Solve a Holy Knight's tour: Difference between revisions
m (elided the superfluous "1" for the example.) |
m (→{{header|Phix}}: added syntax colouring, marked p2js compatible) |
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=={{header|Phix}}== |
=={{header|Phix}}== |
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Tweaked the knights tour algorithm (to use a limit variable rather than size*size). Bit slow on the second one... |
Tweaked the knights tour algorithm (to use a limit variable rather than size*size). Bit slow on the second one... (hence omitted under pwa/p2js) |
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<lang Phix> |
<!--<lang Phix>(phixonline)--> |
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<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span> |
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<span style="color: #004080;">sequence</span> <span style="color: #000000;">board</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">warnsdorffs</span> |
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integer size, limit, nchars |
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string fmt, blank |
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<span style="color: #004080;">integer</span> <span style="color: #000000;">size</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">limit</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">nchars</span> |
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<span style="color: #004080;">string</span> <span style="color: #000000;">fmt</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">blank</span> |
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constant ROW = 1, COL = 2 |
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constant moves = {{-1,-2},{-2,-1},{-2,1},{-1,2},{1,2},{2,1},{2,-1},{1,-2}} |
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<span style="color: #008080;">constant</span> <span style="color: #000000;">ROW</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">COL</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> |
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<span style="color: #000000;">moves</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},{-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},{-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},{-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">}}</span> |
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function onboard(integer row, integer col) |
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return row>=1 and row<=size and col>=nchars and col<=nchars*size |
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<span style="color: #008080;">function</span> <span style="color: #000000;">onboard</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">row</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">col</span><span style="color: #0000FF;">)</span> |
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end function |
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<span style="color: #008080;">return</span> <span style="color: #000000;">row</span><span style="color: #0000FF;">>=</span><span style="color: #000000;">1</span> <span style="color: #008080;">and</span> <span style="color: #000000;">row</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">size</span> <span style="color: #008080;">and</span> <span style="color: #000000;">col</span><span style="color: #0000FF;">>=</span><span style="color: #000000;">nchars</span> <span style="color: #008080;">and</span> <span style="color: #000000;">col</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">nchars</span><span style="color: #0000FF;">*</span><span style="color: #000000;">size</span> |
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span> |
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procedure init_warnsdorffs() |
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integer nrow,ncol |
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<span style="color: #008080;">procedure</span> <span style="color: #000000;">init_warnsdorffs</span><span style="color: #0000FF;">()</span> |
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for row=1 to size do |
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<span style="color: #008080;">for</span> <span style="color: #000000;">row</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">size</span> <span style="color: #008080;">do</span> |
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for col=nchars to nchars*size by nchars do |
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<span style="color: #008080;">for</span> <span style="color: #000000;">col</span><span style="color: #0000FF;">=</span><span style="color: #000000;">nchars</span> <span style="color: #008080;">to</span> <span style="color: #000000;">nchars</span><span style="color: #0000FF;">*</span><span style="color: #000000;">size</span> <span style="color: #008080;">by</span> <span style="color: #000000;">nchars</span> <span style="color: #008080;">do</span> |
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for move=1 to length(moves) do |
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<span style="color: #008080;">for</span> <span style="color: #000000;">move</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">moves</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span> |
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nrow = row+moves[move][ROW] |
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<span style="color: #004080;">integer</span> <span style="color: #000000;">nrow</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">row</span><span style="color: #0000FF;">+</span><span style="color: #000000;">moves</span><span style="color: #0000FF;">[</span><span style="color: #000000;">move</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ROW</span><span style="color: #0000FF;">],</span> |
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ncol = col+moves[move][COL]*nchars |
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<span style="color: #000000;">ncol</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">col</span><span style="color: #0000FF;">+</span><span style="color: #000000;">moves</span><span style="color: #0000FF;">[</span><span style="color: #000000;">move</span><span style="color: #0000FF;">][</span><span style="color: #000000;">COL</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">nchars</span> |
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if onboard(nrow,ncol) then |
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<span style="color: #008080;">if</span> <span style="color: #000000;">onboard</span><span style="color: #0000FF;">(</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> |
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-- (either of these would work) |
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<span style="color: #000000;">warnsdorffs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span> |
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warnsdorffs[row][col] += 1 |
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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end for |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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end for |
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<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span> |
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end for |
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end procedure |
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<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">(),</span> |
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<span style="color: #000000;">t1</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()+</span><span style="color: #000000;">1</span> |
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atom t0 = time() |
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<span style="color: #004080;">integer</span> <span style="color: #000000;">tries</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">backtracks</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span> |
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integer tries = 0, backtracks = 0 |
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<span style="color: #008080;">function</span> <span style="color: #000000;">solve</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">row</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">col</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> |
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atom t1 = time()+1 |
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<span style="color: #008080;">if</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()></span><span style="color: #000000;">t1</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()!=</span><span style="color: #004600;">JS</span> <span style="color: #008080;">then</span> |
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function solve(integer row, integer col, integer n) |
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<span style="color: #0000FF;">?{</span><span style="color: #000000;">row</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">col</span><span style="color: #0000FF;">/</span><span style="color: #000000;">nchars</span><span style="color: #0000FF;">),</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">tries</span><span style="color: #0000FF;">}</span> |
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integer nrow, ncol |
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<span style="color: #7060A8;">puts</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">board</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">)&</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">)</span> |
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if time()>t1 then |
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<span style="color: #000000;">t1</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()+</span><span style="color: #000000;">1</span> |
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?{row,floor(col/nchars),n,tries} |
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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puts(1,join(board,"\n")) |
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<span style="color: #000000;">tries</span><span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span> |
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t1 = time()+1 |
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<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">></span><span style="color: #000000;">limit</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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-- if wait_key()='!' then ?9/0 end if |
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<span style="color: #004080;">sequence</span> <span style="color: #000000;">wmoves</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span> |
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end if |
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<span style="color: #004080;">integer</span> <span style="color: #000000;">nrow</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">ncol</span> |
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tries+= 1 |
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<span style="color: #008080;">for</span> <span style="color: #000000;">move</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">moves</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span> |
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if n>limit then return 1 end if |
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<span style="color: #000000;">nrow</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">row</span><span style="color: #0000FF;">+</span><span style="color: #000000;">moves</span><span style="color: #0000FF;">[</span><span style="color: #000000;">move</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ROW</span><span style="color: #0000FF;">]</span> |
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sequence wmoves = {} |
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<span style="color: #000000;">ncol</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">col</span><span style="color: #0000FF;">+</span><span style="color: #000000;">moves</span><span style="color: #0000FF;">[</span><span style="color: #000000;">move</span><span style="color: #0000FF;">][</span><span style="color: #000000;">COL</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">nchars</span> |
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for move=1 to length(moves) do |
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<span style="color: #008080;">if</span> <span style="color: #000000;">onboard</span><span style="color: #0000FF;">(</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">)</span> |
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nrow = row+moves[move][ROW] |
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<span style="color: #008080;">and</span> <span style="color: #000000;">board</span><span style="color: #0000FF;">[</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">' '</span> <span style="color: #008080;">then</span> |
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ncol = col+moves[move][COL]*nchars |
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<span style="color: #000000;">wmoves</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">wmoves</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">warnsdorffs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">],</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">})</span> |
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if onboard(nrow,ncol) |
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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and board[nrow][ncol]=' ' then |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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wmoves = append(wmoves,{warnsdorffs[nrow][ncol],nrow,ncol}) |
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<span style="color: #000000;">wmoves</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sort</span><span style="color: #0000FF;">(</span><span style="color: #000000;">wmoves</span><span style="color: #0000FF;">)</span> |
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end if |
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<span style="color: #000080;font-style:italic;">-- avoid creating orphans</span> |
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end for |
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<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">wmoves</span><span style="color: #0000FF;">)<</span><span style="color: #000000;">2</span> <span style="color: #008080;">or</span> <span style="color: #000000;">wmoves</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]></span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> |
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wmoves = sort(wmoves) |
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<span style="color: #008080;">for</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">wmoves</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span> |
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-- avoid creating orphans |
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<span style="color: #0000FF;">{?,</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">wmoves</span><span style="color: #0000FF;">[</span><span style="color: #000000;">m</span><span style="color: #0000FF;">]</span> |
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if length(wmoves)<2 or wmoves[2][1]>1 then |
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<span style="color: #000000;">warnsdorffs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span> |
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for m=1 to length(wmoves) do |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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{?,nrow,ncol} = wmoves[m] |
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<span style="color: #008080;">for</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">wmoves</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span> |
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warnsdorffs[nrow][ncol] -= 1 |
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<span style="color: #0000FF;">{?,</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">wmoves</span><span style="color: #0000FF;">[</span><span style="color: #000000;">m</span><span style="color: #0000FF;">]</span> |
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end for |
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<span style="color: #004080;">integer</span> <span style="color: #000000;">scol</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ncol</span><span style="color: #0000FF;">-</span><span style="color: #000000;">nchars</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> |
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for m=1 to length(wmoves) do |
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<span style="color: #000000;">board</span><span style="color: #0000FF;">[</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">][</span><span style="color: #000000;">scol</span><span style="color: #0000FF;">..</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fmt</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> |
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{?,nrow,ncol} = wmoves[m] |
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<span style="color: #008080;">if</span> <span style="color: #000000;">solve</span><span style="color: #0000FF;">(</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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board[nrow][ncol-nchars+1..ncol] = sprintf(fmt,n) |
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<span style="color: #000000;">backtracks</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span> |
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if solve(nrow,ncol,n+1) then return 1 end if |
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<span style="color: #000000;">board</span><span style="color: #0000FF;">[</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">][</span><span style="color: #000000;">scol</span><span style="color: #0000FF;">..</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">blank</span> |
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backtracks += 1 |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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board[nrow][ncol-nchars+1..ncol] = blank |
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<span style="color: #008080;">for</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">wmoves</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span> |
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end for |
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<span style="color: #0000FF;">{?,</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">wmoves</span><span style="color: #0000FF;">[</span><span style="color: #000000;">m</span><span style="color: #0000FF;">]</span> |
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for m=1 to length(wmoves) do |
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<span style="color: #000000;">warnsdorffs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">nrow</span><span style="color: #0000FF;">][</span><span style="color: #000000;">ncol</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span> |
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{?,nrow,ncol} = wmoves[m] |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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warnsdorffs[nrow][ncol] += 1 |
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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end for |
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<span style="color: #008080;">return</span> <span style="color: #000000;">0</span> |
|||
end if |
|||
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span> |
|||
return 0 |
|||
end function |
|||
<span style="color: #008080;">procedure</span> <span style="color: #000000;">holyknight</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">)</span> |
|||
<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">split</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #008000;">'\n'</span><span style="color: #0000FF;">)</span> |
|||
procedure holyknight(sequence s) |
|||
<span style="color: #000000;">size</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">)</span> |
|||
integer y, ch, sx, sy |
|||
<span style="color: #000000;">nchars</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">" %d"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">size</span><span style="color: #0000FF;">*</span><span style="color: #000000;">size</span><span style="color: #0000FF;">))</span> |
|||
s = split(s,'\n') |
|||
<span style="color: #000000;">fmt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">" %%%dd"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">nchars</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> |
|||
size = length(s) |
|||
<span style="color: #000000;">blank</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">' '</span><span style="color: #0000FF;">,</span><span style="color: #000000;">nchars</span><span style="color: #0000FF;">)</span> |
|||
nchars = length(sprintf(" %d",size*size)) |
|||
<span style="color: #000000;">board</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">' '</span><span style="color: #0000FF;">,</span><span style="color: #000000;">size</span><span style="color: #0000FF;">*</span><span style="color: #000000;">nchars</span><span style="color: #0000FF;">),</span><span style="color: #000000;">size</span><span style="color: #0000FF;">)</span> |
|||
fmt = sprintf(" %%%dd",nchars-1) |
|||
<span style="color: #000000;">limit</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> |
|||
blank = repeat(' ',nchars) |
|||
<span style="color: #004080;">integer</span> <span style="color: #000000;">sx</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">sy</span> |
|||
board = repeat(repeat(' ',size*nchars),size) |
|||
<span style="color: #008080;">for</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">size</span> <span style="color: #008080;">do</span> |
|||
limit = 1 |
|||
<span style="color: #004080;">integer</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">nchars</span> |
|||
for x=1 to size do |
|||
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">size</span> <span style="color: #008080;">do</span> |
|||
y = nchars |
|||
<span style="color: #004080;">integer</span> <span style="color: #000000;">ch</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">j</span><span style="color: #0000FF;">></span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">[</span><span style="color: #000000;">x</span><span style="color: #0000FF;">])?</span><span style="color: #008000;">'-'</span><span style="color: #0000FF;">:</span><span style="color: #000000;">s</span><span style="color: #0000FF;">[</span><span style="color: #000000;">x</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">])</span> |
|||
for j=1 to size do |
|||
<span style="color: #008080;">if</span> <span style="color: #000000;">ch</span><span style="color: #0000FF;">=</span><span style="color: #008000;">' '</span> <span style="color: #008080;">then</span> |
|||
if j>length(s[x]) then |
|||
<span style="color: #000000;">ch</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">'-'</span> |
|||
ch = '-' |
|||
<span style="color: #008080;">elsif</span> <span style="color: #000000;">ch</span><span style="color: #0000FF;">=</span><span style="color: #008000;">'0'</span> <span style="color: #008080;">then</span> |
|||
else |
|||
<span style="color: #000000;">ch</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">' '</span> |
|||
ch = s[x][j] |
|||
<span style="color: #000000;">limit</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span> |
|||
end if |
|||
<span style="color: #008080;">elsif</span> <span style="color: #000000;">ch</span><span style="color: #0000FF;">=</span><span style="color: #008000;">'1'</span> <span style="color: #008080;">then</span> |
|||
if ch=' ' then |
|||
<span style="color: #000000;">sx</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x</span> |
|||
ch = '-' |
|||
<span style="color: #000000;">sy</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">y</span> |
|||
elsif ch='0' then |
|||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
|||
<span style="color: #000000;">board</span><span style="color: #0000FF;">[</span><span style="color: #000000;">x</span><span style="color: #0000FF;">][</span><span style="color: #000000;">y</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">ch</span> |
|||
limit += 1 |
|||
<span style="color: #000000;">y</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">nchars</span> |
|||
elsif ch='1' then |
|||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
|||
sx = x |
|||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
|||
sy = y |
|||
<span style="color: #000000;">warnsdorffs</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">size</span><span style="color: #0000FF;">*</span><span style="color: #000000;">nchars</span><span style="color: #0000FF;">),</span><span style="color: #000000;">size</span><span style="color: #0000FF;">)</span> |
|||
end if |
|||
<span style="color: #000000;">init_warnsdorffs</span><span style="color: #0000FF;">()</span> |
|||
board[x][y] = ch |
|||
<span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span> |
|||
y += nchars |
|||
<span style="color: #000000;">tries</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span> |
|||
end for |
|||
<span style="color: #000000;">backtracks</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span> |
|||
end for |
|||
<span style="color: #000000;">t1</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()+</span><span style="color: #000000;">1</span> |
|||
warnsdorffs = repeat(repeat(0,size*nchars),size) |
|||
<span style="color: #008080;">if</span> <span style="color: #000000;">solve</span><span style="color: #0000FF;">(</span><span style="color: #000000;">sx</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sy</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> |
|||
init_warnsdorffs() |
|||
<span style="color: #7060A8;">puts</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">board</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">))</span> |
|||
t0 = time() |
|||
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\nsolution found in %d tries, %d backtracks (%3.2fs)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">tries</span><span style="color: #0000FF;">,</span><span style="color: #000000;">backtracks</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">})</span> |
|||
tries = 0 |
|||
<span style="color: #008080;">else</span> |
|||
backtracks = 0 |
|||
<span style="color: #7060A8;">puts</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"no solutions found\n"</span><span style="color: #0000FF;">)</span> |
|||
t1 = time()+1 |
|||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
|||
if solve(sx,sy,2) then |
|||
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span> |
|||
puts(1,join(board,"\n")) |
|||
printf(1,"\nsolution found in %d tries, %d backtracks (%3.2fs)\n",{tries,backtracks,time()-t0}) |
|||
<span style="color: #008080;">constant</span> <span style="color: #000000;">board1</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""" |
|||
else |
|||
000 |
|||
puts(1,"no solutions found\n") |
|||
0 00 |
|||
0000000 |
|||
end procedure |
|||
000 0 0 |
|||
0 0 000 |
|||
constant board1 = """ |
|||
1000000 |
|||
000 |
|||
00 0 |
|||
000"""</span> |
|||
0000000 |
|||
<span style="color: #000000;">holyknight</span><span style="color: #0000FF;">(</span><span style="color: #000000;">board1</span><span style="color: #0000FF;">)</span> |
|||
0 0 000 |
|||
1000000 |
|||
<span style="color: #008080;">constant</span> <span style="color: #000000;">board2</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""" |
|||
00 0 |
|||
-----1-0----- |
|||
000""" |
|||
-----0-0----- |
|||
----00000---- |
|||
holyknight(board1) |
|||
-----000----- |
|||
--0--0-0--0-- |
|||
constant board2 = """ |
|||
00000---00000 |
|||
-----1-0----- |
|||
---- |
--00-----00-- |
||
00000---00000 |
|||
----- |
--0--0-0--0-- |
||
-- |
-----000----- |
||
----00000---- |
|||
-- |
-----0-0----- |
||
-----0-0-----"""</span> |
|||
00000---00000 |
|||
--0--0-0--0-- |
|||
<span style="color: #008080;">if</span> <span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()!=</span><span style="color: #004600;">JS</span> <span style="color: #008080;">then</span> |
|||
-----000----- |
|||
<span style="color: #000000;">holyknight</span><span style="color: #0000FF;">(</span><span style="color: #000000;">board2</span><span style="color: #0000FF;">)</span> |
|||
----00000---- |
|||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
|||
-----0-0----- |
|||
-----0-0-----""" |
|||
<span style="color: #0000FF;">{}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">wait_key</span><span style="color: #0000FF;">()</span> |
|||
<!--</lang>--> |
|||
holyknight(board2) |
|||
{} = wait_key()</lang> |
|||
{{out}} |
{{out}} |
||
<pre> |
<pre> |
Revision as of 17:08, 23 January 2022
You are encouraged to solve this task according to the task description, using any language you may know.
