), R-(Lg, Cl)) :- LgM1 is Lg - 1, LgM2 is Lg - 2, LgP1 is Lg + 1, LgP2 is Lg + 2, ClM1 is Cl - 1, ClM2 is Cl - 2, ClP1 is Cl + 1, ClP2 is Cl + 2, include(possible_move(NbCol, L), [(LgP1, ClM2), (LgP2, ClM1), (LgP2, ClP1),(LgP1, ClP2), (LgM1, ClM2), (LgM2, ClM1), (LgM2, ClP1),(LgM1, ClP2)], Res), length(Res, Len), ( Len = 0 -> R = 1000; R = Len).
% test if a place is enabled possible_move(NbCol, L, (Lg, Cl)) :- % move must be legal Lg >= 0, Cl >= 0, Lg < NbCol, Cl < NbCol, Pos is Lg * NbCol + Cl, % place must be free nth0(Pos, L, V), var(V).
display(_, _, []).
display(N, N, L) :-
nl,
display(N, 0, L).
display(N, M, [H | T]) :- writef('%3r', [H]), M1 is M + 1, display(N, M1, T). </lang>
Output :
?- knight(8). 1 16 31 40 3 18 21 56 30 39 2 17 42 55 4 19 15 32 41 46 53 20 57 22 38 29 48 43 58 45 54 5 33 14 37 52 47 60 23 62 28 49 34 59 44 63 6 9 13 36 51 26 11 8 61 24 50 27 12 35 64 25 10 7 true . ?- knight(20). 1 40 81 90 3 42 77 94 5 44 73 102 7 46 69 62 9 48 51 60 82 89 2 41 92 95 4 43 76 101 6 45 72 103 8 47 68 61 10 49 39 80 91 96 153 78 93 100 129 74 109 104 123 70 111 120 63 50 59 52 88 83 154 79 98 159 152 75 108 105 128 71 110 121 124 67 112 119 64 11 155 38 97 160 157 200 99 162 151 130 107 122 127 132 141 118 125 66 53 58 84 87 156 199 176 161 158 201 106 163 150 131 142 145 126 133 140 113 12 65 37 182 85 178 207 198 175 164 173 216 143 166 149 222 139 146 117 134 57 54 86 179 206 197 204 177 208 217 202 165 172 221 144 167 148 223 138 55 114 13 183 36 181 212 209 218 203 174 215 220 227 170 281 224 303 168 147 116 135 56 180 211 196 205 230 213 238 219 228 171 280 225 302 169 282 343 304 137 14 115 35 184 231 210 237 246 229 214 279 226 301 298 283 342 367 308 347 344 305 136 232 195 236 245 234 239 278 247 300 297 284 359 366 309 348 345 368 307 350 15 185 34 233 240 261 248 287 296 285 358 299 310 341 378 365 384 349 346 369 306 194 241 250 235 244 277 260 313 294 311 360 373 364 383 354 379 370 385 16 351 33 186 243 262 249 288 295 286 361 316 357 340 377 372 395 386 353 380 333 388 242 193 254 251 276 259 314 293 312 321 374 363 398 355 382 371 394 387 352 17 187 32 263 258 267 252 289 322 315 362 317 356 339 376 399 396 381 334 389 332 192 255 190 253 264 275 268 271 292 323 320 375 326 397 338 335 390 393 18 21 31 188 257 266 29 270 273 290 27 318 327 324 25 336 329 400 23 20 331 392 256 191 30 189 274 265 28 269 272 291 26 319 328 325 24 337 330 391 22 19 true .
Alternative version
<lang prolog>:- initialization(main).
board_size(8).
in_board(X*Y) :- board_size(N), between(1,N,Y), between(1,N,X).
% express jump-graph in dynamic "move"-rules
make_graph :-
findall(_, (in_board(P), assert_moves(P)), _).
% where
assert_moves(P) :-
findall(_, (can_move(P,Q), asserta(move(P,Q))), _).
can_move(X*Y,Q) :-
( one(X,X1), two(Y,Y1) ; two(X,X1), one(Y,Y1) )
, Q = X1*Y1, in_board(Q)
. % where
one(M,N) :- succ(M,N) ; succ(N,M).
two(M,N) :- N is M + 2 ; N is M - 2.
hamiltonian(P,Pn) :-
board_size(N), Size is N * N , hamiltonian(P,Size,[],Ps), enumerate(Size,Ps,Pn) . % where enumerate(_, [] , [] ). enumerate(N, [P|Ps], [N:P|Pn]) :- succ(M,N), enumerate(M,Ps,Pn).
hamiltonian(P,N,Ps,Res) :-
N =:= 1 -> Res = [P|Ps]
; warnsdorff(Ps,P,Q), succ(M,N)
, hamiltonian(Q,M,[P|Ps],Res)
.
% where
warnsdorff(Ps,P,Q) :-
moves(Ps,P,Qs), maplist(next_moves(Ps), Qs, Xs)
, keysort(Xs,Ys), member(_-Q,Ys)
.
next_moves(Ps,Q,L-Q) :- moves(Ps,Q,Rs), length(Rs,L).
moves(Ps,P,Qs) :-
findall(Q, (move(P,Q), \+ member(Q,Ps)), Qs).
show_path(Pn) :- findall(_, (in_board(P), show_cell(Pn,P)), _).
% where
show_cell(Pn,X*Y) :-
member(N:X*Y,Pn), format('%3.0d',[N]), board_size(X), nl.
main :- make_graph, hamiltonian(5*3,Pn), show_path(Pn), halt.</lang>
- Output:
5 18 35 22 3 16 55 24 36 21 4 17 54 23 2 15 19 6 59 34 1 14 25 56 60 37 20 53 62 57 32 13 7 52 61 58 33 30 63 26 38 49 40 29 64 45 12 31 41 8 51 48 43 10 27 46 50 39 42 9 28 47 44 1
20x20 board runs in: time: 0.91 memory: 68608.
