Knight's tour

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Task
Knight's tour
You are encouraged to solve this task according to the task description, using any language you may know.

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

Cf.


Contents

[edit] Ada

First, we specify a naive implementation the package Knigths_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.

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;

Here is the implementation:

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;

Here is the main program:

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;

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.

 
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

Its implementation is as follows.

   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;

The modification for the main program is trivial:

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;

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

[edit] ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.win32
# 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
 
)
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

[edit] AutoHotkey

Library: GDIP
#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
}
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.

[edit] BBC BASIC

Knights tour bbc.gif
      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%))

[edit] 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);
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

[edit] 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.

#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;
}

[edit] C++

Works with: C++11

Uses Warnsdorff's rule and (iterative) backtracking if that fails.

#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;
}

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

[edit] C#

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;
}
}
}

[edit] 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.

 
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
 

output

 
> 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
 

[edit] D

[edit] Fast Version

Translation of: C++
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();
}
}
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

[edit] Shorter Version

Translation of: Haskell
import std.stdio, std.math, std.algorithm, std.range;
 
alias Sq = Tuple!(int,"x", int,"y"); /// Square.
 
const(Sq[]) knightTour(in Sq[] board, in Sq[] moves) /*pure nothrow*/ {
enum findMoves = (in Sq sq) /*nothrow*/ =>
cartesianProduct([1, -1, 2, -2], [1, -1, 2, -2])
.filter!(ij => ij[0].abs != ij[1].abs)
.map!(ij => Sq(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;
//const newSq = newMoves.reduce!(min!(m=>findMoves(m).walkLength));
immutable newSq = newMoves
.map!(s => tuple(findMoves(s).walkLength, s))
.reduce!min[1];
return board.knightTour(moves ~ newSq);
}
 
void main(in string[] args) {
enum toSq = (in string xy) => Sq(xy[0] - '`', xy[1] - '0');
immutable toAlg = (in Sq 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!Sq.array;
writefln("%(%-(%s -> %)\n%)",
board.knightTour([sq]).map!toAlg.chunks(8));
}
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

[edit] Erlang

Again I use backtracking. It seemed easier this time.

 
-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].
 
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

[edit] 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
}
}

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

[edit] Haskell

 
import System (getArgs)
import Data.Char (ord, chr)
import Data.List (minimumBy, (\\), intercalate, sort)
import Data.Ord (comparing)
 
type Square = (Int, Int)
 
board :: [Square]
board = [ (x,y) | x <- [1..8], y <- [1..8] ]
 
knightMoves :: Square -> [Square]
knightMoves (x,y) = filter (flip 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
| candMoves == [] = reverse moves
| otherwise = knightTour $ newSquare : moves
where newSquare = minimumBy (comparing (length . findMoves)) candMoves
candMoves = findMoves $ head moves
findMoves sq = knightMoves sq \\ moves
 
main :: IO ()
main = do
sq <- fmap (toSq . head) getArgs
printTour $ map toAlg $ knightTour [sq]
where toAlg (x,y) = [chr (x + 96), chr (y + 48)]
toSq [x,y] = ((ord x) - 96, (ord y) - 48)
printTour [] = return ()
printTour tour = do
putStrLn $ intercalate " -> " $ take 8 tour
printTour $ drop 8 tour
 

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

[edit] 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.

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

The following can be used when debugging to validate the board structure and to image the available moves on the board.

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


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  h
Two 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

[edit] J

Solution:
The Knight's tour essay on the Jwiki shows a couple of solutions including one using Warnsdorffs algorithm.

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
)

Example Use:

   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...

[edit] Java

Works with: Java version 7
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();
}
}
}
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 

[edit] Kotlin

Translation of: Haskell
import java.util.ArrayList
 
class Square(val x : Int, val y : Int) {
fun equals(s : Square) : Boolean = s.x == x && s.y == y
}
 
class Pair<T>(val a : T, val b : T)
 
val board = Array<Square>(8 * 8, {Square(it / 8 + 1, it % 8 + 1)})
val axisMoves = array(1, 2, -1, -2)
 
fun allPairs<T>(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.a) != Math.abs(it.b)}
fun onBoard(s : Square) = board.any {it equals s}
return moves map {Square(s.x + it.a, s.y + it.b)} filter {onBoard(it)}
}
 
fun knightTour(moves : List<Square>) : List<Square> {
fun findMoves(s : Square) = knightMoves(s) filterNot {m -> moves any {it equals m}}
val newSquare = findMoves(moves.last()) minBy {findMoves(it).size}
return if (newSquare == null) moves else knightTour(moves + newSquare)
}
 
fun knightTourFrom(start : Square) = knightTour(array(start).toList())
 
fun main(args : Array<String>) {
var col = 0
for (move in knightTourFrom(Square(1, 1))) {
System.out.print("${move.x},${move.y}")
System.out.print(if (col == 7) "\n" else " ")
col = (col + 1) % 8
}
}
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

[edit] Locomotive Basic

Influenced by the Python version, although computed tours are different.

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

Knights tour Locomotive Basic.png

[edit] 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

[edit] Mathematica

Solution

 
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]]
 

Usage

 
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"}
 

Analysis

vertexLabels replaces the default vertex (i.e. square) names of the chessboard with the standard algebraic names "a1", "a2",...,"h8".

 
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"}
 
 

knightsGraph creates a graph of the solution space.

 
knightsGraph = KnightTourGraph[i, i, VertexLabels -> vertexLabels, ImagePadding -> 15];
 

KnightsTour-3.png

Find a Hamiltonian cycle (a path that visits each square exactly one time.)

 
hamiltonianCycle = ((FindHamiltonianCycle[knightsGraph] /. UndirectedEdge -> DirectedEdge) /. labels)[[1]];
 

Find the end square:

 
end = Cases[hamiltonianCycle, (x_ \[DirectedEdge] start) :> x][[1]];
 

Find shortest path from the start square to the end square.