Chess coaches have been known to inflict a kind of torture on beginners by taking a chess board, placing pennies on some squares and requiring that a Knight's tour be constructed that avoids the squares with pennies.
This kind of knight's tour puzzle is similar to Hidato.
The present task is to produce a solution to such problems. At least demonstrate your program by solving the following:
- Example
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
Note that the zeros represent the available squares, not the pennies.
Extra credit is available for other interesting examples.
- Related tasks
- A* search algorithm
- Knight's tour
- N-queens problem
- Solve a Hidato puzzle
- Solve a Hopido puzzle
- Solve a Numbrix puzzle
- Solve the no connection puzzle
11l
<lang 11l>V moves = [
[-1, -2], [1, -2], [-1, 2], [1, 2], [-2, -1], [-2, 1], [2, -1], [2, 1]
]
F solve(&pz, sz, sx, sy, idx, cnt)
I idx > cnt R 1
L(i) 0 .< :moves.len V x = sx + :moves[i][0] V y = sy + :moves[i][1] I sz > x & x > -1 & sz > y & y > -1 & pz[x][y] == 0 pz[x][y] = idx I 1 == solve(&pz, sz, x, y, idx + 1, cnt) R 1 pz[x][y] = 0 R 0
F find_solution(pz, sz)
V p = [[-1] * sz] * sz V idx = 0 V x = 0 V y = 0 V cnt = 0 L(j) 0 .< sz L(i) 0 .< sz I pz[idx] == ‘x’ p[i][j] = 0 cnt++ E I pz[idx] == ‘s’ p[i][j] = 1 cnt++ x = i y = j idx++
I 1 == solve(&p, sz, x, y, 2, cnt) L(j) 0 .< sz L(i) 0 .< sz I p[i][j] != -1 print(‘ #02’.format(p[i][j]), end' ‘’) E print(‘ ’, end' ‘’) print() E print(‘Cannot solve this puzzle!’)
find_solution(‘.xxx.....x.xx....xxxxxxxxxx..x.xx.x..xxxsxxxxxx...xx.x.....xxx..’, 8) print() find_solution(‘.....s.x..........x.x.........xxxxx.........xxx.......x..x.x..x..xxxxx...xxxxx..xx.....xx..xxxxx...xxxxx..x..x.x..x.......xxx.........xxxxx.........x.x..........x.x.....’, 13)</lang>
- Output:
17 14 29 28 18 15 13 16 27 30 19 32 07 25 02 11 06 20 12 26 31 08 33 01 24 03 10 05 34 21 36 23 09 04 35 22 01 05 10 12 02 13 04 09 06 08 11 14 36 03 07 16 35 42 33 44 37 15 20 27 22 25 38 41 17 24 39 34 43 32 45 19 28 21 26 23 40 31 29 18 46 51 56 52 55 30 47 50 48 54 53 49
Ada
This solution uses the package Knights_Tour from Knight's Tour#Ada. The board is quadratic, the size of the board is read from the command line and the board itself is read from the standard input. For the board itself, Space and Minus indicate a no-go (i.e., a coin on the board), all other characters represent places the knight must visit. A '1' represents the start point. Ill-formatted input will crash the program.
<lang Ada>with Knights_Tour, Ada.Text_IO, Ada.Command_Line;
procedure Holy_Knight is
Size: Positive := Positive'Value(Ada.Command_Line.Argument(1)); package KT is new Knights_Tour(Size => Size); Board: KT.Tour := (others => (others => Natural'Last)); Start_X, Start_Y: KT.Index:= 1; -- default start place (1,1) S: String(KT.Index); I: Positive := KT.Index'First;
begin
-- read the board from standard input while not Ada.Text_IO.End_Of_File and I <= Size loop S := Ada.Text_IO.Get_Line; for J in KT.Index loop if S(J) = ' ' or S(J) = '-' then Board(I,J) := Natural'Last; elsif S(J) = '1' then Start_X := I; Start_Y := J; Board(I,J) := 1; else Board(I,J) := 0; end if; end loop; I := I + 1; end loop;
-- print the board Ada.Text_IO.Put_Line("Start Configuration (Length:" & Natural'Image(KT.Count_Moves(Board)) & "):"); KT.Tour_IO(Board, Width => 1); Ada.Text_IO.New_Line;
-- search for the tour and print it Ada.Text_IO.Put_Line("Tour:"); KT.Tour_IO(KT.Warnsdorff_Get_Tour(Start_X, Start_Y, Board));
end Holy_Knight;</lang>
- Output:
>holy_knight 8 < standard_problem.txt Start Configuration (Length: 36): --000--- --0-00-- -0000000 000--0-0 0-0--000 1000000- --00-0-- ---000-- Tour: - - 30 15 20 - - - - - 21 - 29 16 - - - 33 14 31 22 19 6 17 13 36 23 - - 28 - 8 34 - 32 - - 7 18 5 1 12 35 24 27 4 9 - - - 2 11 - 25 - - - - - 26 3 10 - -
Extra Credit
The Holy_Knight program can immediately be used to tackle "more interesting" problems, such as those from New Knight's Tour Puzzles and Graphs. Here is one sample solution:
>holy_knight 13 < problem10.txt Start Configuration (Length: 56): -----1-0----- -----0-0----- ----00000---- -----000----- --0--0-0--0-- 00000---00000 --00-----00-- 00000---00000 --0--0-0--0-- -----000----- ----00000---- -----0-0----- -----0-0----- Tour: - - - - - 1 - 27 - - - - - - - - - - 56 - 2 - - - - - - - - - 24 3 28 55 26 - - - - - - - - - 54 25 4 - - - - - - - 50 - - 23 - 29 - - 6 - - 51 20 47 22 53 - - - 5 30 9 32 7 - - 52 49 - - - - - 33 36 - - 19 48 21 46 17 - - - 37 10 31 8 35 - - 18 - - 45 - 11 - - 34 - - - - - - - 16 41 38 - - - - - - - - - 42 39 44 15 12 - - - - - - - - - 14 - 40 - - - - - - - - - - 43 - 13 - - - - -
ALGOL 68
Uses a modified version of the Knight's Tour#ALGOL 68 solution. <lang algol68># directions for moves # INT nne = 1, ne = 2, se = 3, sse = 4; INT ssw = 5, sw = 6, nw = 7, nnw = 8;
INT lowest move = nne; INT highest move = nnw;
- the vertical position changes of the moves #
[]INT offset v = ( -2, -1, 1, 2, 2, 1, -1, -2 );
- the horizontal position changes of the moves #
[]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, ne, se, sse, ssw, sw, nw or # # nnw # );
- get the size of the board - must be between 4 and 8 #
INT board size = 8;
- the board #
[ board size, board size ]SQUARE board;
- starting position #
INT start row := 1; INT start col := 1;
- the tour will be complete when we have made as many moves #
- as there are free squares in the initial board #
INT final move := 0;
- initialise the board setting the free squares from the supplied pttern #
- the pattern has the rows in revers order #
PROC initialise board = ( []STRING pattern )VOID:
BEGIN INT pattern row := UPB board; FOR row FROM 1 LWB board TO 1 UPB board DO FOR col FROM 2 LWB board TO 2 UPB board DO IF pattern[ pattern row ][ col ] = "-" THEN # can't use this square # board[ row, col ] := ( -1, -1 ) ELSE # available square # board[ row, col ] := ( 0, 0 ); final move +:= 1; IF pattern[ pattern row ][ col ] = "1" THEN # have the start position # start row := row; start col := col FI FI OD; pattern row -:= 1 OD END; # initialise board #
- statistics #
INT iterations := 0; INT backtracks := 0;
- prints the board #
PROC print tour = VOID: BEGIN
# format "number" into at least two characters # PROC n2 = ( INT number )STRING: IF number < 0 THEN " -" ELIF number < 10 AND number >= 0 THEN " " + whole( number, 0 ) ELSE whole( number, 0 ) FI; # n2 # 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( ( n2( row ) ) ); print( ( "|" ) );
FOR col FROM 2 LWB board TO 2 UPB board DO print( ( " " ) ); print( ( n2( move OF board[ row, col ] ) ) ) OD; print( ( newline ) ) OD
END; # print tour #
- 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 = BOOL: BEGIN
BOOL result := TRUE; INT move number := 1; INT row := start row; INT col := start col; INT direction := lowest move - 1; # the first move is to place the knight on the starting square # board[ row, col ] := ( move number, lowest move - 1 ); # attempt to find a sequence of moves that will reach each square once # WHILE move number < final move AND result DO IF direction < highest move THEN # try the next move from this position # direction +:= 1; INT new row = row + offset v[ direction ]; INT new col = col + offset h[ direction ]; IF new row <= 1 UPB board AND new row >= 1 LWB board AND new col <= 2 UPB board AND new col >= 2 LWB board THEN # the move is legal, check the new square is unused # IF move OF board[ new row, new col ] = 0 THEN # can move here # iterations +:= 1; row := new row; col := new col; move number +:= 1; board[ row, col ] := ( move number, direction ); direction := lowest move - 1 FI FI 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; row -:= offset v[ direction OF board[ curr row, curr col ] ]; col -:= offset h[ direction OF board[ curr row, curr col ] ]; # determine which direction to try next # direction := direction OF board[ curr row, curr col ]; # reset the square we just backtracked from # board[ curr row, curr col ] := ( 0, 0 ) FI FI OD; result
END; # find tour #
main:(
initialise board( ( "-000----" , "-0-00---" , "-0000000" , "000--0-0" , "0-0--000" , "1000000-" , "--00-0--" , "---000--" ) ); IF find tour THEN # found a solution # print tour ELSE # couldn't find a solution # print( ( "Solution not found", newline ) ) FI; print( ( iterations, " iterations, ", backtracks, " backtracks", newline ) )
)</lang>
- Output:
a b c d e f g h ________________________ 8| - 21 34 25 - - - - 7| - 24 - 20 7 - - - 6| - 35 22 33 26 11 6 9 5| 23 32 19 - - 8 - 12 4| 36 - 16 - - 27 10 5 3| 1 18 31 28 15 4 13 - 2| - - 2 17 - 29 - - 1| - - - 30 3 14 - - +578929 iterations, +578894 backtracks
Bracmat
This solution can handle different input formats: the widths of the first and the other columns are computed. The cell were to start from should have a unique value, but this value is not prescribed. Non-empty cells (such as the start cell) should contain a character that is different from '-', '.' or white space. The puzzle solver itself is only a few lines long. <lang bracmat>( ( Holy-Knight
= begin colWidth crumbs non-empty pairs path parseLine , display isolateStartCell minDistance numberElementsAndSort , parseBoard reverseList rightAlign solve strlen . "'non-empty' is a pattern that is used several times in bigger patterns." & ( non-empty = = %@ : ~( "." | "-" | " " | \t | \r | \n ) ) & ( reverseList = a L . :?L & whl'(!arg:%?a ?arg&!a !L:?L) & !L ) & (strlen=e.@(!arg:? [?e)&!e) & ( rightAlign = string width . !arg:(?width,?string) & !width+-1*strlen$!string:?width & whl ' ( !width+-1:~<0:?width & " " !string:?string ) & str$!string ) & ( minDistance = board pat1 pat2 minWidth pos1 pos2 pattern . !arg:(?board,(=?pat1),(=?pat2)) & -1:?minWidth & "Construct a pattern using a template. The pattern finds the smallest distance between any two columns in the input. Assumption: all columns have the same width and columns are separated by one or more spaces. The function can also be used to find the width of the first column by letting pat1 match a new line." & ' ( ? ( $pat1 [?pos1 (? " "|`) ()$pat2 [?pos2 ? & !pos2+-1*!pos1 : ( <!minWidth | ?&!minWidth:<0 ) : ?minWidth & ~ ) ) : (=?pattern) & "'pattern', by design, always fails. The interesting part is a side effect: the column width." & (@(!board:!pattern)|!minWidth) ) & ( numberElementsAndSort = a sum n . 0:?sum:?n & "An evaluated sum is always sorted. The terms are structured so the sorting order is by row and then by column (both part of 'a')." & whl ' ( !arg:%?a ?arg & 1+!n:?n & (!a,!n)+!sum:?sum ) & "return the sorted list (sum) and also the size of a field that can contain the highest number." & (!sum.strlen$!n+1) ) & ( parseLine = line row columnWidth width col , bins val A M Z cell validPat . !arg:(?line,?row,?width,?columnWidth,?bins) & 0:?col & "Find the cells and create a pair [row,col] for each. Put each pair in a bin. There are as many bins as there are different values in cells." & '(? ($!non-empty:?val) ?) : (=?validPat) & whl ' ( @(!line:?cell [!width ?line) & ( @(!cell:!validPat) & ( !bins:?A (!val.?M) ?Z & !A (!val.(!row.!col) !M) !Z | (!val.!row.!col) !bins ) : ?bins | ) & !columnWidth:?width & 1+!col:?col ) & !bins ) & ( parseBoard = board firstColumnWidth columnWidth,row bins line . !arg:?board & ( minDistance $ (str$(\r \n !arg),(=\n),!non-empty) , minDistance$(!arg,!non-empty,!non-empty) ) : (?firstColumnWidth,?columnWidth) & 0:?row & :?bins & whl ' ( @(!board:?line \n ?board) & parseLine $ (!line,!row,!firstColumnWidth,!columnWidth,!bins) : ?bins & (!bins:|1+!row:?row) ) & parseLine $ (!board,!row,!firstColumnWidth,!columnWidth,!bins) : ?bins ) & "Find the first bin with only one pair. Return this pair and the combined pairs in all remaining bins." & ( isolateStartCell = A begin Z valuedPairs pairs . !arg:?A (?.? [1:?begin) ?Z & !A !Z:?arg & :?pairs & whl ' ( !arg:(?.?valuedPairs) ?arg & !valuedPairs !pairs:?pairs ) & (!begin.!pairs) ) & ( display = board solution row col x y n colWidth . !arg:(?board,?solution,?colWidth) & out$!board & 0:?row & -1:?col & whl ' ( !solution:((?y.?x),?n)+?solution & whl ' ( !row:<!y & 1+!row:?row & -1:?col & put$\n ) & whl ' ( 1+!col:?col:<!x & put$(rightAlign$(!colWidth,)) ) & put$(rightAlign$(!colWidth,!n)) ) & put$\n ) & ( solve = A Z x y crumbs pairs X Y solution . !arg:((?y.?x),?crumbs,?pairs) & ( !pairs:&(!y.!x) !crumbs | !pairs : ?A ( (?Y.?X) ?Z & (!x+-1*!X)*(!y+-1*!Y) : (2|-2) & solve $ ( (!Y.!X) , (!y.!x) !crumbs , !A !Z ) : ?solution ) & !solution ) ) & ( isolateStartCell$(parseBoard$!arg):(?begin.?pairs) | out$"Sorry, I cannot identify a start cell."&~ ) & solve$(!begin,,!pairs):?crumbs & numberElementsAndSort$(reverseList$!crumbs) : (?path.?colWidth) & display$(!arg,!path,!colWidth) )
& "
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 " "
1-0-----
0-0-----
00000----
000-----
--0--0-0--0-- 00000---00000 --00-----00-- 00000---00000 --0--0-0--0--
000-----
00000----
0-0-----
0-0-----"
: ?boards
& whl'(!boards:%?board ?boards&Holy-Knight$!board) & done );</lang> Output:
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 21 30 19 36 22 29 31 20 35 18 23 28 25 15 34 17 26 8 32 14 9 24 27 1 16 33 10 13 4 7 2 5 11 12 3 6 -----1-0----- -----0-0----- ----00000---- -----000----- --0--0-0--0-- 00000---00000 --00-----00-- 00000---00000 --0--0-0--0-- -----000----- ----00000---- -----0-0----- -----0-0----- 1 27 26 56 30 55 2 25 28 24 29 54 36 31 3 50 37 34 39 32 23 53 4 47 6 51 22 35 49 52 21 38 33 40 19 9 46 5 48 7 20 41 45 8 18 43 10 42 11 14 17 44 16 12 13 15
C#
The same solver can solve Hidato, Holy Knight's Tour, Hopido and Numbrix puzzles.