Python
Knights tour using Warnsdorffs algorithm <lang python>import copy
boardsize=6 _kmoves = ((2,1), (1,2), (-1,2), (-2,1), (-2,-1), (-1,-2), (1,-2), (2,-1))
def chess2index(chess, boardsize=boardsize):
'Convert Algebraic chess notation to internal index format'
chess = chess.strip().lower()
x = ord(chess[0]) - ord('a')
y = boardsize - int(chess[1:])
return (x, y)
def boardstring(board, boardsize=boardsize):
r = range(boardsize)
lines =
for y in r:
lines += '\n' + ','.join('%2i' % board[(x,y)] if board[(x,y)] else ' '
for x in r)
return lines
def knightmoves(board, P, boardsize=boardsize):
Px, Py = P
kmoves = set((Px+x, Py+y) for x,y in _kmoves)
kmoves = set( (x,y)
for x,y in kmoves
if 0 <= x < boardsize
and 0 <= y < boardsize
and not board[(x,y)] )
return kmoves
def accessibility(board, P, boardsize=boardsize):
access = []
brd = copy.deepcopy(board)
for pos in knightmoves(board, P, boardsize=boardsize):
brd[pos] = -1
access.append( (len(knightmoves(brd, pos, boardsize=boardsize)), pos) )
brd[pos] = 0
return access
def knights_tour(start, boardsize=boardsize, _debug=False):
board = {(x,y):0 for x in range(boardsize) for y in range(boardsize)}
move = 1
P = chess2index(start, boardsize)
board[P] = move
move += 1
if _debug:
print(boardstring(board, boardsize=boardsize))
while move <= len(board):
P = min(accessibility(board, P, boardsize))[1]
board[P] = move
move += 1
if _debug:
print(boardstring(board, boardsize=boardsize))
input('\n%2i next: ' % move)
return board
if __name__ == '__main__':
while 1:
boardsize = int(input('\nboardsize: '))
if boardsize < 5:
continue
start = input('Start position: ')
board = knights_tour(start, boardsize)
print(boardstring(board, boardsize=boardsize))</lang>
- Sample runs
boardsize: 5 Start position: c3 19,12,17, 6,21 2, 7,20,11,16 13,18, 1,22, 5 8, 3,24,15,10 25,14, 9, 4,23 boardsize: 8 Start position: h8 38,41,18, 3,22,27,16, 1 19, 4,39,42,17, 2,23,26 40,37,54,21,52,25,28,15 5,20,43,56,59,30,51,24 36,55,58,53,44,63,14,29 9, 6,45,62,57,60,31,50 46,35, 8,11,48,33,64,13 7,10,47,34,61,12,49,32 boardsize: 10 Start position: e6 29, 4,57,24,73, 6,95,10,75, 8 58,23,28, 5,94,25,74, 7,100,11 3,30,65,56,27,72,99,96, 9,76 22,59, 2,63,68,93,26,81,12,97 31,64,55,66, 1,82,71,98,77,80 54,21,60,69,62,67,92,79,88,13 49,32,53,46,83,70,87,42,91,78 20,35,48,61,52,45,84,89,14,41 33,50,37,18,47,86,39,16,43,90 36,19,34,51,38,17,44,85,40,15 boardsize: 200 Start position: a1 510,499,502,101,508,515,504,103,506,5021 ... 195,8550,6691,6712,197,6704,201,6696,199 501,100,509,514,503,102,507,5020,5005,10 ... 690,6713,196,8553,6692,6695,198,6703,202 498,511,500,4989,516,5019,5004,505,5022, ... ,30180,8559,6694,6711,8554,6705,200,6697 99,518,513,4992,5003,4990,5017,5044,5033 ... 30205,8552,30181,8558,6693,6702,203,6706 512,497,4988,517,5018,5001,5034,5011,504 ... 182,30201,30204,8555,6710,8557,6698,6701 519,98,4993,5002,4991,5016,5043,5052,505 ... 03,30546,30183,30200,30185,6700,6707,204 496,4987,520,5015,5000,5035,5012,5047,51 ... 4,30213,30202,31455,8556,6709,30186,6699 97,522,4999,4994,5013,5042,5051,5060,505 ... 7,31456,31329,30184,30199,30190,205,6708 4986,495,5014,521,5036,4997,5048,5101,50 ... 1327,31454,30195,31472,30187,30198,30189 523,96,4995,4998,5041,5074,5061,5050,507 ... ,31330,31471,31328,31453,30196,30191,206 ... 404,731,704,947,958,1013,966,1041,1078,1 ... 9969,39992,39987,39996,39867,39856,39859 5,706,735,960,955,972,957,1060,1025,106 ... ,39978,39939,39976,39861,39990,297,39866 724,403,730,705,946,967,1012,971,1040,10 ... 9975,39972,39991,39868,39863,39860,39855 707, 4,723,736,729,956,973,996,1061,1026 ... ,39850,39869,39862,39973,39852,39865,298 402,725,708,943,968,945,970,1011,978,997 ... 6567,39974,39851,39864,36571,39854,36573 3,722,737,728,741,942,977,974,995,1010, ... ,39800,39849,36570,39853,36574,299,14088 720,401,726,709,944,969,742,941,980,975, ... ,14091,36568,36575,14084,14089,36572,843 711, 2,721,738,727,740,715,976,745,940,9 ... 65,36576,14083,14090,36569,844,14087,300 400,719,710,713,398,717,746,743,396,981, ... ,849,304,14081,840,847,302,14085,842,845 1,712,399,718,739,714,397,716,747,744,3 ... 4078,839,848,303,14082,841,846,301,14086
The 200x200 example warmed my study in its computation but did return a tour.
P.S. There is a slight deviation to a strict interpretation of Warnsdorffs algorithm in that as a conveniance, tuples of the length of the night moves followed by the position are minimized so knights moves with the same length will try and break the ties based on their minimum x,y position. In practice, it seems to give comparable results to the original algorithm.
boardsize: 5 Start position: a3 Traceback (most recent call last):
File "rosettacodekt.py", line 65, in <module> board = knights_tour(start, boardsize) File "rosettacodekt.py", line 51, in knights_tour P = min(accessibility(board, P, boardsize))[1]
ValueError: min() arg is an empty sequence
R
Based on a slight modification of Warnsdorff's algorithm, in that if a dead-end is reached, the program backtracks to the next best move.
<lang r>#!/usr/bin/Rscript
- M x N Chess Board.