 
FindShortestPath[g, start, end]]
 


[edit] 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.

 
/*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;
 

Produces:

 
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
 

and

 
/*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;
 

Produces:

 
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
 

[edit] Perl

Knight's tour using Warnsdorffs algorithm

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'));
}

Sample output (start square c3):

Perl knights tour.png

[edit] Perl 6

Translation of: Perl
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');
}

(Output identical to Perl's above.)

[edit] 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) )

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)

[edit] PostScript

You probably shouldn't send this to a printer. Solution using Warnsdorffs algorithm.

%!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

[edit] Prolog

Works with: SWI-Prolog

Knights tour using Warnsdorffs algorithm

% 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).
 

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 .

[edit] Alternative version

Works with: GNU 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.
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.

[edit] Python

Knights tour using Warnsdorffs algorithm

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))
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.

[edit] 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.

#!/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)

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

[edit] 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)))
 
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

[edit] REXX

This REXX version is modeled after the XPL0 example.
The boardsize may be specified as well as the knight's starting position.

/*REXX program solves the knight's tour  problem for a  NxN  chessboard.*/
parse arg N sRank sFile . /*boardsize, starting position. */
if N=='' | N==',' then N=8 /*Boardsize specified? Default. */
if sRank=='' then sRank=N /*starting rank given? Default. */
if sFile=='' then sFile=1 /* " file " " */
!=left('', 9*(n<18)) /*used for indentation of board. */
NN=N**2; NxN='a ' N"x"N ' chessboard' /* [↓] r=Rank, f=File.*/
@.=; do r=1 for N; do f=1 for N; @.r.f=' '; end /*f*/; end /*r*/
/*[↑] zero the NxN chessboard.*/
Kr = '2 1 -1 -2 -2 -1 1 2' /*legal "rank" move for a knight.*/
Kf = '1 2 2 1 -1 -2 -2 -1' /* " "file" " " " " */
do i=1 for words(Kr) /*legal knight moves*/
Kr.i = word(Kr,i); Kf.i = word(Kf,i)
end /*i*/ /*for fast indexing.*/
@.sRank.sFile=1 /*the knight's starting position.*/
if \move(2,sRank,sFile) & ,
\(N==1) then say "No knight's tour solution for" NxN'.'
else say "A solution for the knight's tour on" NxN':'
_=substr(copies("┼───",N),2); say; say  ! translate('┌'_"┐", '┬', "┼")
do r=N for N by -1; if r\==N then say ! '├'_"┤"; L=@.
do f=1 for N; L=L'│'centre(@.r.f,3) /*preserve squareness.*/
end /*f*/ /*done with a rank on chessboard.*/
say ! L'│' /*show a rank of the chessboard.*/
end /*r*/ /*80 cols can view 19x19 chessbrd*/
say  ! translate('└'_"┘", '┴', "┼") /*show the last rank of the board*/
exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────MOVE subroutine─────────────────────*/
move: procedure expose @. Kr. Kf. N NN; parse arg #,rank,file; b=' '
do t=1 for 8; nr=rank+Kr.t; nf=file+Kf.t
if @.nr.nf==b then do; @.nr.nf=# /*Kn move.*/
if #==NN then return 1 /*last mv?*/
if move(#+1,nr,nf) then return 1 /* " " */
@.nr.nf=b /*undo the above move. */
end /*try different move. */
end /*t*/ /* [↑] all moves tried.*/
return 0 /*the tour not possible.*/

output

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 │64 │
          ├───┼───┼───┼───┼───┼───┼───┼───┤
          │50 │43 │30 │61 │14 │63 │28 │ 7 │
          └───┴───┴───┴───┴───┴───┴───┴───┘

[edit] Ruby

Knights tour using Warnsdorffs rule

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)

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:

[edit] 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]}
}
}

Demonstrating:

set kt [KnightsTour new]
$kt constructRandom
$kt print
if {[$kt isClosed]} {
puts "This is a closed tour"
} else {
puts "This is an open tour"
}

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:

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"
}

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

[edit] 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");
]

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 

[edit] XSLT 3.0

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…

 
<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">
<!-- Solves the tour for any specified piece. -->
<!-- Outputs either a full solution of 64 squares, of if fail,
a copy of the $state input. -->
<xsl:param name="state" as="item()+" />
<xsl:variable name="compute-possible-moves"
select="$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>
 

And now for the style-sheet to solve the Knight’s tour…

 
<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>
 
<!-- Add templates for other piece types if you want to solve
their tours too. Solve by calling tour:solve-tour() . -->
 
</xsl:stylesheet>
 

So an input like this…

 
<tt>
<knight>
<square file="1" rank="1" />
</knight>
</tt>
 

…should be transformed in something like this…

 
<tt>
<knight>
<square file="1" rank="1" />
<square file="2" rank="3" />
<square file="1" rank="5" />
... etc for 64 squares.
</knight>
</tt>
 

[edit] 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.fp(Void.Read,7),
fcn(ns){vm.arglist.apply("%2s".fmt).concat(",")+"\n"});
}
}
b:=Board(); b.knightsTour(3,3);
b.println();
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:

[[(x,y); [0..7]; [0..7]; 
{b:=Board(); n:=b.knightsTour(x,y); if(n!=64) b.println(">>>",x,",",y)} ]];
Output:
Nada
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