The input can be an array of strings if each cell is one character. The length of the first row must be the number of columns in the puzzle.
Any non-numeric value indicates a no-go.
If there are cells that require more characters, then a 2-dimensional array of ints must be used. Any number < 0 indicates a no-go.
The puzzle can be made circular (the end cell must connect to the start cell). In that case, no start cell needs to be given.
<lang csharp>using System.Collections;
using System.Collections.Generic;
using static System.Console;
using static System.Math;
using static System.Linq.Enumerable;
public class Solver {
private static readonly (int dx, int dy)[] //other puzzle types elided knightMoves = {(1,-2),(2,-1),(2,1),(1,2),(-1,2),(-2,1),(-2,-1),(-1,-2)};
private (int dx, int dy)[] moves; public static void Main() { var knightSolver = new Solver(knightMoves); Print(knightSolver.Solve(true, ".000....", ".0.00...", ".0000000", "000..0.0", "0.0..000", "1000000.", "..00.0..", "...000.."));
Print(knightSolver.Solve(true, ".....0.0.....", ".....0.0.....", "....00000....", ".....000.....", "..0..0.0..0..", "00000...00000", "..00.....00..", "00000...00000", "..0..0.0..0..", ".....000.....", "....00000....", ".....0.0.....", ".....0.0....." )); }
public Solver(params (int dx, int dy)[] moves) => this.moves = moves;
public int[,] Solve(bool circular, params string[] puzzle) { var (board, given, count) = Parse(puzzle); return Solve(board, given, count, circular); }
public int[,] Solve(bool circular, int[,] puzzle) { var (board, given, count) = Parse(puzzle); return Solve(board, given, count, circular); }
private int[,] Solve(int[,] board, BitArray given, int count, bool circular) { var (height, width) = (board.GetLength(0), board.GetLength(1)); bool solved = false; for (int x = 0; x < height && !solved; x++) { solved = Range(0, width).Any(y => Solve(board, given, circular, (height, width), (x, y), count, (x, y), 1)); if (solved) return board; } return null; }
private bool Solve(int[,] board, BitArray given, bool circular, (int h, int w) size, (int x, int y) start, int last, (int x, int y) current, int n) { var (x, y) = current; if (x < 0 || x >= size.h || y < 0 || y >= size.w) return false; if (board[x, y] < 0) return false; if (given[n - 1]) { if (board[x, y] != n) return false; } else if (board[x, y] > 0) return false; board[x, y] = n; if (n == last) { if (!circular || AreNeighbors(start, current)) return true; } for (int i = 0; i < moves.Length; i++) { var move = moves[i]; if (Solve(board, given, circular, size, start, last, (x + move.dx, y + move.dy), n + 1)) return true; } if (!given[n - 1]) board[x, y] = 0; return false;
bool AreNeighbors((int x, int y) p1, (int x, int y) p2) => moves.Any(m => (p2.x + m.dx, p2.y + m.dy).Equals(p1)); }
private static (int[,] board, BitArray given, int count) Parse(string[] input) { (int height, int width) = (input.Length, input[0].Length); int[,] board = new int[height, width]; int count = 0; for (int x = 0; x < height; x++) { string line = input[x]; for (int y = 0; y < width; y++) { board[x, y] = y < line.Length && char.IsDigit(line[y]) ? line[y] - '0' : -1; if (board[x, y] >= 0) count++; } } BitArray given = Scan(board, count, height, width); return (board, given, count); }
private static (int[,] board, BitArray given, int count) Parse(int[,] input) { (int height, int width) = (input.GetLength(0), input.GetLength(1)); int[,] board = new int[height, width]; int count = 0; for (int x = 0; x < height; x++) for (int y = 0; y < width; y++) if ((board[x, y] = input[x, y]) >= 0) count++; BitArray given = Scan(board, count, height, width); return (board, given, count); }
private static BitArray Scan(int[,] board, int count, int height, int width) { var given = new BitArray(count + 1); for (int x = 0; x < height; x++) for (int y = 0; y < width; y++) if (board[x, y] > 0) given[board[x, y] - 1] = true; return given; }
private static void Print(int[,] board) { if (board == null) { WriteLine("No solution"); } else { int w = board.Cast<int>().Where(i => i > 0).Max(i => (int?)Ceiling(Log10(i+1))) ?? 1; string e = new string('-', w); foreach (int x in Range(0, board.GetLength(0))) WriteLine(string.Join(" ", Range(0, board.GetLength(1)) .Select(y => board[x, y] < 0 ? e : board[x, y].ToString().PadLeft(w, ' ')))); } WriteLine(); }
}</lang>
- Output:
-- 23 32 21 -- -- -- -- -- 16 -- 24 31 -- -- -- -- 33 22 15 20 25 30 27 17 36 19 -- -- 28 -- 8 34 -- 14 -- -- 9 26 29 1 18 35 10 13 4 7 -- -- -- 2 5 -- 11 -- -- -- -- -- 12 3 6 -- -- -- -- -- -- -- 1 -- 37 -- -- -- -- -- -- -- -- -- -- 34 -- 56 -- -- -- -- -- -- -- -- -- 2 55 38 33 36 -- -- -- -- -- -- -- -- -- 32 35 54 -- -- -- -- -- -- -- 28 -- -- 3 -- 39 -- -- 48 -- -- 29 6 25 4 31 -- -- -- 53 40 51 42 47 -- -- 30 27 -- -- -- -- -- 49 46 -- -- 7 26 5 24 9 -- -- -- 45 52 41 50 43 -- -- 8 -- -- 23 -- 15 -- -- 44 -- -- -- -- -- -- -- 10 19 22 -- -- -- -- -- -- -- -- -- 18 21 16 11 14 -- -- -- -- -- -- -- -- -- 12 -- 20 -- -- -- -- -- -- -- -- -- -- 17 -- 13 -- -- -- -- --
C++
<lang cpp>
- include <vector>
- include <sstream>
- include <iostream>
- include <iterator>
- include <stdlib.h>
- include <string.h>
using namespace std;
struct node {
int val; unsigned char neighbors;
};
class nSolver { public:
nSolver() {
dx[0] = -1; dy[0] = -2; dx[1] = -1; dy[1] = 2; dx[2] = 1; dy[2] = -2; dx[3] = 1; dy[3] = 2; dx[4] = -2; dy[4] = -1; dx[5] = -2; dy[5] = 1; dx[6] = 2; dy[6] = -1; dx[7] = 2; dy[7] = 1;
}
void solve( vector<string>& puzz, int max_wid ) {
if( puzz.size() < 1 ) return; wid = max_wid; hei = static_cast<int>( puzz.size() ) / wid; int len = wid * hei, c = 0; max = len; arr = new node[len]; memset( arr, 0, len * sizeof( node ) );
for( vector<string>::iterator i = puzz.begin(); i != puzz.end(); i++ ) { if( ( *i ) == "*" ) { max--; arr[c++].val = -1; continue; } arr[c].val = atoi( ( *i ).c_str() ); c++; }
solveIt(); c = 0; for( vector<string>::iterator i = puzz.begin(); i != puzz.end(); i++ ) { if( ( *i ) == "." ) { ostringstream o; o << arr[c].val; ( *i ) = o.str(); } c++; } delete [] arr;
}
private:
bool search( int x, int y, int w ) {
if( w > max ) return true;
node* n = &arr[x + y * wid]; n->neighbors = getNeighbors( x, y );
for( int d = 0; d < 8; d++ ) { if( n->neighbors & ( 1 << d ) ) { int a = x + dx[d], b = y + dy[d]; if( arr[a + b * wid].val == 0 ) { arr[a + b * wid].val = w; if( search( a, b, w + 1 ) ) return true; arr[a + b * wid].val = 0; } } } return false;
}
unsigned char getNeighbors( int x, int y ) {
unsigned char c = 0; int a, b; for( int xx = 0; xx < 8; xx++ ) { a = x + dx[xx], b = y + dy[xx]; if( a < 0 || b < 0 || a >= wid || b >= hei ) continue; if( arr[a + b * wid].val > -1 ) c |= ( 1 << xx ); } return c;
}
void solveIt() {
int x, y, z; findStart( x, y, z ); if( z == 99999 ) { cout << "\nCan't find start point!\n"; return; } search( x, y, z + 1 );
}
void findStart( int& x, int& y, int& z ) {
z = 99999; for( int b = 0; b < hei; b++ ) for( int a = 0; a < wid; a++ ) if( arr[a + wid * b].val > 0 && arr[a + wid * b].val < z ) { x = a; y = b; z = arr[a + wid * b].val; }
}
int wid, hei, max, dx[8], dy[8]; node* arr;
};
int main( int argc, char* argv[] ) {
int wid; string p; //p = "* . . . * * * * * . * . . * * * * . . . . . . . . . . * * . * . . * . * * . . . 1 . . . . . . * * * . . * . * * * * * . . . * *"; wid = 8; p = "* * * * * 1 * . * * * * * * * * * * . * . * * * * * * * * * . . . . . * * * * * * * * * . . . * * * * * * * . * * . * . * * . * * . . . . . * * * . . . . . * * . . * * * * * . . * * . . . . . * * * . . . . . * * . * * . * . * * . * * * * * * * . . . * * * * * * * * * . . . . . * * * * * * * * * . * . * * * * * * * * * * . * . * * * * * "; wid = 13; istringstream iss( p ); vector<string> puzz; copy( istream_iterator<string>( iss ), istream_iterator<string>(), back_inserter<vector<string> >( puzz ) ); nSolver s; s.solve( puzz, wid ); int c = 0; for( vector<string>::iterator i = puzz.begin(); i != puzz.end(); i++ ) {
if( ( *i ) != "*" && ( *i ) != "." ) { if( atoi( ( *i ).c_str() ) < 10 ) cout << "0"; cout << ( *i ) << " ";
}
else cout << " "; if( ++c >= wid ) { cout << endl; c = 0; }
} cout << endl << endl; return system( "pause" );
} </lang>
- Output:
17 14 29 28 18 15 13 16 27 30 19 32 07 25 02 11 06 20 12 26 31 08 33 01 24 03 10 05 34 21 36 23 09 04 35 22 01 05 10 12 02 13 04 09 06 08 11 14 34 03 07 16 7 30 39 28 35 15 56 49 54 51 36 33 17 52 1 38 29 40 27 19 48 55 50 53 32 41 47 18 26 23 20 42 21 44 25 46 24 22 43 45
D
From the refactored C++ version with more precise typing, and some optimizations. The HolyKnightPuzzle struct is created at compile-time, so its pre-conditions can catch most malformed puzzles at compile-time. <lang d>import std.stdio, std.conv, std.string, std.range, std.algorithm,
std.typecons, std.typetuple;
struct HolyKnightPuzzle {
private alias InputCellBaseType = char; private enum InputCell : InputCellBaseType { available = '#', unavailable = '.', start='1' } private alias Cell = uint; private enum : Cell { unknownCell = 0, unavailableCell = Cell.max, startCell=1 } // Special Cell values.
// Neighbors, [shift row, shift column]. static struct P { int x, y; } alias shifts = TypeTuple!(P(-2, -1), P(2, -1), P(-2, 1), P(2, 1), P(-1, -2), P(1, -2), P(-1, 2), P(1, 2));
immutable size_t gridWidth, gridHeight; private immutable Cell nAvailableCells; private /*immutable*/ const InputCell[] flatPuzzle; private Cell[] grid; // Flattened mutable game grid.