M = 8; N = 8; board = matrix(0, nrow = M, ncol = N)
- Get/Set value on a board position.
getboard = function (position) { board[position[1], position[2]] } setboard = function (position, x) { board[position[1], position[2]] <<- x }
- (Relative) Hops of a Knight.
hops = cbind(c(-2, -1), c(-1, -2), c(+1, -2), c(+2, -1),
c(+2, +1), c(+1, +2), c(-1, +2), c(-2, +1))
- Validate a move.
valid = function (move) {
all(1 <= move & move <= c(M, N)) && (getboard(move) == 0)
}
- Moves possible from a given position.
explore = function (position) {
moves = position + hops cbind(moves[, apply(moves, 2, valid)])
}
- Possible moves sorted according to their Wornsdorff cost.
candidates = function (position) {
moves = explore(position)
# No candidate moves available.
if (ncol(moves) == 0) { return(moves) }
wcosts = apply(moves, 2, function (position) { ncol(explore(position)) })
cbind(moves[, order(wcosts)])
}
- Recursive function for touring the chess board.
knightTour = function (position, moveN) {
# Tour Complete.
if (moveN > (M * N)) {
print(board)
quit()
}
# Available moves.
moves = candidates(position)
# None possible. Backtrack.
if (ncol(moves) == 0) { return() }
# Make a move, and continue the tour.
apply(moves, 2, function (position) {
setboard(position, moveN)
knightTour(position, moveN + 1)
setboard(position, 0)
})
}
- User Input: Starting position (in algebraic notation).
square = commandArgs(trailingOnly = TRUE)
- Convert into board co-ordinates.
row = M + 1 - as.integer(substr(square, 2, 2)) ascii = function (ch) { as.integer(charToRaw(ch)) } col = 1 + ascii(substr(square, 1, 1)) - ascii('a') position = c(row, col)
- Begin tour.
setboard(position, 1); knightTour(position, 2)</lang>
Output:
./knight.R e3
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
[1,] 6 9 24 55 62 11 26 29
[2,] 23 54 7 10 25 28 63 12
[3,] 8 5 50 61 56 59 30 27
[4,] 37 22 53 58 43 48 13 64
[5,] 4 51 38 49 60 57 44 31
[6,] 21 36 19 52 1 42 47 14
[7,] 18 3 34 39 16 45 32 41
[8,] 35 20 17 2 33 40 15 46
Racket
<lang racket>
- lang racket
(define N 8) (define nexts ; construct the graph
(let ([ds (for*/list ([x 2] [x* '(+1 -1)] [y* '(+1 -1)])
(cons (* x* (+ 1 x)) (* y* (- 2 x))))])
(for*/vector ([i N] [j N])
(filter values (for/list ([d ds])
(let ([i (+ i (car d))] [j (+ j (cdr d))])
(and (< -1 i N) (< -1 j N) (+ j (* N i)))))))))
(define (tour x y)
(define xy (+ x (* N y)))
(let loop ([seen (list xy)] [ns (vector-ref nexts xy)] [n (sub1 (* N N))])
(if (zero? n) (reverse seen)
(for/or ([next (sort (map (λ(n) (cons n (remq* seen (vector-ref nexts n)))) ns)
< #:key length #:cache-keys? #t)])
(loop (cons (car next) seen) (cdr next) (sub1 n))))))
(define (draw tour)
(define v (make-vector (* N N)))
(for ([n tour] [i (in-naturals 1)]) (vector-set! v n i))
(for ([i N])
(displayln (string-join (for/list ([j (in-range i (* N N) N)])
(~a (vector-ref v j) #:width 2 #:align 'right))
" "))))
(draw (tour (random N) (random N))) </lang>
- Output:
56 11 36 33 52 13 38 17 35 32 55 12 37 16 51 14 10 57 34 53 48 45 18 39 31 54 43 64 41 50 15 46 60 9 58 49 44 47 40 19 27 30 61 42 63 22 1 4 8 59 28 25 6 3 20 23 29 26 7 62 21 24 5 2
REXX
This REXX version is modeled after the XPL0 example.
The size of the chessboard may be specified as well as the knight's starting position.
This is an open tour solution. (See this task's discussion page for an explanation, the section is The 7x7 problem.) <lang rexx>/*REXX program solves the knight's tour problem for a (general) NxN chessboard.*/ parse arg N sRank sFile . /*obtain optional arguments from the CL*/ if N== | N=="," then N=8 /*No boardsize specified? Use default.*/ if sRank== | sRank=="," then sRank=N /*No starting rank given? " " */ if sFile== | sFile=="," then sFile=1 /* " " file " " " */ NN=N**2; NxN='a ' N"x"N ' chessboard' /*file [↓] [↓] r=rank */ @.=; do r=1 for N; do f=1 for N; @.r.f=.; end /*f*/; end /*r*/ beg= '-1-' /*[↑] create an empty NxN chessboard.*/
Kr = '2 1 -1 -2 -2 -1 1 2' /*the legal "rank" moves for a knight.*/
Kf = '1 2 2 1 -1 -2 -2 -1' /* " " "file" " " " " */
kr.M=words(Kr) /*number of possible moves for a Knight*/ parse var Kr Kr.1 Kr.2 Kr.3 Kr.4 Kr.5 Kr.6 Kr.7 Kr.8 /*parse the legal moves by hand*/ parse var Kf Kf.1 Kf.2 Kf.3 Kf.4 Kf.5 Kf.6 Kf.7 Kf.8 /* " " " " " " */ @.sRank.sFile= beg /*the knight's starting position. */ @kt= "knight's tour" /*a handy-dandy literal for the SAYs. */ if \move(2, sRank, sFile) & \(N==1) then say 'No' @kt "solution for" NxN'.'
else say 'A solution for the' @kt "on" NxN':'
!=left(, 9 * (n<18) ) /*used for indentation of chessboard. */ _=substr(copies("┼───",N),2); say; say ! translate('┌'_"┐", '┬', "┼") /*a square.*/
/* [↓] build a display for chessboard.*/
do r=N for N by -1; if r\==N then say ! '├'_"┤"; L=@.