@disable this();
this(in string[] rawPuzzle) pure @safe in { assert(!rawPuzzle.empty); assert(!rawPuzzle[0].empty); assert(rawPuzzle.all!(row => row.length == rawPuzzle[0].length)); // Is rectangular. assert(rawPuzzle.join.count(InputCell.start) == 1); // Exactly one start point. } body { //immutable puzzle = rawPuzzle.to!(InputCell[][]); immutable puzzle = rawPuzzle.map!representation.array.to!(InputCell[][]);
gridWidth = puzzle[0].length; gridHeight = puzzle.length; flatPuzzle = puzzle.join;
// This counts the start cell too. nAvailableCells = flatPuzzle.representation.count!(ic => ic != InputCell.unavailable);
grid = flatPuzzle .map!(ic => ic.predSwitch(InputCell.available, unknownCell, InputCell.unavailable, unavailableCell, InputCell.start, startCell)) .array; }
Nullable!(string[][]) solve(size_t width)() pure /*nothrow*/ @safe out(result) { if (!result.isNull) assert(!grid.canFind(unknownCell)); } body { assert(width == gridWidth);
// Find start position. foreach (immutable r; 0 .. gridHeight) foreach (immutable c; 0 .. width) if (grid[r * width + c] == startCell && search!width(r, c, startCell + 1)) { auto result = zip(flatPuzzle, grid) // Not nothrow. //.map!({p, c} => ... .map!(pc => (pc[0] == InputCell.available) ? pc[1].text : InputCellBaseType(pc[0]).text) .array .chunks(width) .array; return typeof(return)(result); }
return typeof(return)(); }
private bool search(size_t width)(in size_t r, in size_t c, in Cell cell) pure nothrow @safe @nogc { if (cell > nAvailableCells) return true; // One solution found.
// This doesn't use the Warnsdorff rule. foreach (immutable sh; shifts) { immutable r2 = r + sh.x, c2 = c + sh.y, pos = r2 * width + c2; // No need to test for >= 0 because uint wraps around. if (c2 < width && r2 < gridHeight && grid[pos] == unknownCell) { grid[pos] = cell; // Try. if (search!width(r2, c2, cell + 1)) return true; grid[pos] = unknownCell; // Restore. } }
return false; }
}
void main() @safe {
// Enum HolyKnightPuzzle to catch malformed puzzles at compile-time. enum puzzle1 = ".###.... .#.##... .####### ###..#.# #.#..### 1######. ..##.#.. ...###..".split.HolyKnightPuzzle;
enum puzzle2 = ".....1.#..... .....#.#..... ....#####.... .....###..... ..#..#.#..#.. #####...##### ..##.....##.. #####...##### ..#..#.#..#.. .....###..... ....#####.... .....#.#..... .....#.#.....".split.HolyKnightPuzzle;
foreach (/*enum*/ puzzle; TypeTuple!(puzzle1, puzzle2)) { //immutable solution = puzzle.solve!(puzzle.gridWidth); enum width = puzzle.gridWidth; immutable solution = puzzle.solve!width; // Solved at run-time. if (solution.isNull) writeln("No solution found for puzzle.\n"); else writefln("One solution:\n%(%-(%2s %)\n%)\n", solution); }
}</lang>
- Output:
One solution: . 17 14 29 . . . . . 28 . 18 15 . . . . 13 16 27 30 19 32 7 25 2 11 . . 6 . 20 12 . 26 . . 31 8 33 1 24 3 10 5 34 21 . . . 36 23 . 9 . . . . . 4 35 22 . . One solution: . . . . . 1 . 5 . . . . . . . . . . 10 . 12 . . . . . . . . . 2 13 4 9 6 . . . . . . . . . 8 11 14 . . . . . . . 34 . . 3 . 7 . . 16 . . 37 30 39 28 35 . . . 15 56 49 54 51 . . 36 33 . . . . . 17 52 . . 31 38 29 40 27 . . . 19 48 55 50 53 . . 32 . . 41 . 47 . . 18 . . . . . . . 26 23 20 . . . . . . . . . 42 21 44 25 46 . . . . . . . . . 24 . 22 . . . . . . . . . . 43 . 45 . . . . .
Run-time about 0.58 seconds with ldc2 compiler (using a switch statement if you don't have the predSwitch yet in Phobos), about 23 times faster than the Haskell entry.
Elixir
This solution uses HLPsolver from here <lang elixir># require HLPsolver
adjacent = [{-1,-2},{-2,-1},{-2,1},{-1,2},{1,2},{2,1},{2,-1},{1,-2}]
""" . . 0 0 0 . . 0 . 0 0 . 0 0 0 0 0 0 0 0 0 0 . . 0 . 0 0 . 0 . . 0 0 0 1 0 0 0 0 0 0 . . 0 0 . 0 . . . 0 0 0 """ |> HLPsolver.solve(adjacent)
"""
_ _ _ _ _ 1 _ 0 _ _ _ _ _ 0 _ 0 _ _ _ _ 0 0 0 0 0 _ _ _ _ _ 0 0 0 _ _ 0 _ _ 0 _ 0 _ _ 0 0 0 0 0 0 _ _ _ 0 0 0 0 0 _ _ 0 0 _ _ _ _ _ 0 0 0 0 0 0 0 _ _ _ 0 0 0 0 0 _ _ 0 _ _ 0 _ 0 _ _ 0 _ _ _ _ _ 0 0 0 _ _ _ _ 0 0 0 0 0 _ _ _ _ _ 0 _ 0 _ _ _ _ _ 0 _ 0
""" |> HLPsolver.solve(adjacent)</lang>
- Output:
Problem: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 Solution: 18 21 36 13 19 22 17 20 35 14 25 6 23 31 12 15 34 26 16 32 7 24 5 1 30 11 8 33 4 27 2 29 9 10 3 28 Problem: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Solution: 1 55 34 36 2 37 56 33 54 32 35 38 28 3 53 44 29 6 25 4 31 39 52 41 50 43 30 27 45 48 7 26 5 24 9 47 40 51 42 49 8 23 15 46 10 19 22 18 21 16 11 14 12 20 17 13
Go
<lang go>package main
import "fmt"
var moves = [][2]int{
{-1, -2}, {1, -2}, {-1, 2}, {1, 2}, {-2, -1}, {-2, 1}, {2, -1}, {2, 1},
}
var board1 = " xxx " +
" x xx " + " xxxxxxx" + "xxx x x" + "x x xxx" + "sxxxxxx " + " xx x " + " xxx "
var board2 = ".....s.x....." +
".....x.x....." + "....xxxxx...." + ".....xxx....." + "..x..x.x..x.." + "xxxxx...xxxxx" + "..xx.....xx.." + "xxxxx...xxxxx" + "..x..x.x..x.." + ".....xxx....." + "....xxxxx...." + ".....x.x....." + ".....x.x....."
func solve(pz [][]int, sz, sx, sy, idx, cnt int) bool {
if idx > cnt { return true } for i := 0; i < len(moves); i++ { x := sx + moves[i][0] y := sy + moves[i][1] if (x >= 0 && x < sz) && (y >= 0 && y < sz) && pz[x][y] == 0 { pz[x][y] = idx if solve(pz, sz, x, y, idx+1, cnt) { return true } pz[x][y] = 0 } } return false
}
func findSolution(b string, sz int) {
pz := make([][]int, sz) for i := 0; i < sz; i++ { pz[i] = make([]int, sz) for j := 0; j < sz; j++ { pz[i][j] = -1 } } var x, y, idx, cnt int for j := 0; j < sz; j++ { for i := 0; i < sz; i++ { switch b[idx] { case 'x': pz[i][j] = 0 cnt++ case 's': pz[i][j] = 1 cnt++ x, y = i, j } idx++ } }
if solve(pz, sz, x, y, 2, cnt) { for j := 0; j < sz; j++ { for i := 0; i < sz; i++ { if pz[i][j] != -1 { fmt.Printf("%02d ", pz[i][j]) } else { fmt.Print("-- ") } } fmt.Println() } } else { fmt.Println("Cannot solve this puzzle!") }
}
func main() {
findSolution(board1, 8) fmt.Println() findSolution(board2, 13)
}</lang>
- Output:
-- 17 14 29 -- -- -- -- -- 28 -- 18 15 -- -- -- -- 13 16 27 30 19 32 07 25 02 11 -- -- 06 -- 20 12 -- 26 -- -- 31 08 33 01 24 03 10 05 34 21 -- -- -- 36 23 -- 09 -- -- -- -- -- 04 35 22 -- -- -- -- -- -- -- 01 -- 05 -- -- -- -- -- -- -- -- -- -- 10 -- 12 -- -- -- -- -- -- -- -- -- 02 13 04 09 06 -- -- -- -- -- -- -- -- -- 08 11 14 -- -- -- -- -- -- -- 36 -- -- 03 -- 07 -- -- 16 -- -- 35 42 33 44 37 -- -- -- 15 20 27 22 25 -- -- 38 41 -- -- -- -- -- 17 24 -- -- 39 34 43 32 45 -- -- -- 19 28 21 26 23 -- -- 40 -- -- 31 -- 29 -- -- 18 -- -- -- -- -- -- -- 46 51 56 -- -- -- -- -- -- -- -- -- 52 55 30 47 50 -- -- -- -- -- -- -- -- -- 48 -- 54 -- -- -- -- -- -- -- -- -- -- 53 -- 49 -- -- -- -- --
Haskell
<lang Haskell>import Data.Array
(Array, (//), (!), assocs, elems, bounds, listArray)
import Data.Foldable (forM_) import Data.List (intercalate, transpose) import Data.Maybe
type Position = (Int, Int)
type KnightBoard = Array Position (Maybe Int)
toSlot :: Char -> Maybe Int toSlot '0' = Just 0 toSlot '1' = Just 1 toSlot _ = Nothing
toString :: Maybe Int -> String toString Nothing = replicate 3 ' ' toString (Just n) = replicate (3 - length nn) ' ' ++ nn
where nn = show n
chunksOf :: Int -> [a] -> a chunksOf _ [] = [] chunksOf n xs =
let (chunk, rest) = splitAt n xs in chunk : chunksOf n rest
showBoard :: KnightBoard -> String showBoard board =
intercalate "\n" . map concat . transpose . chunksOf (height + 1) . map toString $ elems board where (_, (_, height)) = bounds board
toBoard :: [String] -> KnightBoard toBoard strs = board
where height = length strs width = minimum (length <$> strs) board = listArray ((0, 0), (width - 1, height - 1)) . map toSlot . concat . transpose $ take width <$> strs
add
:: Num a => (a, a) -> (a, a) -> (a, a)
add (a, b) (x, y) = (a + x, b + y)
within
:: Ord a => ((a, a), (a, a)) -> (a, a) -> Bool
within ((a, b), (c, d)) (x, y) = a <= x && x <= c && b <= y && y <= d
-- Enumerate valid moves given a board and a knight's position. validMoves :: KnightBoard -> Position -> [Position] validMoves board position = filter isValid plausible
where bound = bounds board plausible = add position <$> [(1, 2), (2, 1), (2, -1), (-1, 2), (-2, 1), (1, -2), (-1, -2), (-2, -1)] isValid pos = within bound pos && maybe False (== 0) (board ! pos)
isSolved :: KnightBoard -> Bool isSolved = all (maybe True (0 /=))
-- Solve the knight's tour with a simple Depth First Search. solveKnightTour :: KnightBoard -> Maybe KnightBoard solveKnightTour board = solve board 1 initPosition
where initPosition = fst $ head $ filter ((== Just 1) . snd) $ assocs board solve boardA depth position = let boardB = boardA // [(position, Just depth)] in if isSolved boardB then Just boardB else listToMaybe $ mapMaybe (solve boardB $ depth + 1) $ validMoves boardB position
tourExA :: [String] tourExA =
[ " 000 " , " 0 00 " , " 0000000" , "000 0 0" , "0 0 000" , "1000000 " , " 00 0 " , " 000 " ]
tourExB :: [String] tourExB =
[ "-----1-0-----" , "-----0-0-----" , "----00000----" , "-----000-----" , "--0--0-0--0--" , "00000---00000" , "--00-----00--" , "00000---00000" , "--0--0-0--0--" , "-----000-----" , "----00000----" , "-----0-0-----" , "-----0-0-----" ]
main :: IO () main =
forM_ [tourExA, tourExB] (\board -> case solveKnightTour $ toBoard board of Nothing -> putStrLn "No solution.\n" Just solution -> putStrLn $ showBoard solution ++ "\n")</lang>
- Output:
19 26 17 36 20 25 31 18 27 16 21 6 23 35 28 15 24 8 30 32 7 22 5 1 34 29 14 11 4 9 2 33 13 12 3 10 1 31 32 28 56 27 2 33 30 34 29 26 48 55 3 24 47 52 45 54 35 25 4 11 6 23 36 49 9 22 51 46 53 44 37 21 12 5 10 7 50 43 13 8 38 41 20 42 19 16 39 14 40 18 17 15
As requested, in an attempt to make this solution faster, the following is a version that replaces the Array with an STUArray (unboxed and mutable), and yields a speedup of 4.2. No speedups were gained until move validation was inlined with the logic in `solve'. This seems to point to the list consing as the overhead for time and allocation, although profiling did show that about 25% of the time in the immutable version was spent creating arrays. Perhaps a more experienced Haskeller could provide insight on how to further optimize this or what optimizations were frivolous (barring a different algorithm or search heuristic, and jumping into C, unless those are the only way). <lang Haskell>{-# LANGUAGE FlexibleContexts #-}
import Control.Monad (forM_)
import qualified Data.Array.Unboxed as AU
import Control.Monad.ST (ST, runST)
import Data.Array.Base (unsafeFreeze)
import Data.List (intercalate, transpose)
import Data.Array.ST
(STUArray, readArray, writeArray, newListArray)
type Position = (Int, Int)
type KnightBoard = AU.UArray Position Int
toSlot :: Char -> Int toSlot '0' = 0 toSlot '1' = 1 toSlot _ = -1
toString :: Int -> String toString (-1) = replicate 3 ' ' toString n = replicate (3 - length nn) ' ' ++ nn
where nn = show n
chunksOf :: Int -> [a] -> a chunksOf _ [] = [] chunksOf n xs = uncurry ((. chunksOf n) . (:)) (splitAt n xs)
showBoard :: KnightBoard -> String showBoard board =
intercalate "\n" . map concat . transpose . chunksOf (height + 1) . map toString $ AU.elems board where (_, (_, height)) = AU.bounds board
toBoard :: [String] -> KnightBoard toBoard strs = board
where height = length strs width = minimum (length <$> strs) board = AU.listArray ((0, 0), (width - 1, height - 1)) . map toSlot . concat . transpose $ take width <$> strs
add
:: Num a => (a, a) -> (a, a) -> (a, a)
add (a, b) (x, y) = (a + x, b + y)
within
:: Ord a => ((a, a), (a, a)) -> (a, a) -> Bool
within ((a, b), (c, d)) (x, y) = a <= x && x <= c && b <= y && y <= d
-- Solve the knight's tour with a simple Depth First Search. solveKnightTour :: KnightBoard -> Maybe KnightBoard solveKnightTour board =
runST $ do let assocs = AU.assocs board bounds = AU.bounds board array <- newListArray bounds (AU.elems board) :: ST s (STUArray s Position Int) let initPosition = fst $ head $ filter ((== 1) . snd) assocs maxDepth = fromIntegral $ 1 + length (filter ((== 0) . snd) assocs) offsets = [ (1, 2) , (2, 1) , (2, -1) , (-1, 2) , (-2, 1) , (1, -2) , (-1, -2) , (-2, -1) ] solve depth position = if within bounds position then do oldValue <- readArray array position if oldValue == 0 then do writeArray array position depth if depth == maxDepth then return True -- This mapM-any combo can be reduced to a string of ||'s -- with the goal of removing the allocation overhead due to consing -- which the compiler may not be able to optimize out. else do results <- mapM (solve (depth + 1) . add position) offsets if or results then return True else do writeArray array position oldValue return False else return False else return False writeArray array initPosition 0 result <- solve 1 initPosition farray <- unsafeFreeze array return $ if result then Just farray else Nothing
tourExA :: [String] tourExA =
[ " 000 " , " 0 00 " , " 0000000" , "000 0 0" , "0 0 000" , "1000000 " , " 00 0 " , " 000 " ]
tourExB :: [String] tourExB =
[ "-----1-0-----" , "-----0-0-----" , "----00000----" , "-----000-----" , "--0--0-0--0--" , "00000---00000" , "--00-----00--" , "00000---00000" , "--0--0-0--0--" , "-----000-----" , "----00000----" , "-----0-0-----" , "-----0-0-----" ]
main :: IO () main =
forM_ [tourExA, tourExB] (\board -> case solveKnightTour $ toBoard board of Nothing -> putStrLn "No solution.\n" Just solution -> putStrLn $ showBoard solution ++ "\n")</lang>
This version is similar to the previous one but:
- the working code is cleaned up slightly with minor optimisations here and there
- only valid board fields are taken into consideration: previously a huge amount of time was wasted on constantly verifying if moves were valid rather than building only valid moves to start with
- vector is used instead of array to take advantage of any fusion
This results in 117x speedup over the very first version. This speed up comes from a smarter traversal rather than from minor code optimisations.