do f=1 for N; ?=@.r.f; if ?==NN then ?='end'; L=L'│'center(?, 3) /*is "end"?*/
end /*f*/ /*done with rank of the chessboard.*/
say ! translate(L'│', , .) /*display a " " " " */
end /*r*/ /*19x19 chessboard can be shown 80 cols*/
say ! translate('└'_"┘", '┴', "┼") /*show the last rank of the chessboard.*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ move: procedure expose @. Kr. Kf. NN; parse arg #,rank,file /*obtain move,rank,file.*/
do t=1 for Kr.M; nr=rank+Kr.t; nf=file+Kf.t /*position of the knight*/
if @.nr.nf==. then do; @.nr.nf=# /*Empty? Knight can move*/
if #==NN then return 1 /*is this the last move?*/
if move(#+1,nr,nf) then return 1 /* " " " " " */
@.nr.nf=. /*undo the above move. */
end /*try different move. */
end /*t*/ /* [↑] all moves tried.*/
return 0 /*tour is not possible. */</lang>
output when using the default input:
A solution for the knight's tour on a 8x8 chessboard:
┌───┬───┬───┬───┬───┬───┬───┬───┐
│-1-│38 │55 │34 │ 3 │36 │19 │22 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│54 │47 │ 2 │37 │20 │23 │ 4 │17 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│39 │56 │33 │46 │35 │18 │21 │10 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│48 │53 │40 │57 │24 │11 │16 │ 5 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│59 │32 │45 │52 │41 │26 │ 9 │12 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│44 │49 │58 │25 │62 │15 │ 6 │27 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│31 │60 │51 │42 │29 │ 8 │13 │end│
├───┼───┼───┼───┼───┼───┼───┼───┤
│50 │43 │30 │61 │14 │63 │28 │ 7 │
└───┴───┴───┴───┴───┴───┴───┴───┘
Ruby
Knights tour using Warnsdorffs rule <lang ruby>class Board
Cell = Struct.new(:value, :adj) do
def self.end=(end_val)
@@end = end_val
end
def try(seq_num)
self.value = seq_num
return true if seq_num==@@end
a = []
adj.each_with_index do |cell, n|
a << [wdof(cell.adj)*10+n, cell] if cell.value.zero?
end
a.sort.each {|_, cell| return true if cell.try(seq_num+1)}
self.value = 0
false
end
def wdof(adj)
adj.count {|cell| cell.value.zero?}
end
end
def initialize(rows, cols)
@rows, @cols = rows, cols
unless defined? ADJACENT # default move (Knight)
eval("ADJACENT = [[-1,-2],[-2,-1],[-2,1],[-1,2],[1,2],[2,1],[2,-1],[1,-2]]")
end
frame = ADJACENT.flatten.map(&:abs).max
@board = Array.new(rows+frame) do |i|
Array.new(cols+frame) do |j|
(i<rows and j<cols) ? Cell.new(0) : nil # frame (Sentinel value : nil)
end
end
rows.times do |i|
cols.times do |j|
@board[i][j].adj = ADJACENT.map{|di,dj| @board[i+di][j+dj]}.compact
end
end
Cell.end = rows * cols
@format = " %#{(rows * cols).to_s.size}d"
end
def solve(sx, sy)
if (@rows*@cols).odd? and (sx+sy).odd?
puts "No solution"
else
puts (@board[sx][sy].try(1) ? to_s : "No solution")
end
end
def to_s
(0...@rows).map do |x|
(0...@cols).map{|y| @format % @board[x][y].value}.join
end
end
end
def knight_tour(rows=8, cols=rows, sx=rand(rows), sy=rand(cols))
puts "\nBoard (%d x %d), Start:[%d, %d]" % [rows, cols, sx, sy] Board.new(rows, cols).solve(sx, sy)
end
knight_tour(8,8,3,1)
knight_tour(5,5,2,2)
knight_tour(4,9,0,0)
knight_tour(5,5,0,1)
knight_tour(12,12,1,1)</lang> Which produces:
Board (8 x 8), Start:[3, 1] 23 20 3 32 25 10 5 8 2 35 24 21 4 7 26 11 19 22 33 36 31 28 9 6 34 1 50 29 48 37 12 27 51 18 53 44 61 30 47 38 54 43 56 49 58 45 62 13 17 52 41 60 15 64 39 46 42 55 16 57 40 59 14 63 Board (5 x 5), Start:[2, 2] 19 8 3 14 25 2 13 18 9 4 7 20 1 24 15 12 17 22 5 10 21 6 11 16 23 Board (4 x 9), Start:[0, 0] 1 34 3 28 13 24 9 20 17 4 29 6 33 8 27 18 23 10 35 2 31 14 25 12 21 16 19 30 5 36 7 32 15 26 11 22 Board (5 x 5), Start:[0, 1] No solution Board (12 x 12), Start:[1, 1] 87 24 59 2 89 26 61 4 39 8 31 6 58 1 88 25 60 3 92 27 30 5 38 9 23 86 83 90 103 98 29 62 93 40 7 32 82 57 102 99 84 91 104 97 28 37 10 41 101 22 85 114 105 116 111 94 63 96 33 36 56 81 100 123 128 113 106 117 110 35 42 11 21 122 141 80 115 124 127 112 95 64 109 34 144 55 78 121 142 129 118 107 126 133 12 43 51 20 143 140 79 120 125 138 69 108 65 134 54 73 52 77 130 139 70 119 132 137 44 13 19 50 75 72 17 48 131 68 15 46 135 66 74 53 18 49 76 71 16 47 136 67 14 45
cf. Solve a Holy Knight's tour:
Rust
<lang rust> use std::fmt;
const SIZE: usize = 8; const MOVES: [(i32, i32); 8] = [(2,1), (1,2), (-1,2), (-2,1), (-2,-1), (-1,-2), (1,-2), (2,-1)];
- [derive(Copy, Clone, Eq, PartialEq, PartialOrd, Ord)]
struct Point {
x: i32, y: i32
}
impl Point {
fn mov(&self, &(dx,dy): &(i32, i32)) -> Point {
Point {
x: self.x + dx,
y: self.y + dy
}
}
}
struct Board {
field: [[i32;SIZE];SIZE]
}
impl Board {
fn new() -> Board {
return Board {
field: [[0; SIZE]; SIZE]
};
}
fn available(&self, p: Point) -> bool {
let valid = 0 <= p.x && p.x < SIZE as i32
&& 0 <= p.y && p.y < SIZE as i32;
return valid && self.field[p.x as usize][p.y as usize] == 0;
}
// calculate the number of possible moves
fn count_degree(&self, p: Point) -> i32 {
let mut count = 0;
for dir in MOVES.iter() {
let next = p.mov(dir);
if self.available(next) {
count += 1;
}
}
return count;
}
}
impl fmt::Display for Board {
fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
for row in self.field.iter() {
for x in row.iter(){
try!(write!(f, "{:3} ", x));
}
try!(write!(f, "\n"));
}
Ok(())
}
}
fn knights_tour(x: i32, y: i32) -> Option<Board> {
let mut board = Board::new();
let mut p = Point {x: x, y: y};
let mut step = 1;
board.field[p.x as usize][p.y as usize] = step;
step += 1;
while step <= (SIZE * SIZE) as i32 {
// choose next square by Warnsdorf's rule
let mut candidates = vec![];
for dir in MOVES.iter() {
let adj = p.mov(dir);
if board.available(adj) {
let degree = board.count_degree(adj);
candidates.push((degree, adj));
}
}
match candidates.iter().min() {
Some(&(_, adj)) => // move to next square
p = adj,
None => // can't move
return None
};
board.field[p.x as usize][p.y as usize] = step;
step += 1;
}
return Some(board);
}
fn main() {
let (x, y) = (3, 1);
println!("Board size: {}", SIZE);
println!("Starting position: ({}, {})", x, y);
match knights_tour(x, y) {
Some(b) =>
print!("{}", b),
None =>
println!("Fail!")
}
}
</lang>
- Output:
Board size: 8 Starting position: (3, 1) 23 20 3 32 25 10 5 8 2 33 24 21 4 7 26 11 19 22 51 34 31 28 9 6 50 1 40 29 54 35 12 27 41 18 55 52 61 30 57 36 46 49 44 39 56 53 62 13 17 42 47 60 15 64 37 58 48 45 16 43 38 59 14 63
Scala
<lang Scala> val b=Seq.tabulate(8,8,8,8)((x,y,z,t)=>(1L<<(x*8+y),1L<<(z*8+t),f"${97+z}%c${49+t}%c",(x-z)*(x-z)+(y-t)*(y-t)==5)).flatten.flatten.flatten.filter(_._4).groupBy(_._1) def f(p:Long,s:Long,v:Any){if(-1L!=s)b(p).foreach(x=>if((s&x._2)==0)f(x._2,s|x._2,v+x._3))else println(v)} f(1,1,"a1") </lang>
a1b3a5b7c5a4b2c4a3b1c3a2b4a6b8c6a7b5c7a8b6c8d6e4d2f1e3c2d4e2c1d3e1g2f4d5e7g8h6f5h4g6h8f7d8e6f8d7e5g4h2f3g1h3g5h7f6e8g7h5g3h1f2d1
Scheme
<lang scheme>
- /usr/bin/petite
- encoding
- utf-8
- Author
- Panda
- panbaoxiang@hotmail.com
- Created Time
- Thu 29 Jan 2015 10:18:49 AM CST
- Description
- size of the chessboard
(define X 8) (define Y 8)
- position is an integer that could be decoded into the x coordinate and y coordinate
(define(decode position)
(cons (div position Y) (remainder position Y)))
;;record the paths and number of territories you have conquered
(define dictionary '())
(define counter 1)
;;define the forbiddend territories(conquered and cul-de-sac)
(define forbiddened '())
;;renew when havn't conquered the world.
(define (renew position)
(define possible
(let ((rules (list (+ (* 2 Y) 1 position)
(+ (* 2 Y) -1 position)
(+ (* -2 Y) 1 position)
(+ (* -2 Y) -1 position)
(+ Y 2 position)
(+ Y -2 position)
(- position Y 2)
(- position Y -2))))
(filter (lambda(x) (not (or (member x forbiddened) (< x 0) (>= x (* X Y))))) rules)))
(if (null? possible)
(begin (set! forbiddened (cons (car dictionary) forbiddened))
(set! dictionary (cdr dictionary))
(set! counter (- counter 1))
(car dictionary))
(begin (set! dictionary (cons (car possible) dictionary))
(set! forbiddened dictionary)
(set! counter (+ counter 1))
(car possible))))
- go to search
(define (go position)
(if (= counter (* X Y)) (begin (set! result (reverse dictionary)) (display (map (lambda(x) (decode x)) result))) (go (renew position))))
</lang>
- Output:
(go 35) ((6 . 4) (4 . 5) (6 . 6) (4 . 7) (7 . 0) (5 . 1) (7 . 2) (5 . 3) (7 . 4) (5 . 5) (7 . 6) (5 . 7) (4 . 0) (6 . 1) (4 . 2) (6 . 3) (4 . 4) (6 . 5) (4 . 6) (6 . 7) (5 . 0) (7 . 1) (5 . 2) (7 . 3) (5 . 4) (7 . 5) (5 . 6) (7 . 7) (6 . 0) (4 . 1) (6 . 2) (4 . 3) (2 . 4) (0 . 5) (2 . 6) (0 . 7) (3 . 0) (1 . 1) (3 . 2) (1 . 3) (3 . 4) (1 . 5) (3 . 6) (1 . 7) (0 . 0) (2 . 1) (0 . 2) (2 . 3) (0 . 4) (2 . 5) (0 . 6) (2 . 7) (1 . 0) (3 . 1) (1 . 2) (3 . 3) (1 . 4) (3 . 5) (1 . 6) (3 . 7) (2 . 0) (0 . 1) (2 . 2))
SequenceL
Knights tour using Warnsdorffs rule (No Backtracking) <lang sequenceL> import <Utilities/Sequence.sl>; import <Utilities/Conversion.sl>;
main(args(2)) := let N := stringToInt(args[1]) when size(args) > 0 else 8; M := stringToInt(args[2]) when size(args) > 1 else N; startX := stringToInt(args[3]) when size(args) > 2 else 1; startY := stringToInt(args[4]) when size(args) > 3 else 1; board[i,j] := 0 foreach i within 1 ... N, j within 1 ... M; spacing := size(toString(N*M)) + 1; in join(printRow( tour(setBoard(board, startX, startX, 1), [startX,startY], 2), spacing));
potentialMoves := [[2,1], [2,-1], [1,2], [1,-2], [-1,2], [-1,-2], [-2,1], [-2,-1]];
printRow(row(1), spacing) := join(printSquare(row, spacing)) ++ "\n";
printSquare(val, spacing) := let str := toString(val); in duplicate(' ', spacing - size(str)) ++ str;