<lang Haskell> {-# LANGUAGE FlexibleContexts #-} {-# LANGUAGE LambdaCase #-} {-# LANGUAGE BangPatterns #-} {-# OPTIONS_GHC -ddump-simpl -ddump-to-file -ddump-stg -O2 -fforce-recomp #-} module Main (main) where
import Control.Monad.ST (runST) import Data.List (intercalate, transpose) import qualified Data.Ix as Ix import qualified Data.Vector as V import Data.Vector.Unboxed (Vector) import qualified Data.Vector.Unboxed as U import qualified Data.Vector.Unboxed.Mutable as MU import Data.Foldable (for_)
type Position = ( Int, Int )
type Bounds = (Position, Position)
type KnightBoard = (Bounds, Vector Int)
toSlot :: Char -> Int toSlot '0' = 0 toSlot '1' = 1 toSlot _ = -1
toString :: Int -> String toString (-1) = replicate 3 ' ' toString n = replicate (3 - length nn) ' ' ++ nn
where nn = show n
chunksOf :: Int -> [a] -> a chunksOf _ [] = [] chunksOf n xs = uncurry ((. chunksOf n) . (:)) (splitAt n xs)
showBoard :: KnightBoard -> String showBoard (bounds, board) =
intercalate "\n" . map concat . transpose . chunksOf (height + 1) . map toString $ U.toList board where (_, (_, height)) = bounds
toBoard :: [String] -> KnightBoard toBoard strs = (((0,0),(width-1,height-1)), board)
where height = length strs width = minimum (length <$> strs) board = U.fromListN (width*height) . map toSlot . concat . transpose $ take width <$> strs
-- Solve the knight's tour with a simple Depth First Search. solveKnightTour :: KnightBoard -> Maybe KnightBoard solveKnightTour (bounds@(_,(_,yb)), board) = runST $ do
array <- U.thaw board let maxDepth = U.length $ U.filter (/= (-1)) board Just iniIdx = U.findIndex (==1) board initPosition = mkPos iniIdx !hops = V.generate (U.length board) $ \i -> if board `U.unsafeIndex` i == -1 then U.empty else mkHops (mkPos i)
solve !depth !position = MU.unsafeRead array position >>= \case 0 -> do MU.unsafeWrite array position depth case depth == maxDepth of True -> return True False -> do results <- U.mapM (solve (depth + 1)) (hops `V.unsafeIndex` position) if U.or results then return True else do MU.unsafeWrite array position 0 return False _ -> pure False
MU.unsafeWrite array (Ix.index bounds initPosition) 0 result <- solve 1 (Ix.index bounds initPosition) farray <- U.unsafeFreeze array return $ if result then Just (bounds, farray) else Nothing where offsets = U.fromListN 8 [ (1, 2), (2, 1), (2, -1), (-1, 2), (-2, 1), (1, -2), (-1, -2), (-2, -1) ] mkHops pos = U.filter (\i -> board `U.unsafeIndex` i == 0) $ U.map (Ix.index bounds) $ U.filter (Ix.inRange bounds) $ U.map (add pos) offsets add (x, y) (x', y') = (x + x', y + y') mkPos idx = idx `quotRem` (yb+1)
tourExA :: [String]
tourExA =
[ " 000 " , " 0 00 " , " 0000000" , "000 0 0" , "0 0 000" , "1000000 " , " 00 0 " , " 000 " ]
tourExB :: [String] tourExB =
[ "-----1-0-----" , "-----0-0-----" , "----00000----" , "-----000-----" , "--0--0-0--0--" , "00000---00000" , "--00-----00--" , "00000---00000" , "--0--0-0--0--" , "-----000-----" , "----00000----" , "-----0-0-----" , "-----0-0-----" ]
main :: IO () main =
for_ [tourExA, tourExB] (\board -> do case solveKnightTour $ toBoard board of Nothing -> putStrLn "No solution.\n" Just solution -> putStrLn $ showBoard solution ++ "\n")
</lang>
Icon and Unicon
This is a Unicon-specific solution: <lang unicon>global nCells, cMap, best record Pos(r,c)
procedure main(A)
puzzle := showPuzzle("Input",readPuzzle()) QMouse(puzzle,findStart(puzzle),&null,0) showPuzzle("Output", solvePuzzle(puzzle)) | write("No solution!")
end
procedure readPuzzle()
# Start with a reduced puzzle space p := [[-1],[-1]] nCells := maxCols := 0 every line := !&input do { put(p,[: -1 | -1 | gencells(line) | -1 | -1 :]) maxCols <:= *p[-1] } every put(p, [-1]|[-1]) # Now normalize all rows to the same length every i := 1 to *p do p[i] := [: !p[i] | (|-1\(maxCols - *p[i])) :] return p
end
procedure gencells(s)
static WS, NWS initial { NWS := ~(WS := " \t") cMap := table() # Map to/from internal model cMap["#"] := -1; cMap["_"] := 0 cMap[-1] := " "; cMap[0] := "_" }
s ? while not pos(0) do { w := (tab(many(WS))|"", tab(many(NWS))) | break w := numeric(\cMap[w]|w) if -1 ~= w then nCells +:= 1 suspend w }
end
procedure showPuzzle(label, p)
write(label," with ",nCells," cells:") every r := !p do { every c := !r do writes(right((\cMap[c]|c),*nCells+1)) write() } return p
end
procedure findStart(p)
if \p[r := !*p][c := !*p[r]] = 1 then return Pos(r,c)
end
procedure solvePuzzle(puzzle)
if path := \best then { repeat { loc := path.getLoc() puzzle[loc.r][loc.c] := path.getVal() path := \path.getParent() | break } return puzzle }
end
class QMouse(puzzle, loc, parent, val)
method getVal(); return val; end method getLoc(); return loc; end method getParent(); return parent; end method atEnd(); return nCells = val; end
method visit(r,c) if /best & validPos(r,c) then return Pos(r,c) end
method validPos(r,c) v := val+1 xv := (0 <= puzzle[r][c]) | fail if xv = (v|0) then { # make sure this path hasn't already gone there ancestor := self while xl := (ancestor := \ancestor.getParent()).getLoc() do if (xl.r = r) & (xl.c = c) then fail return } end
initially
val := val+1 if atEnd() then return best := self QMouse(puzzle, visit(loc.r-2,loc.c-1), self, val) QMouse(puzzle, visit(loc.r-2,loc.c+1), self, val) QMouse(puzzle, visit(loc.r-1,loc.c+2), self, val) QMouse(puzzle, visit(loc.r+1,loc.c+2), self, val) QMouse(puzzle, visit(loc.r+2,loc.c+1), self, val) QMouse(puzzle, visit(loc.r+2,loc.c-1), self, val) QMouse(puzzle, visit(loc.r+1,loc.c-2), self, val) QMouse(puzzle, visit(loc.r-1,loc.c-2), self, val)
end</lang>
Sample run:
->hkt <hkt.in Input with 36 cells: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 _ _ _ _ _ _ _ _ _ _ _ _ Output with 36 cells: 19 4 13 12 18 5 25 20 3 14 17 6 31 21 2 11 32 16 26 24 15 30 7 1 22 27 10 35 8 33 36 23 29 28 9 34 ->
J
The simplest J implementation here uses a breadth first search - but that can be memory inefficient so it's worth representing the boards as characters (several orders of magnitude space improvement) and it's worth capping how much memory we allow J to use (2^34 is 16GB):
<lang J>9!:21]2^34
unpack=:verb define
mask=. +./' '~:y board=. (255 0 1{a.) {~ {.@:>:@:"."0 mask#"1 y
)
ex1=:unpack ];._2]0 :0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0
)
solve=:verb define
board=.,:y for_move.1+i.+/({.a.)=,y do. board=. ;move <@knight"2 board end.
)
kmoves=: ,/(2 1,:1 2)*"1/_1^#:i.4
knight=:dyad define
pos=. ($y)#:(,y)i.x{a. moves=. <"1(#~ 0&<: */"1@:* ($y)&>"1)pos+"1 kmoves moves=. (#~ (0{a.)={&y) moves moves y adverb def (':';'y x} m')"0 (x+1){a.
)</lang>
Letting that cook:
<lang J> $~.sol 48422 8 8</lang>
That's 48422 solutions. Here's one of them:
<lang J> (a.i.{.sol){(i.255),__ __ 11 28 13 __ __ __ __ __ 22 __ 10 29 __ __ __ __ 27 12 21 14 9 16 31 23 2 25 __ __ 30 __ 8 26 __ 20 __ __ 15 32 17
1 24 3 34 5 18 7 __
__ __ 36 19 __ 33 __ __ __ __ __ 4 35 6 __ __</lang>
and here's a couple more:
<lang J> (a.i.{:sol){(i.255),__ __ 5 8 31 __ __ __ __ __ 32 __ 6 9 __ __ __ __ 7 4 33 30 23 10 21
3 34 29 __ __ 20 __ 24
36 __ 2 __ __ 11 22 19
1 28 35 12 15 18 25 __
__ __ 16 27 __ 13 __ __ __ __ __ 14 17 26 __ __
(a.i.24211{sol){(i.255),__
__ 11 14 33 __ __ __ __ __ 34 __ 10 13 __ __ __ __ 19 12 15 32 9 6 25 35 16 31 __ __ 24 __ 8 18 __ 20 __ __ 7 26 5
1 36 17 30 27 4 23 __
__ __ 2 21 __ 29 __ __ __ __ __ 28 3 22 __ __</lang>
This is something of a problem, however, because finding all those solutions is slow. And even having to be concerned about a 16GB memory limit for this small of a problem is troubling (and using 64 bit integers, instead of 8 bit characters, to represent board squares, would exceed that limit). Also, you'd get bored, inspecting 48422 boards.
So, let's just find one solution:
<lang J>unpack=:verb define
mask=. +./' '~:y board=. __ 0 1 {~ {.@:>:@:"."0 mask#"1 y
)
ex1=:unpack ];._2]0 :0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0
)
solve1=:verb define
(1,+/0=,y) solve1 ,:y
for_block._10 <\ y do. board=. ;({.x) <@knight"2 ;block if. #board do. if. =/x do. {.board return. else. board=. (1 0+x) solve1 board if. #board do. board return. end. end. end. end. i.0 0
)
kmoves=: ,/(2 1,:1 2)*"1/_1^#:i.4
knight=:dyad define
pos=. ($y)#:(,y)i.x moves=. <"1(#~ 0&<: */"1@:* ($y)&>"1)pos+"1 kmoves moves=. (#~ 0={&y) moves moves y adverb def (':';'y x} m')"0 x+1
)</lang>
Here, we break our problem space up into blocks of no more than 10 boards each, and use recursion to investigate each batch of boards. When we find a solution, we stop there (for each iteration at each level of recursion):
<lang J> solve1 ex1 __ 11 28 13 __ __ __ __ __ 22 __ 10 29 __ __ __ __ 27 12 21 14 9 16 31 23 2 25 __ __ 30 __ 8 26 __ 20 __ __ 15 32 17
1 24 3 34 5 18 7 __
__ __ 36 19 __ 33 __ __ __ __ __ 4 35 6 __ __</lang>
[Why ten boards and not just one board? Because 10 is a nice compromise between amortizing the overhead of each attempt and not trying too much at one time. Most individual attempts will fail, but by splitting up the workload after exceeding 10 possibilities, instead of investigating each possibility individually, we increase the chances that we are investigating something useful. Also, J implementations penalize the performance of algorithms which are overly serial in structure.]