tour(board(2), current(1), move) := let validMoves := validMove(board, current + potentialMoves); numMoves[i] := size(validMove(board, validMoves[i] + potentialMoves)); chosenMove := minPosition(numMoves); in board when move > size(board) * size(board[1]) else [] when size(validMoves) = 0 else [] when move < size(board) * size(board[1]) and numMoves[chosenMove] = 0 else tour(setBoard(board, validMoves[chosenMove][1], validMoves[chosenMove][2], move), validMoves[chosenMove], move + 1);
validMove(board(2), position(1)) := (position when board[position[1], position[2]] = 0) when position[1] >= 1 and position[1] <= size(board) and position[2] >= 1 and position[2] <= size(board);
minPosition(x(1)) := minPositionHelper(x, 2, 1, x[1]); minPositionHelper(x(1), i, minPos, minVal) := minPos when i > size(x) else minPositionHelper(x, i + 1, minPos, minVal) when x[i] > minVal else minPositionHelper(x, i + 1, i, x[i]);
setBoard(board(2), x, y, value)[i,j] := value when x = i and y = j else board[i,j] foreach i within 1 ... size(board), j within 1 ... size(board[1]); </lang>
- Output:
8 X 8 board:
1 16 31 40 3 18 21 56 30 39 2 17 42 55 4 19 15 32 41 46 53 20 57 22 38 29 48 43 58 45 54 5 33 14 37 52 47 60 23 62 28 49 34 59 44 63 6 9 13 36 51 26 11 8 61 24 50 27 12 35 64 25 10 7
20 X 20 board:
1 40 81 90 3 42 77 94 5 44 73 102 7 46 69 62 9 48 51 60 82 89 2 41 92 95 4 43 76 101 6 45 72 103 8 47 68 61 10 49 39 80 91 96 153 78 93 100 129 74 109 104 123 70 111 120 63 50 59 52 88 83 154 79 98 159 152 75 108 105 128 71 110 121 124 67 112 119 64 11 155 38 97 160 157 200 99 162 151 130 107 122 127 132 141 118 125 66 53 58 84 87 156 199 176 161 158 201 106 163 150 131 142 145 126 133 140 113 12 65 37 182 85 178 207 198 175 164 173 216 143 166 149 222 139 146 117 134 57 54 86 179 206 197 204 177 208 217 202 165 172 221 144 167 148 223 138 55 114 13 183 36 181 212 209 218 203 174 215 220 227 170 281 224 303 168 147 116 135 56 180 211 196 205 230 213 238 219 228 171 280 225 302 169 282 343 304 137 14 115 35 184 231 210 237 246 229 214 279 226 301 298 283 342 367 308 347 344 305 136 232 195 236 245 234 239 278 247 300 297 284 359 366 309 348 345 368 307 350 15 185 34 233 240 261 248 287 296 285 358 299 310 341 378 365 384 349 346 369 306 194 241 250 235 244 277 260 313 294 311 360 373 364 383 354 379 370 385 16 351 33 186 243 262 249 288 295 286 361 316 357 340 377 372 395 386 353 380 333 388 242 193 254 251 276 259 314 293 312 321 374 363 398 355 382 371 394 387 352 17 187 32 263 258 267 252 289 322 315 362 317 356 339 376 399 396 381 334 389 332 192 255 190 253 264 275 268 271 292 323 320 375 326 397 338 335 390 393 18 21 31 188 257 266 29 270 273 290 27 318 327 324 25 336 329 400 23 20 331 392 256 191 30 189 274 265 28 269 272 291 26 319 328 325 24 337 330 391 22 19
Sidef
<lang ruby>var board = [] var I = 8 var J = 8 var F = (I*J > 99 ? '%3d' : '%2d')
var (i, j) = (I.irand, J.irand)
func from_algebraic(square) {
if (var match = square.match(/^([a-z])([0-9])\z/)) {
return(I - Num(match[1]), match[0].ord - 'a'.ord)
}
die "Invalid block square: #{square}"
}
func possible_moves(i,j) {
gather {
for ni,nj in [
[i-2,j-1], [i-2,j+1], [i-1,j-2], [i-1,j+2],
[i+1,j-2], [i+1,j+2], [i+2,j-1], [i+2,j+1],
] {
if ((ni ~~ ^I) && (nj ~~ ^J) && !board[ni][nj]) {
take([ni, nj])
}
}
}
}
func to_algebraic(i,j) {
('a'.ord + j).chr + Str(I - i)
}
if (ARGV[0]) {
(i, j) = from_algebraic(ARGV[0])
}
var moves = [] for move in (1 .. I*J) {
moves << to_algebraic(i, j)
board[i][j] = move
var min = [9]
for target in possible_moves(i, j) {
var (ni, nj) = target...
var nxt = possible_moves(ni, nj).len
if (nxt < min[0]) {
min = [nxt, ni, nj]
}
}
(i, j) = min[1,2]
}
say (moves/4 -> map { .join(', ') }.join("\n") + "\n")
for i in ^I {
for j in ^J {
(i%2 == j%2) && print "\e[7m"
F.printf(board[i][j])
print "\e[0m"
}
print "\n"
}</lang>
Tcl
<lang tcl>package require Tcl 8.6; # For object support, which makes coding simpler
oo::class create KnightsTour {
variable width height visited
constructor {{w 8} {h 8}} {
set width $w set height $h set visited {}
}
method ValidMoves {square} {
lassign $square c r set moves {} foreach {dx dy} {-1 -2 -2 -1 -2 1 -1 2 1 2 2 1 2 -1 1 -2} { set col [expr {($c % $width) + $dx}] set row [expr {($r % $height) + $dy}] if {$row >= 0 && $row < $height && $col >=0 && $col < $width} { lappend moves [list $col $row] } } return $moves
}
method CheckSquare {square} {
set moves 0 foreach site [my ValidMoves $square] { if {$site ni $visited} { incr moves } } return $moves
}
method Next {square} {
set minimum 9 set nextSquare {-1 -1} foreach site [my ValidMoves $square] { if {$site ni $visited} { set count [my CheckSquare $site] if {$count < $minimum} { set minimum $count set nextSquare $site } elseif {$count == $minimum} { set nextSquare [my Edgemost $nextSquare $site] } } } return $nextSquare
}
method Edgemost {a b} {
lassign $a ca ra lassign $b cb rb # Calculate distances to edge set da [expr {min($ca, $width - 1 - $ca, $ra, $height - 1 - $ra)}] set db [expr {min($cb, $width - 1 - $cb, $rb, $height - 1 - $rb)}] if {$da < $db} {return $a} else {return $b}
}
method FormatSquare {square} {
lassign $square c r format %c%d [expr {97 + $c}] [expr {1 + $r}]
}
method constructFrom {initial} {
while 1 { set visited [list $initial] set square $initial while 1 { set square [my Next $square] if {$square eq {-1 -1}} { break } lappend visited $square } if {[llength $visited] == $height*$width} { return } puts stderr "rejecting path of length [llength $visited]..." }
}
method constructRandom {} {
my constructFrom [list \ [expr {int(rand()*$width)}] [expr {int(rand()*$height)}]]
}
method print {} {
set s " " foreach square $visited { puts -nonewline "$s[my FormatSquare $square]" if {[incr i]%12} { set s " -> " } else { set s "\n -> " } } puts ""
}
method isClosed {} {
set a [lindex $visited 0] set b [lindex $visited end] expr {$a in [my ValidMoves $b]}
}
}</lang> Demonstrating: <lang tcl>set kt [KnightsTour new] $kt constructRandom $kt print if {[$kt isClosed]} {
puts "This is a closed tour"
} else {
puts "This is an open tour"
}</lang> Sample output:
e6 -> f8 -> h7 -> g5 -> h3 -> g1 -> e2 -> c1 -> a2 -> b4 -> a6 -> b8 -> d7 -> b6 -> a8 -> c7 -> e8 -> g7 -> h5 -> g3 -> h1 -> f2 -> d1 -> b2 -> a4 -> c3 -> b1 -> a3 -> b5 -> a7 -> c8 -> e7 -> g8 -> h6 -> f7 -> h8 -> g6 -> h4 -> g2 -> f4 -> d5 -> f6 -> g4 -> h2 -> f1 -> e3 -> f5 -> d6 -> e4 -> d2 -> c4 -> a5 -> b7 -> d8 -> c6 -> e5 -> f3 -> e1 -> d3 -> c5 -> b3 -> a1 -> c2 -> d4 This is a closed tour
The above code supports other sizes of boards and starting from nominated locations: <lang tcl>set kt [KnightsTour new 7 7] $kt constructFrom {0 0} $kt print if {[$kt isClosed]} {
puts "This is a closed tour"
} else {
puts "This is an open tour"
}</lang> Which could produce this output:
a1 -> c2 -> e1 -> g2 -> f4 -> g6 -> e7 -> f5 -> g7 -> e6 -> g5 -> f7 -> d6 -> b7 -> a5 -> b3 -> c1 -> a2 -> b4 -> a6 -> c7 -> b5 -> a7 -> c6 -> d4 -> e2 -> g1 -> f3 -> d2 -> f1 -> g3 -> e4 -> f2 -> g4 -> f6 -> d7 -> e5 -> d3 -> c5 -> a4 -> b2 -> d1 -> e3 -> d5 -> b6 -> c4 -> a3 -> b1 -> c3 This is an open tour
XPL0
<lang XPL0>int Board(8+2+2, 8+2+2); \board array with borders int LegalX, LegalY; \arrays of legal moves def IntSize=4; \number of bytes in an integer (4 or 2) include c:\cxpl\codes; \intrinsic 'code' declarations
func Try(I, X, Y); \Make a tentative move from X,Y
int I, X, Y;
int K, U, V;
[for K:= 0 to 8-1 do \for all possible moves...
[U:= X + LegalX(K); \U and V are next square
V:= Y + LegalY(K);
if Board(U,V) = 0 then \if square has not been visited then
[Board(U,V):= I; \ mark square with sequence number
if I = 8*8 then return true;
if Try(I+1, U, V) then return true \led to solution?
else Board(U,V):= 0; \no, undo tenative move
];
];
return false; ]; \Try
int I, J;
[LegalX:= [2, 1, -1, -2, -2, -1, 1, 2];
LegalY:= [1, 2, 2, 1, -1, -2, -2, -1];
for J:= 0 to 8+2+2-1 do \set up surrounding border for speed
for I:= 0 to 8+2+2-1 do
Board(I,J):= 1;
for J:= 0 to 8+2+2-1 do \reposition Board(0,0) to Board(2,2)
Board(J):= Board(J) + 2*IntSize;
Board:= Board + 2*IntSize; for J:= 0 to 8-1 do \empty board
for I:= 0 to 8-1 do
Board(I,J):= 0;
Text(0, "Starting square (1-8,1-8): "); I:= IntIn(0)-1; J:= IntIn(0)-1; Board(I,J):= 1; \starting location is 0,0
if Try(2, I, J) then \try to find second square
[for J:= 0 to 8-1 do \draw board with knight's move sequence
[for I:= 0 to 8-1 do
[if Board(I,J) < 10 then ChOut(0, ^ );
IntOut(0, Board(I,J));
ChOut(0, ^ );
];
CrLf(0);
];
]
else Text(0, "No Solution.^M^J"); ]</lang>
Example output:
Starting square (1-8,1-8): 1 1 1 38 59 36 43 48 57 52 60 35 2 49 58 51 44 47 39 32 37 42 3 46 53 56 34 61 40 27 50 55 4 45 31 10 33 62 41 26 23 54 18 63 28 11 24 21 14 5 9 30 19 16 7 12 25 22 64 17 8 29 20 15 6 13
XSLT
This solution is for XSLT 3.0 Working Draft 10 (July 2012). This solution, originally reported on this blog post, will be updated or removed when the final version of XSLT 3.0 is released.