With this tool in hand, we can now attempt bigger problems:
<lang J>ex2=:unpack ];._2]0 :0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
)</lang>
Finding a solution for this looks like:
<lang J> solve1 ex2 __ __ __ __ __ 1 __ 5 __ __ __ __ __ __ __ __ __ __ 6 __ 46 __ __ __ __ __ __ __ __ __ 48 45 2 7 4 __ __ __ __ __ __ __ __ __ 8 47 44 __ __ __ __ __ __ __ 56 __ __ 49 __ 3 __ __ 42 __ __ 13 52 11 50 9 __ __ __ 43 38 31 36 33 __ __ 14 55 __ __ __ __ __ 41 34 __ __ 53 12 51 10 15 __ __ __ 39 30 37 32 35 __ __ 54 __ __ 23 __ 29 __ __ 40 __ __ __ __ __ __ __ 16 19 22 __ __ __ __ __ __ __ __ __ 24 21 26 17 28 __ __ __ __ __ __ __ __ __ 18 __ 20 __ __ __ __ __ __ __ __ __ __ 25 __ 27 __ __ __ __ __</lang>
Java
<lang java>import java.util.*;
public class HolyKnightsTour {
final static String[] board = { " xxx ", " x xx ", " xxxxxxx", "xxx x x", "x x xxx", "1xxxxxx ", " xx x ", " xxx "};
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 = 2;
public static void main(String[] args) { int row = 0, col = 0;
grid = new int[base][base];
for (int r = 0; r < base; r++) { Arrays.fill(grid[r], -1); for (int c = 2; c < base - 2; c++) { if (r >= 2 && r < base - 2) { if (board[r - 2].charAt(c - 2) == 'x') { grid[r][c] = 0; total++; } if (board[r - 2].charAt(c - 2) == '1') { row = r; col = c; } } } }
grid[row][col] = 1;
if (solve(row, col, 2)) printResult(); }
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, (a, b) -> a[2] - b[2]);
for (int[] nb : nbrs) { r = nb[0]; c = nb[1]; grid[r][c] = count; if (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) - 1; 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 void printResult() { for (int[] row : grid) { for (int i : row) { if (i == -1) System.out.printf("%2s ", ' '); else System.out.printf("%2d ", i); } System.out.println(); } }
}</lang>
19 26 21 28 18 25 33 20 27 22 17 24 7 29 2 35 8 16 34 32 23 6 9 1 30 3 36 13 10 15 12 31 5 4 11 14
JavaScript
ES6
By composition of generic functions, cacheing degree-sorted moves for each node. <lang JavaScript>(() => {
'use strict';
// problems :: String const problems = [ [ " 000 " // , " 0 00 " // , " 0000000" // , "000 0 0" // , "0 0 000" // , "1000000 " // , " 00 0 " // , " 000 " // ], [ "-----1-0-----" // , "-----0-0-----" // , "----00000----" // , "-----000-----" // , "--0--0-0--0--" // , "00000---00000" // , "--00-----00--" // , "00000---00000" // , "--0--0-0--0--" // , "-----000-----" // , "----00000----" // , "-----0-0-----" // , "-----0-0-----" // ] ];
// GENERIC FUNCTIONS ------------------------------------------------------
// comparing :: (a -> b) -> (a -> a -> Ordering) const comparing = f => (x, y) => { const a = f(x), b = f(y); return a < b ? -1 : a > b ? 1 : 0 };
// concat :: a -> [a] | [String] -> String const concat = xs => xs.length > 0 ? (() => { const unit = typeof xs[0] === 'string' ? : []; return unit.concat.apply(unit, xs); })() : [];
// charColRow :: Char -> [String] -> Maybe (Int, Int) const charColRow = (c, rows) => foldr((a, xs, iRow) => a.nothing ? (() => { const mbiCol = elemIndex(c, xs); return mbiCol.nothing ? mbiCol : { just: [mbiCol.just, iRow], nothing: false }; })() : a, { nothing: true }, rows);
// 2 or more arguments // curry :: Function -> Function const curry = (f, ...args) => { const go = xs => xs.length >= f.length ? (f.apply(null, xs)) : function () { return go(xs.concat(Array.from(arguments))); }; return go([].slice.call(args, 1)); };
// elem :: Eq a => a -> [a] -> Bool const elem = (x, xs) => xs.indexOf(x) !== -1;
// elemIndex :: Eq a => a -> [a] -> Maybe Int const elemIndex = (x, xs) => { const i = xs.indexOf(x); return { nothing: i === -1, just: i }; };
// enumFromTo :: Int -> Int -> [Int] const enumFromTo = (m, n) => Array.from({ length: Math.floor(n - m) + 1 }, (_, i) => m + i);
// filter :: (a -> Bool) -> [a] -> [a] const filter = (f, xs) => xs.filter(f);
// findIndex :: (a -> Bool) -> [a] -> Maybe Int const findIndex = (f, xs) => { for (var i = 0, lng = xs.length; i < lng; i++) { if (f(xs[i])) return { nothing: false, just: i }; } return { nothing: true }; };
// foldl :: (b -> a -> b) -> b -> [a] -> b const foldl = (f, a, xs) => xs.reduce(f, a);
// foldr (a -> b -> b) -> b -> [a] -> b const foldr = (f, a, xs) => xs.reduceRight(f, a);
// groupBy :: (a -> a -> Bool) -> [a] -> a const groupBy = (f, xs) => { const dct = xs.slice(1) .reduce((a, x) => { const h = a.active.length > 0 ? a.active[0] : undefined, blnGroup = h !== undefined && f(h, x); return { active: blnGroup ? a.active.concat([x]) : [x], sofar: blnGroup ? a.sofar : a.sofar.concat([a.active]) }; }, { active: xs.length > 0 ? [xs[0]] : [], sofar: [] }); return dct.sofar.concat(dct.active.length > 0 ? [dct.active] : []); };
// intercalate :: String -> [a] -> String const intercalate = (s, xs) => xs.join(s);
// intersectBy::(a - > a - > Bool) - > [a] - > [a] - > [a] const intersectBy = (eq, xs, ys) => (xs.length > 0 && ys.length > 0) ? xs.filter(x => ys.some(curry(eq)(x))) : [];
// justifyRight :: Int -> Char -> Text -> Text const justifyRight = (n, cFiller, strText) => n > strText.length ? ( (cFiller.repeat(n) + strText) .slice(-n) ) : strText;
// length :: [a] -> Int const length = xs => xs.length;
// map :: (a -> b) -> [a] -> [b] const map = (f, xs) => xs.map(f);
// mappendComparing :: [(a -> b)] -> (a -> a -> Ordering) const mappendComparing = fs => (x, y) => fs.reduce((ord, f) => { if (ord !== 0) return ord; const a = f(x), b = f(y); return a < b ? -1 : a > b ? 1 : 0 }, 0);
// maximumBy :: (a -> a -> Ordering) -> [a] -> a const maximumBy = (f, xs) => xs.reduce((a, x) => a === undefined ? x : ( f(x, a) > 0 ? x : a ), undefined);
// min :: Ord a => a -> a -> a const min = (a, b) => b < a ? b : a;
// replicate :: Int -> a -> [a] const replicate = (n, a) => { let v = [a], o = []; if (n < 1) return o; while (n > 1) { if (n & 1) o = o.concat(v); n >>= 1; v = v.concat(v); } return o.concat(v); };
// sortBy :: (a -> a -> Ordering) -> [a] -> [a] const sortBy = (f, xs) => xs.slice() .sort(f);
// splitOn :: String -> String -> [String] const splitOn = (s, xs) => xs.split(s);
// take :: Int -> [a] -> [a] const take = (n, xs) => xs.slice(0, n);
// unlines :: [String] -> String const unlines = xs => xs.join('\n');
// until :: (a -> Bool) -> (a -> a) -> a -> a const until = (p, f, x) => { let v = x; while (!p(v)) v = f(v); return v; };
// zip :: [a] -> [b] -> [(a,b)] const zip = (xs, ys) => xs.slice(0, Math.min(xs.length, ys.length)) .map((x, i) => [x, ys[i]]);
// zipWith :: (a -> b -> c) -> [a] -> [b] -> [c] const zipWith = (f, xs, ys) => Array.from({ length: min(xs.length, ys.length) }, (_, i) => f(xs[i], ys[i]));
// HOLY KNIGHT's TOUR FUNCTIONS -------------------------------------------
// kmoves :: (Int, Int) -> [(Int, Int)] const kmoves = ([x, y]) => map( ([a, b]) => [a + x, b + y], [ [1, 2], [1, -2], [-1, 2], [-1, -2], [2, 1], [2, -1], [-2, 1], [-2, -1] ]);
// rowPosns :: Int -> String -> [(Int, Int)] const rowPosns = (iRow, s) => { return foldl((a, x, i) => (elem(x, ['0', '1']) ? ( a.concat([ [i, iRow] ]) ) : a), [], splitOn(, s)); };
// hash :: (Int, Int) -> String const hash = ([col, row]) => col.toString() + '.' + row.toString();
// Start node, and degree-sorted cache of moves from each node // All node references are hash strings (for this cache)
// problemModel :: String -> {cache: {nodeKey: [nodeKey], start:String}} const problemModel = boardLines => { const steps = foldl((a, xs, i) => a.concat(rowPosns(i, xs)), [], boardLines), courseMoves = (xs, [x, y]) => intersectBy( ([a, b], [c, d]) => a === c && b === d, kmoves([x, y]), xs ), maybeStart = charColRow('1', boardLines); return { start: maybeStart.nothing ? : hash(maybeStart.just), boardWidth: boardLines.length > 0 ? boardLines[0].length : 0, stepCount: steps.length, cache: (() => { const moveCache = foldl((a, xy) => ( a[hash(xy)] = map(hash, courseMoves(steps, xy)), a ), {}, steps), lstMoves = Object.keys(moveCache), dctDegree = foldl((a, k) => (a[k] = moveCache[k].length, a), {}, lstMoves);
return foldl((a, k) => ( a[k] = sortBy(comparing(x => dctDegree[x]), moveCache[k]), a ), {}, lstMoves); })() }; };
// firstSolution :: {nodeKey: [nodeKey]} -> Int -> // nodeKey -> nodeKey -> [nodeKey] -> // -> {path::[nodeKey], pathLen::Int, found::Bool} const firstSolution = (dctMoves, intTarget, strStart, strNodeKey, path) => { const intPath = path.length, moves = dctMoves[strNodeKey];
if ((intTarget - intPath) < 2 && elem(strStart, moves)) { return { nothing: false, just: [strStart, strNodeKey].concat(path), pathLen: intTarget }; }
const nexts = filter(k => !elem(k, path), moves), intNexts = nexts.length, lstFullPath = [strNodeKey].concat(path);
// Until we find a full path back to start return until( x => (x.nothing === false || x.i >= intNexts), x => { const idx = x.i, dctSoln = firstSolution( dctMoves, intTarget, strStart, nexts[idx], lstFullPath ); return { i: idx + 1, nothing: dctSoln.nothing, just: dctSoln.just, pathLen: dctSoln.pathLen }; }, { nothing: true, just: [], i: 0 } ); };
// maybeTour :: [String] -> { // nothing::Bool, Just::[nodeHash], i::Int: pathLen::Int } const maybeTour = trackLines => { const dctModel = problemModel(trackLines), strStart = dctModel.start; return strStart !== ? firstSolution( dctModel.cache, dctModel.stepCount, strStart, strStart, [] ) : { nothing: true }; };
// showLine :: Int -> Int -> String -> Maybe (Int, Int) -> // [(Int, Int, String)] -> String const showLine = curry((intCell, strFiller, maybeStart, xs) => { const blnSoln = maybeStart.nothing, [startCol, startRow] = blnSoln ? [0, 0] : maybeStart.just; return foldl((a, [iCol, iRow, sVal], i, xs) => ({ col: iCol + 1, txt: a.txt + concat(replicate((iCol - a.col) * intCell, strFiller)) + justifyRight( intCell, strFiller, (blnSoln ? sVal : ( iRow === startRow && iCol === startCol ? '1' : '0') ) ) }), { col: 0, txt: }, xs ) .txt });
// solutionString :: [String] -> Int -> String const solutionString = (boardLines, iProblem) => { const dtePre = Date.now(), intCols = boardLines.length > 0 ? boardLines[0].length : 0, soln = maybeTour(boardLines), intMSeconds = Date.now() - dtePre;
if (soln.nothing) return 'No solution found …';
const kCol = 0, kRow = 1, kSeq = 2, steps = soln.just, lstTriples = zipWith((h, n) => { const [col, row] = map( x => parseInt(x, 10), splitOn('.', h) ); return [col, row, n.toString()]; }, steps, enumFromTo(1, soln.pathLen)), cellWidth = length(maximumBy( comparing(x => length(x[kSeq])), lstTriples )[kSeq]) + 1, lstGroups = groupBy( (a, b) => a[kRow] === b[kRow], sortBy( mappendComparing([x => x[kRow], x => x[kCol]]), lstTriples )), startXY = take(2, lstTriples[0]), strMap = 'PROBLEM ' + (parseInt(iProblem, 10) + 1) + '.\n\n' + unlines(map(showLine(cellWidth, ' ', { nothing: false, just: startXY }), lstGroups)), strSoln = 'First solution found in c. ' + intMSeconds + ' milliseconds:\n\n' + unlines(map(showLine(cellWidth, ' ', { nothing: true, just: startXY }), lstGroups)) + '\n\n';
console.log(strSoln); return strMap + '\n\n' + strSoln; };
// TEST ------------------------------------------------------------------- return unlines(map(solutionString, problems));
})();</lang>
- Output:
(Executed in Atom editor, using 'Script' package).
PROBLEM 1. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 First solution found in c. 21 milliseconds: 25 14 23 8 26 15 13 24 7 22 27 16 31 9 36 11 30 28 12 6 21 32 17 1 10 35 20 3 18 29 2 5 33 34 19 4 PROBLEM 2. 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 First solution found in c. 7084 milliseconds: 1 3 50 52 56 53 2 49 4 48 51 54 46 55 5 10 45 42 35 40 47 11 6 13 8 15 44 37 9 16 43 36 41 34 39 19 12 7 14 17 38 33 27 18 26 23 20 32 21 30 25 28 24 22 31 29 [Finished in 7.2s]
Julia
Uses the Hidato puzzle solver module, which has its source code listed here in the Hadato task. <lang julia>using .Hidato # Note that the . here means to look locally for the module rather than in the libraries
const holyknight = """
. 0 0 0 . . . . . 0 . 0 0 . . . . 0 0 0 0 0 0 0 0 0 0 . . 0 . 0 0 . 0 . . 0 0 0 1 0 0 0 0 0 0 . . . 0 0 . 0 . . . . . 0 0 0 . . """
const knightmoves = [[-2, -1], [-2, 1], [-1, -2], [-1, 2], [1, -2], [1, 2], [2, -1], [2, 1]]
board, maxmoves, fixed, starts = hidatoconfigure(holyknight) printboard(board, " 0", " ") hidatosolve(board, maxmoves, knightmoves, fixed, starts[1][1], starts[1][2], 1) printboard(board)
</lang>
- Output:
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 07 4 17 16 8 5 9 6 3 18 25 20 23 31 2 15 22 26 10 30 19 24 21 1 32 11 14 29 34 27 36 33 13 12 35 28
Kotlin
<lang scala>// version 1.1.3
val moves = arrayOf(
intArrayOf(-1, -2), intArrayOf( 1, -2), intArrayOf(-1, 2), intArrayOf(1, 2), intArrayOf(-2, -1), intArrayOf(-2, 1), intArrayOf( 2, -1), intArrayOf(2, 1)
)
val board1 =
" xxx " + " x xx " + " xxxxxxx" + "xxx x x" + "x x xxx" + "sxxxxxx " + " xx x " + " xxx "
val board2 =
".....s.x....." + ".....x.x....." + "....xxxxx...." + ".....xxx....." + "..x..x.x..x.." + "xxxxx...xxxxx" + "..xx.....xx.." + "xxxxx...xxxxx" + "..x..x.x..x.." + ".....xxx....." + "....xxxxx...." + ".....x.x....." + ".....x.x....."
fun solve(pz: Array<IntArray>, sz: Int, sx: Int, sy: Int, idx: Int, cnt: Int): Boolean {
if (idx > cnt) return true for (i in 0 until moves.size) { val x = sx + moves[i][0] val y = sy + moves[i][1] if ((x in 0 until sz) && (y in 0 until sz) && pz[x][y] == 0) { pz[x][y] = idx if (solve(pz, sz, x, y, idx + 1, cnt)) return true pz[x][y] = 0 } } return false
}
fun findSolution(b: String, sz: Int) {
val pz = Array(sz) { IntArray(sz) { -1 } } var x = 0 var y = 0 var idx = 0 var cnt = 0 for (j in 0 until sz) { for (i in 0 until sz) { if (b[idx] == 'x') { pz[i][j] = 0 cnt++ } else if (b[idx] == 's') { pz[i][j] = 1 cnt++ x = i y = j } idx++ } }
if (solve(pz, sz, x, y, 2, cnt)) { for (j in 0 until sz) { for (i in 0 until sz) { if (pz[i][j] != -1) print("%02d ".format(pz[i][j])) else print("-- ") } println() } } else println("Cannot solve this puzzle!")