First we build a generic package for solving any kind of tour over the chess board. Here it is… <lang> <xsl:package xsl:version="3.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:fn="http://www.w3.org/2005/xpath-functions" xmlns:tour="http://www.seanbdurkin.id.au/tour" name="tour:tours">
<xsl:stylesheet>
<xsl:function name="tour:manufacture-square"
as="element(square)" visibility="public">
<xsl:param name="rank" as="xs:integer" />
<xsl:param name="file" as="xs:integer" />
<square file="$file" rank="$rank" />
</xsl:function>
<xsl:function name="tour:on-board" as="xs:boolean" visibility="public">
<xsl:param name="rank" as="xs:integer" />
<xsl:param name="file" as="xs:integer" />
<xsl:copy-of select="($rank ge 1) and ($rank le 8) and
($file ge 1) and ($file le 8)" />
</xsl:function>
<xsl:function name="tour:solve-tour" as="item()*" visibility="public">
<xsl:param name="state" as="item()+" />
<xsl:variable name="compute-possible-moves"
select="$state[. instance of function(*)]"
as="function(element(square)) as element(square)*">
<xsl:variable name="way-points" select="$state/self::square" />
<xsl:choose>
<xsl:when test="count($way-points) eq 64">
<xsl:sequence ="$state" />
</xsl:when>
<xsl:otherwise>
<xsl:sequence select="
let $try-move := function( $state as item()*, $move as item()) as item()*)
{
if $state/self::square[@file=$move/@file]
[@rank=$move/@rank]
then $state
else tour:solve-tour( ( $state, $move) )
},
$possible-moves := $compute-possible-moves( $way-points[last()])
return if empty( $possible-moves) then $state
else fn:fold-left( $try-move, $state, $possible-moves)" />
</xsl:otherwise>
</xsl:choose>
</xsl:variable></xsl:function>
</xsl:stylesheet>
<xsl:expose component="function"
names="tour:manufacture-square tour:on-board tour:solve-tour" visibility="public" />
</xsl:package> </lang>
And now for the style-sheet to solve the Knight’s tour…
<lang> <xsl:stylesheet version="3.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:fn="http://www.w3.org/2005/xpath-functions" xmlns:tour="http://www.seanbdurkin.id.au/tour" exclude-result-prefixes="xsl fn xs tour">
<xsl:use-package name="tour:tours" /> <xsl:output indent="yes" encoding="UTF-8" omit-xml-declaration="yes" /> <xsl:mode on-no-match="shallow-copy" streamable="yes"/>
<xsl:template match="knight[square]">
<xsl:variable name="error">
<error>Failed to find solution to Knight's Tour.</error>
</xsl:variable>
<xsl:copy>
<xsl:copy-of select="
let $final-state := tour:solve-tour((
function( $piece-position as element(square)) as element(square)*
{ (: This function defines a knight's move. :)
let $r0 := number( $piece-position/@rank),
let $f0 := number( $piece-position/@file),
for $r in -2..2, $f in -2..2 return
if (abs($r) + abs($f) eq 3) and
tour:on-board($r+$r0, $f+$f0) then
tour:manufacture-square($r+$r0, $f+$f0)
else ()
}
, current()/square)),
$solution := $final-state/self::square
return if count($solution) eq 64 then $solution
else $error/*" />
</xsl:copy>
</xsl:template>
</xsl:stylesheet>
</lang>
So an input like this…
<lang>
<knight> <square file="1" rank="1" /> </knight>
</lang>
…should be transformed in something like this…
<lang>
<knight> <square file="1" rank="1" /> <square file="2" rank="3" /> <square file="1" rank="5" /> ... etc for 64 squares. </knight>
</lang>
zkl
<lang zkl> // Use Warnsdorff's rule to perform a knights tour of a 8x8 board in
// linear time. // See Pohl, Ira (July 1967), // "A method for finding Hamilton paths and Knight's tours" // http://portal.acm.org/citation.cfm?id=363463 // Uses back tracking as a tie breaker (for the few cases in a 8x8 tour)
class Board{
var[const]deltas=(dx,dy); T(-2,2); T(-1,1); _.extend( (dx,dy); T(-1,1); T(-2,2); _); fcn init{ var board=L(); (0).pump(64,board.append.fpM("1-",Void)); // fill board with Void } fcn idx(x,y) { x*8+y } fcn isMoveOK(x,y){ (0<=x<8) and (0<=y<8) and Void==board[idx(x,y)] } fcn gyrate(x,y,f){ // walk all legal moves from (a,b) deltas.pump(List,'wrap([(dx,dy)]){ x+=dx; y+=dy; if(isMoveOK(x,y)) f(x,y); else Void.Skip }); } fcn count(x,y){ n:=Ref(0); gyrate(x,y,n.inc); n.value } fcn moves(x,y){ gyrate(x,y,fcn(x,y){ T(x,y,count(x,y)) })} fcn knightsTour(x=0,y=0,n=1){ // using Warnsdorff's rule board[idx(x,y)]=n; while(m:=moves(x,y)){ min:=m.reduce('wrap(pc,[(_,_,c)]){ (pc<c) and pc or c },9);
m=m.filter('wrap([(_,_,c)]){ c==min }); // moves with same min moves if(m.len()>1){ // tie breaker time, may need to backtrack bs:=board.copy(); if (64==m.pump(Void,'wrap([(a,b)]){ board[idx(a,b)]=n; n2:=knightsTour(a,b,n+1); if (n2==64) return(Void.Stop,n2); // found a solution board=bs.copy(); })) return(64); return(0);
}
else{ x,y=m[0]; n+=1; board[idx(x,y)]=n; }
} //while
return(n);
}
fcn toString{ board.pump(String,T(Void.Read,7),
fcn(ns){ vm.arglist.apply("%2s".fmt).concat(",")+"\n" });
}
}</lang>
<lang zkl>b:=Board(); b.knightsTour(3,3); b.println();</lang>
- Output:
3,34, 5,54,19,36,29,50 6,21, 2,35,56,49,18,37 33, 4,55,20,53,30,51,28 22, 7,32, 1,48,57,38,17 11,44,23,62,31,52,27,58 8,63,10,45,60,47,16,39 43,12,61,24,41,14,59,26 64, 9,42,13,46,25,40,15
Check that a solution for all squares is found: <lang zkl>[[(x,y); [0..7]; [0..7];
{ b:=Board(); n:=b.knightsTour(x,y); if(n!=64) b.println(">>>",x,",",y) } ]];</lang>
- Output:
Nada
- Programming Tasks
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