}
fun main(args: Array<String>) {
findSolution(board1, 8) println() findSolution(board2, 13)
}</lang>
- Output:
-- 17 14 29 -- -- -- -- -- 28 -- 18 15 -- -- -- -- 13 16 27 30 19 32 07 25 02 11 -- -- 06 -- 20 12 -- 26 -- -- 31 08 33 01 24 03 10 05 34 21 -- -- -- 36 23 -- 09 -- -- -- -- -- 04 35 22 -- -- -- -- -- -- -- 01 -- 05 -- -- -- -- -- -- -- -- -- -- 10 -- 12 -- -- -- -- -- -- -- -- -- 02 13 04 09 06 -- -- -- -- -- -- -- -- -- 08 11 14 -- -- -- -- -- -- -- 36 -- -- 03 -- 07 -- -- 16 -- -- 35 42 33 44 37 -- -- -- 15 20 27 22 25 -- -- 38 41 -- -- -- -- -- 17 24 -- -- 39 34 43 32 45 -- -- -- 19 28 21 26 23 -- -- 40 -- -- 31 -- 29 -- -- 18 -- -- -- -- -- -- -- 46 51 56 -- -- -- -- -- -- -- -- -- 52 55 30 47 50 -- -- -- -- -- -- -- -- -- 48 -- 54 -- -- -- -- -- -- -- -- -- -- 53 -- 49 -- -- -- -- --
Lua
<lang lua> local p1, p1W = ".xxx.....x.xx....xxxxxxxxxx..x.xx.x..xxxsxxxxxx...xx.x.....xxx..", 8 local p2, p2W = ".....s.x..........x.x.........xxxxx.........xxx.......x..x.x..x..xxxxx...xxxxx..xx.....xx..xxxxx...xxxxx..x..x.x..x.......xxx.........xxxxx.........x.x..........x.x.....", 13 local puzzle, movesCnt, wid = {}, 0, 0 local moves = { { -1, -2 }, { 1, -2 }, { -1, 2 }, { 1, 2 },
{ -2, -1 }, { -2, 1 }, { 2, -1 }, { 2, 1 } }
function isValid( x, y )
return( x > 0 and x <= wid and y > 0 and y <= wid and puzzle[x + y * wid - wid] == 0 )
end function solve( x, y, s )
if s > movesCnt then return true end local test, a, b for i = 1, #moves do test = false a = x + moves[i][1]; b = y + moves[i][2] if isValid( a, b ) then puzzle[a + b * wid - wid] = s if solve( a, b, s + 1 ) then return true end puzzle[a + b * wid - wid] = 0 end end return false
end function printSolution()
local lp for j = 1, wid do for i = 1, wid do lp = puzzle[i + j * wid - wid] if lp == -1 then io.write( " " ) else io.write( string.format( " %.2d", lp ) ) end end print() end print( "\n" )
end local sx, sy function fill( pz, w )
puzzle = {}; wid = w; movesCnt = #pz local lp for i = 1, #pz do lp = pz:sub( i, i ) if lp == "x" then table.insert( puzzle, 0 ) elseif lp == "." then table.insert( puzzle, -1 ); movesCnt = movesCnt - 1 else table.insert( puzzle, 1 ) sx = 1 + ( i - 1 ) % wid; sy = math.floor( ( i + wid - 1 ) / wid ) end end
end -- entry point -- print( "\n\n" ); fill( p1, p1W ); if solve( sx, sy, 2 ) then printSolution() end print( "\n\n" ); fill( p2, p2W ); if solve( sx, sy, 2 ) then printSolution() end </lang>
- Output:
17 14 29 28 18 15 13 16 27 30 19 32 07 25 02 11 06 20 12 26 31 08 33 01 24 03 10 05 34 21 36 23 09 04 35 22 01 05 10 12 02 13 04 09 06 08 11 14 36 03 07 16 35 42 33 44 37 15 20 27 22 25 38 41 17 24 39 34 43 32 45 19 28 21 26 23 40 31 29 18 46 51 56 52 55 30 47 50 48 54 53 49
Mathematica/Wolfram Language
Outputs coordinates and a visualization of the tour: <lang Mathematica>puzzle = " 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0";
puzzle = StringSplit[puzzle, "\n"]; puzzle = StringTake[#, {1, -1, 2}] & /@ puzzle; pos0 = Join @@ Table[{i, #} & /@ StringPosition[puzzlei, "0"]All, 1, {i, Length@puzzle}]; pos1 = Join @@ Table[{i, #} & /@ StringPosition[puzzlei, "1"]All, 1, {i, Length@puzzle}];
allpoints = Join[pos1, pos0]; validmoves = Select[Subsets[allpoints, {2}], Differences /* Norm /* EqualTo[Sqrt[5]]]; g = Graph[UndirectedEdge @@@ validmoves]; e = VertexList[g]; order = FindShortestTour[g]2 Graphics[{Red, Disk[#, 0.2] & /@ e, Black, BlockMap[Arrow, eorder, 2, 1]}]</lang>
- Output:
{{6,1},{4,2},{6,3},{8,4},{7,6},{6,4},{5,6},{3,5},{1,4},{2,2},{4,3},{5,1},{3,2},{2,4},{1,2},{3,3},{4,1},{6,2},{7,4},{8,6},{6,7},{4,8},{3,6},{5,7},{3,8},{4,6},{6,5},{5,3},{3,4},{1,3},{2,5},{3,7},{5,8},{6,6},{8,5},{7,3},{6,1}} [Visualization of the tour]
Nim
In this version, the board is described as an array of strings rather than a string in the Go version (so, we don’t have to specify the size). The way to initialize is also different and even if the Moves where in the same order, the solution would be different. Changing the order of the Moves may have a great impact on performance, but there is no best order. The order we have chosen provides excellent performance with the two examples: less than 20 ms on our laptop. But with another order, it took more than 2 seconds!
<lang Nim>import sequtils, strformat
const Moves = [[-1, -2], [1, -2], [-2, -1], [2, -1], [-2, 1], [2, 1], [-1, 2], [1, 2]]
proc solve(pz: var seq[seq[int]]; sx, sy, idx, count: Natural): bool =
if idx > count: return true
var x, y: int for move in Moves: x = sx + move[0] y = sy + move[1] if x in 0..pz.high and y in 0..pz.high and pz[x][y] == 0: pz[x][y] = idx if pz.solve(x, y, idx + 1, count): return true pz[x][y] = 0
proc findSolution(board: openArray[string]) =
let sz = board.len var pz = newSeqWith(sz, repeat(-1, sz))
var count = 0 var x, y: int for i in 0..<sz: for j in 0..<sz: case board[i][j] of 'x': pz[i][j] = 0 inc count of 's': pz[i][j] = 1 inc count (x, y) = (i, j) else: discard
if pz.solve(x, y, 2, count): for i in 0..<sz: for j in 0..<sz: if pz[i][j] != -1: stdout.write &"{pz[i][j]:02} " else: stdout.write "-- " stdout.write '\n'
when isMainModule:
const
Board1 = [" xxx ", " x xx ", " xxxxxxx", "xxx x x", "x x xxx", "sxxxxxx ", " xx x ", " xxx "]
Board2 = [".....s.x.....", ".....x.x.....", "....xxxxx....", ".....xxx.....", "..x..x.x..x..", "xxxxx...xxxxx", "..xx.....xx..", "xxxxx...xxxxx", "..x..x.x..x..", ".....xxx.....", "....xxxxx....", ".....x.x.....", ".....x.x....."]
Board1.findSolution() echo() Board2.findSolution()</lang>
- Output:
-- 13 06 15 -- -- -- -- -- 08 -- 12 31 -- -- -- -- 05 14 07 16 27 32 29 09 02 11 -- -- 30 -- 26 04 -- 22 -- -- 17 28 33 01 10 03 18 21 34 25 -- -- -- 36 23 -- 19 -- -- -- -- -- 20 35 24 -- -- -- -- -- -- -- 01 -- 55 -- -- -- -- -- -- -- -- -- -- 50 -- 48 -- -- -- -- -- -- -- -- -- 02 47 54 51 56 -- -- -- -- -- -- -- -- -- 52 49 46 -- -- -- -- -- -- -- 14 -- -- 03 -- 53 -- -- 44 -- -- 09 06 11 04 13 -- -- -- 45 38 33 40 43 -- -- 08 15 -- -- -- -- -- 35 42 -- -- 07 10 05 12 17 -- -- -- 37 32 39 34 41 -- -- 16 -- -- 23 -- 31 -- -- 36 -- -- -- -- -- -- -- 18 21 24 -- -- -- -- -- -- -- -- -- 22 25 28 19 30 -- -- -- -- -- -- -- -- -- 20 -- 26 -- -- -- -- -- -- -- -- -- -- 27 -- 29 -- -- -- -- --
Perl
We perform a brute-force search. As an enhancement, we unroll the search by one level and use Parallel::ForkManager to search the top-level sub-trees concurrently, subject to the number of cores of course. We implement the search with explicit recursion, which impacts performance but improves readability and provides a use case for the "local" keyword. <lang perl>package KT_Locations;
- A sequence of locations on a 2-D board whose order might or might not
- matter. Suitable for representing a partial tour, a complete tour, or the
- required locations to visit.
use strict; use overload '""' => "as_string"; use English;
- 'locations' must be a reference to an array of 2-element array references,
- where the first element is the rank index and the second is the file index.
use Class::Tiny qw(N locations); use List::Util qw(all);
sub BUILD {
my $self = shift; $self->{N} //= 8; $self->{N} >= 3 or die "N must be at least 3"; all {ref($ARG) eq 'ARRAY' && scalar(@{$ARG}) == 2} @{$self->{locations}} or die "At least one element of 'locations' is invalid"; return;
}
sub as_string {
my $self = shift; my %idxs; my $idx = 1; foreach my $loc (@{$self->locations}) { $idxs{join(q{K},@{$loc})} = $idx++; } my $str; { my $w = int(log(scalar(@{$self->locations}))/log(10.)) + 2; my $fmt = "%${w}d"; my $N = $self->N; my $non_tour = q{ } x ($w-1) . q{-}; for (my $r=0; $r<$N; $r++) { for (my $f=0; $f<$N; $f++) { my $k = join(q{K}, $r, $f); $str .= exists($idxs{$k}) ? sprintf($fmt, $idxs{$k}) : $non_tour; } $str .= "\n"; } } return $str;
}
sub as_idx_hash {
my $self = shift; my $N = $self->N; my $result; foreach my $pair (@{$self->locations}) { my ($r, $f) = @{$pair}; $result->{$r * $N + $f}++; } return $result;
}
package KnightsTour; use strict;
- If supplied, 'str' is parsed to set 'N', 'start_location', and
- 'locations_to_visit'. 'legal_move_idxs' is for improving performance.
use Class::Tiny qw( N start_location locations_to_visit str legal_move_idxs ); use English; use Parallel::ForkManager; use Time::HiRes qw( gettimeofday tv_interval );
sub BUILD {
my $self = shift; if ($self->{str}) { my ($n, $sl, $ltv) = _parse_input_string($self->{str}); $self->{N} = $n; $self->{start_location} = $sl; $self->{locations_to_visit} = $ltv; } $self->{N} //= 8; $self->{N} >= 3 or die "N must be at least 3"; exists($self->{start_location}) or die "Must supply start_location"; die "start_location is invalid" if ref($self->{start_location}) ne 'ARRAY' || scalar(@{$self->{start_location}}) != 2; exists($self->{locations_to_visit}) or die "Must supply locations_to_visit"; ref($self->{locations_to_visit}) eq 'KT_Locations' or die "locations_to_visit must be a KT_Locations instance"; $self->{N} == $self->{locations_to_visit}->N or die "locations_to_visit has mismatched board size"; $self->precompute_legal_moves(); return;
}
sub _parse_input_string {
my @rows = split(/[\r\n]+/s, shift); my $N = scalar(@rows); my ($start_location, @to_visit); for (my $r=0; $r<$N; $r++) { my $row_r = $rows[$r]; for (my $f=0; $f<$N; $f++) { my $c = substr($row_r, $f, 1); if ($c eq '1') { $start_location = [$r, $f]; } elsif ($c eq '0') { push @to_visit, [$r, $f]; } } } $start_location or die "No starting location provided"; return ($N, $start_location, KT_Locations->new(N => $N, locations => \@to_visit));
}
sub precompute_legal_moves {
my $self = shift; my $N = $self->{N}; my $ktl_ixs = $self->{locations_to_visit}->as_idx_hash(); for (my $r=0; $r<$N; $r++) { for (my $f=0; $f<$N; $f++) { my $k = $r * $N + $f; $self->{legal_move_idxs}->{$k} = _precompute_legal_move_idxs($r, $f, $N, $ktl_ixs); } } return;
}
sub _precompute_legal_move_idxs {
my ($r, $f, $N, $ktl_ixs) = @ARG; my $r_plus_1 = $r + 1; my $r_plus_2 = $r + 2; my $r_minus_1 = $r - 1; my $r_minus_2 = $r - 2; my $f_plus_1 = $f + 1; my $f_plus_2 = $f + 2; my $f_minus_1 = $f - 1; my $f_minus_2 = $f - 2; my @result = grep { exists($ktl_ixs->{$ARG}) } map { $ARG->[0] * $N + $ARG->[1] } grep {$ARG->[0] >= 0 && $ARG->[0] < $N && $ARG->[1] >= 0 && $ARG->[1] < $N} ([$r_plus_2, $f_minus_1], [$r_plus_2, $f_plus_1], [$r_minus_2, $f_minus_1], [$r_minus_2, $f_plus_1], [$r_plus_1, $f_plus_2], [$r_plus_1, $f_minus_2], [$r_minus_1, $f_plus_2], [$r_minus_1, $f_minus_2]); return \@result;
}
sub find_tour {
my $self = shift; my $num_to_visit = scalar(@{$self->locations_to_visit->locations}); my $N = $self->N; my $start_loc_idx = $self->start_location->[0] * $N + $self->start_location->[1]; my $visited; for (my $i=0; $i<$N*$N; $i++) { vec($visited, $i, 1) = 0; } vec($visited, $start_loc_idx, 1) = 1; # We unwind the search by one level and use Parallel::ForkManager to search # the top-level sub-trees concurrently, assuming there are enough cores. my @next_loc_idxs = @{$self->legal_move_idxs->{$start_loc_idx}}; my $pm = new Parallel::ForkManager(scalar(@next_loc_idxs)); foreach my $next_loc_idx (@next_loc_idxs) { $pm->start and next; # Do the fork my $t0 = [gettimeofday]; vec($visited, $next_loc_idx, 1) = 1; # (The fork cloned $visited.) my $tour = _find_tour_helper($N, $num_to_visit - 1, $next_loc_idx, $visited, $self->legal_move_idxs); my $elapsed = tv_interval($t0); my ($r, $f) = _idx_to_rank_and_file($next_loc_idx, $N); if (defined $tour) { my @tour_locs = map { [_idx_to_rank_and_file($ARG, $N)] } ($start_loc_idx, $next_loc_idx, split(/\s+/s, $tour)); my $kt_locs = KT_Locations->new(N => $N, locations => \@tour_locs); print "Found a tour after first move ($r, $f) ", "in $elapsed seconds:\n", $kt_locs, "\n"; } else { print "No tour found after first move ($r, $f). ", "Took $elapsed seconds.\n"; } $pm->finish; # Do the exit in the child process } $pm->wait_all_children; return;
}
sub _idx_to_rank_and_file {
my ($idx, $N) = @ARG; my $f = $idx % $N; my $r = ($idx - $f) / $N; return ($r, $f);
}
sub _find_tour_helper {
my ($N, $num_to_visit, $current_loc_idx, $visited, $legal_move_idxs) = @ARG;
# The performance hot spot. local *inner_helper = sub { my ($num_to_visit, $current_loc_idx, $visited) = @ARG; if ($num_to_visit == 0) { return q{ }; # Solution found. } my @next_loc_idxs = @{$legal_move_idxs->{$current_loc_idx}}; my $num_to_visit2 = $num_to_visit - 1; foreach my $loc_idx2 (@next_loc_idxs) { next if vec($visited, $loc_idx2, 1); my $visited2 = $visited; vec($visited2, $loc_idx2, 1) = 1; my $recursion = inner_helper($num_to_visit2, $loc_idx2, $visited2); return $loc_idx2 . q{ } . $recursion if defined $recursion; } return; };
return inner_helper($num_to_visit, $current_loc_idx, $visited);
}
package main; use strict;
solve_size_8_problem(); solve_size_13_problem(); exit 0;
sub solve_size_8_problem {
my $problem = <<"END_SIZE_8_PROBLEM";
--000--- --0-00-- -0000000 000--0-0 0-0--000 1000000- --00-0-- ---000-- END_SIZE_8_PROBLEM
my $kt = KnightsTour->new(str => $problem); print "Finding a tour for an 8x8 problem...\n"; $kt->find_tour(); return;
}
sub solve_size_13_problem {
my $problem = <<"END_SIZE_13_PROBLEM";
1-0-----
0-0-----
00000----
000-----
--0--0-0--0-- 00000---00000 --00-----00-- 00000---00000 --0--0-0--0--
000-----
00000----
0-0-----
0-0-----
END_SIZE_13_PROBLEM
my $kt = KnightsTour->new(str => $problem); print "Finding a tour for a 13x13 problem...\n"; $kt->find_tour(); return;
}</lang>
- Output:
The timings shown below were obtained on a Dell Optiplex 9020 with 4 cores.
...>holy_knights_tour.pl Finding a tour for an 8x8 problem... Found a tour after first move (6, 2) in 0.018372 seconds: - - 18 31 16 - - - - - 23 - 33 30 - - - 19 32 17 24 15 34 29 7 22 5 - - 28 - 26 20 - 8 - - 25 14 35 1 6 21 4 11 36 27 - - - 2 9 - 13 - - - - - 12 3 10 - - Found a tour after first move (4, 2) in 0.010491 seconds: - - 30 23 20 - - - - - 9 - 31 22 - - - 29 24 21 10 19 32 15 25 8 27 - - 16 - 18 28 - 2 - - 11 14 33 1 26 7 12 5 34 17 - - - 36 3 - 13 - - - - - 6 35 4 - - Found a tour after first move (3, 1) in 0.048164 seconds: - - 28 11 14 - - - - - 13 - 9 30 - - - 27 10 29 12 15 18 31 23 2 25 - - 8 - 16 26 - 22 - - 17 32 19 1 24 3 34 5 20 7 - - - 36 21 - 33 - - - - - 4 35 6 - - Finding a tour for a 13x13 problem... Found a tour after first move (2, 6) in 78.827185 seconds: - - - - - 1 - 21 - - - - - - - - - - 22 - 6 - - - - - - - - - 4 7 2 23 20 - - - - - - - - - 24 5 8 - - - - - - - 34 - - 3 - 19 - - 56 - - 35 30 37 28 25 - - - 9 18 11 16 13 - - 26 33 - - - - - 55 14 - - 31 36 29 38 27 - - - 53 10 17 12 15 - - 32 - - 39 - 45 - - 54 - - - - - - - 46 49 52 - - - - - - - - - 40 51 42 47 44 - - - - - - - - - 48 - 50 - - - - - - - - - - 41 - 43 - - - - - Found a tour after first move (2, 4) in 100.327934 seconds: - - - - - 1 - 23 - - - - - - - - - - 24 - 20 - - - - - - - - - 2 19 4 25 22 - - - - - - - - - 26 21 18 - - - - - - - 36 - - 3 - 5 - - 12 - - 37 32 39 30 27 - - - 17 6 15 8 13 - - 28 35 - - - - - 11 56 - - 33 38 31 40 29 - - - 55 16 7 14 9 - - 34 - - 41 - 47 - - 10 - - - - - - - 48 51 54 - - - - - - - - - 42 53 44 49 46 - - - - - - - - - 50 - 52 - - - - - - - - - - 43 - 45 - - - - - Found a tour after first move (1, 7) in 1443.340089 seconds: - - - - - 1 - 21 - - - - - - - - - - 22 - 2 - - - - - - - - - 18 3 16 23 20 - - - - - - - - - 24 19 4 - - - - - - - 34 - - 17 - 15 - - 56 - - 35 30 37 28 25 - - - 5 14 7 12 9 - - 26 33 - - - - - 55 10 - - 31 36 29 38 27 - - - 53 6 13 8 11 - - 32 - - 39 - 45 - - 54 - - - - - - - 46 49 52 - - - - - - - - - 40 51 42 47 44 - - - - - - - - - 48 - 50 - - - - - - - - - - 41 - 43 - - - - -
Phix
Tweaked the knights tour algorithm (to use a limit variable rather than size*size). Bit slow on the second one... (hence omitted under pwa/p2js)
with javascript_semantics sequence board, warnsdorffs integer size, limit, nchars string fmt, blank constant ROW = 1, COL = 2, moves = {{-1,-2},{-2,-1},{-2,1},{-1,2},{1,2},{2,1},{2,-1},{1,-2}} function onboard(integer row, integer col) return row>=1 and row<=size and col>=nchars and col<=nchars*size end function procedure init_warnsdorffs() for row=1 to size do for col=nchars to nchars*size by nchars do for move=1 to length(moves) do integer nrow = row+moves[move][ROW], ncol = col+moves[move][COL]*nchars if onboard(nrow,ncol) then warnsdorffs[nrow][ncol] += 1 end if end for end for end for end procedure atom t0 = time(), t1 = time()+1 integer tries = 0, backtracks = 0 function solve(integer row, integer col, integer n) if time()>t1 and platform()!=JS then ?{row,floor(col/nchars),n,tries} puts(1,join(board,"\n")&"\n") t1 = time()+1 end if tries+= 1 if n>limit then return 1 end if sequence wmoves = {} integer nrow, ncol for move=1 to length(moves) do nrow = row+moves[move][ROW] ncol = col+moves[move][COL]*nchars if onboard(nrow,ncol) and board[nrow][ncol]=' ' then wmoves = append(wmoves,{warnsdorffs[nrow][ncol],nrow,ncol}) end if end for wmoves = sort(wmoves) -- avoid creating orphans if length(wmoves)<2 or wmoves[2][1]>1 then for m=1 to length(wmoves) do {?,nrow,ncol} = wmoves[m] warnsdorffs[nrow][ncol] -= 1 end for for m=1 to length(wmoves) do {?,nrow,ncol} = wmoves[m] integer scol = ncol-nchars+1 board[nrow][scol..ncol] = sprintf(fmt,n) if solve(nrow,ncol,n+1) then return 1 end if backtracks += 1 board[nrow][scol..ncol] = blank end for for m=1 to length(wmoves) do {?,nrow,ncol} = wmoves[m] warnsdorffs[nrow][ncol] += 1 end for end if return 0 end function procedure holyknight(sequence s) s = split(s,'\n') size = length(s) nchars = length(sprintf(" %d",size*size)) fmt = sprintf(" %%%dd",nchars-1) blank = repeat(' ',nchars) board = repeat(repeat(' ',size*nchars),size) limit = 1 integer sx, sy for x=1 to size do integer y = nchars for j=1 to size do integer ch = iff(j>length(s[x])?'-':s[x][j]) if ch=' ' then ch = '-' elsif ch='0' then ch = ' ' limit += 1 elsif ch='1' then sx = x sy = y end if board[x][y] = ch y += nchars end for end for warnsdorffs = repeat(repeat(0,size*nchars),size) init_warnsdorffs() t0 = time() tries = 0 backtracks = 0 t1 = time()+1 if solve(sx,sy,2) then puts(1,join(board,"\n")) printf(1,"\nsolution found in %d tries, %d backtracks (%3.2fs)\n",{tries,backtracks,time()-t0}) else puts(1,"no solutions found\n") end if end procedure constant board1 = """ 000 0 00 0000000 000 0 0 0 0 000 1000000 00 0 000""" holyknight(board1) constant board2 = """ -----1-0----- -----0-0----- ----00000---- -----000----- --0--0-0--0-- 00000---00000 --00-----00-- 00000---00000 --0--0-0--0-- -----000----- ----00000---- -----0-0----- -----0-0-----""" if platform()!=JS then holyknight(board2) end if {} = wait_key()
- Output:
- 21 4 19 - - - - - 18 - 22 5 - - - - 15 20 3 32 23 6 9 17 2 33 - - 8 - 24 14 - 16 - - 31 10 7 1 34 13 30 27 36 25 - - - 28 35 - 11 - - - - - 12 29 26 - - solution found in 31718 tries, 31682 backtracks (0.11s) - - - - - 1 - 55 - - - - - - - - - - 8 - 2 - - - - - - - - - 6 3 54 9 56 - - - - - - - - - 10 7 4 - - - - - - - 12 - - 5 - 53 - - 46 - - 13 16 23 18 11 - - - 45 52 43 50 41 - - 14 21 - - - - - 47 40 - - 15 22 17 24 19 - - - 39 44 51 42 49 - - 20 - - 25 - 33 - - 48 - - - - - - - 32 35 38 - - - - - - - - - 26 37 28 31 34 - - - - - - - - - 30 - 36 - - - - - - - - - - 27 - 29 - - - - - solution found in 61341542 tries, 61341486 backtracks (180.56s)
Python
<lang python> from sys import stdout moves = [
[-1, -2], [1, -2], [-1, 2], [1, 2], [-2, -1], [-2, 1], [2, -1], [2, 1]
]
def solve(pz, sz, sx, sy, idx, cnt):
if idx > cnt: return 1
for i in range(len(moves)): x = sx + moves[i][0] y = sy + moves[i][1] if sz > x > -1 and sz > y > -1 and pz[x][y] == 0: pz[x][y] = idx if 1 == solve(pz, sz, x, y, idx + 1, cnt): return 1 pz[x][y] = 0
return 0
def find_solution(pz, sz):
p = [[-1 for j in range(sz)] for i in range(sz)] idx = x = y = cnt = 0 for j in range(sz): for i in range(sz): if pz[idx] == "x": p[i][j] = 0 cnt += 1 elif pz[idx] == "s": p[i][j] = 1 cnt += 1 x = i y = j idx += 1
if 1 == solve(p, sz, x, y, 2, cnt): for j in range(sz): for i in range(sz): if p[i][j] != -1: stdout.write(" {:0{}d}".format(p[i][j], 2)) else: stdout.write(" ") print() else: print("Cannot solve this puzzle!")
- entry point
find_solution(".xxx.....x.xx....xxxxxxxxxx..x.xx.x..xxxsxxxxxx...xx.x.....xxx..", 8) print() find_solution(".....s.x..........x.x.........xxxxx.........xxx.......x..x.x..x..xxxxx...xxxxx..xx.....xx..xxxxx...xxxxx..x..x.x..x.......xxx.........xxxxx.........x.x..........x.x.....", 13) </lang>
- Output:
17 14 29 28 18 15 13 16 27 30 19 32 07 25 02 11 06 20 12 26 31 08 33 01 24 03 10 05 34 21 36 23 09 04 35 22 01 05 10 12 02 13 04 09 06 08 11 14 36 03 07 16 35 42 33 44 37 15 20 27 22 25 38 41 17 24 39 34 43 32 45 19 28 21 26 23 40 31 29 18 46 51 56 52 55 30 47 50 48 54 53 49
Racket
This solution uses the module "hidato-family-solver.rkt" from Solve a Numbrix puzzle#Racket. The difference between the two is essentially the neighbourhood function.
It solves the tasked problem, as well as the "extra credit" from #Ada.
<lang racket>#lang racket (require "hidato-family-solver.rkt")
(define knights-neighbour-offsets
'((+1 +2) (-1 +2) (+1 -2) (-1 -2) (+2 +1) (-2 +1) (+2 -1) (-2 -1)))
(define solve-a-knights-tour (solve-hidato-family knights-neighbour-offsets))
(displayln
(puzzle->string (solve-a-knights-tour #(#(_ 0 0 0 _ _ _ _) #(_ 0 _ 0 0 _ _ _) #(_ 0 0 0 0 0 0 0) #(0 0 0 _ _ 0 _ 0) #(0 _ 0 _ _ 0 0 0) #(1 0 0 0 0 0 0 _) #(_ _ 0 0 _ 0 _ _) #(_ _ _ 0 0 0 _ _)))))
(newline)
(displayln
(puzzle->string (solve-a-knights-tour #(#(- - - - - 1 - 0 - - - - -) #(- - - - - 0 - 0 - - - - -) #(- - - - 0 0 0 0 0 - - - -) #(- - - - - 0 0 0 - - - - -) #(- - 0 - - 0 - 0 - - 0 - -) #(0 0 0 0 0 - - - 0 0 0 0 0) #(- - 0 0 - - - - - 0 0 - -) #(0 0 0 0 0 - - - 0 0 0 0 0) #(- - 0 - - 0 - 0 - - 0 - -) #(- - - - - 0 0 0 - - - - -) #(- - - - 0 0 0 0 0 - - - -) #(- - - - - 0 - 0 - - - - -) #(- - - - - 0 - 0 - - - - -)))))</lang>
- Output:
_ 13 30 23 _ _ _ _ _ 24 _ 14 31 _ _ _ _ 29 12 25 22 15 32 7 11 26 21 _ _ 6 _ 16 28 _ 10 _ _ 33 8 5 1 20 27 34 9 4 17 _ _ _ 2 19 _ 35 _ _ _ _ _ 36 3 18 _ _ _ _ _ _ _ 1 _ 51 _ _ _ _ _ _ _ _ _ _ 50 _ 2 _ _ _ _ _ _ _ _ _ 56 3 52 49 54 _ _ _ _ _ _ _ _ _ 48 55 4 _ _ _ _ _ _ _ 46 _ _ 5 _ 53 _ _ 24 _ _ 45 8 11 6 47 _ _ _ 23 30 19 28 21 _ _ 44 9 _ _ _ _ _ 25 22 _ _ 43 10 7 12 41 _ _ _ 31 18 29 20 27 _ _ 42 _ _ 13 _ 17 _ _ 26 _ _ _ _ _ _ _ 40 37 32 _ _ _ _ _ _ _ _ _ 36 33 14 39 16 _ _ _ _ _ _ _ _ _ 38 _ 34 _ _ _ _ _ _ _ _ _ _ 35 _ 15 _ _ _ _ _
Raku
(formerly Perl 6)
This uses a Warnsdorff solver, which cuts down the number of tries by more than a factor of six over the brute force approach. This same solver is used in:
- Solve a Hidato puzzle
- Solve a Hopido puzzle
- Solve a Holy Knight's tour
- Solve a Numbrix puzzle
- Solve the no connection puzzle
<lang perl6>my @adjacent =
[ -2, -1], [ -2, 1], [-1,-2], [-1,+2], [+1,-2], [+1,+2], [ +2, -1], [ +2, 1];
put "\n" xx 60;
solveboard q:to/END/;
. 0 0 0 . 0 . 0 0 . 0 0 0 0 0 0 0 0 0 0 . . 0 . 0 0 . 0 . . 0 0 0 1 0 0 0 0 0 0 . . 0 0 . 0 . . . 0 0 0 END
sub solveboard($board) {
my $max = +$board.comb(/\w+/); my $width = $max.chars;
my @grid; my @known; my @neigh; my @degree; @grid = $board.lines.map: -> $line { [ $line.words.map: { /^_/ ?? 0 !! /^\./ ?? Rat !! $_ } ] } sub neighbors($y,$x --> List) { eager gather for @adjacent { my $y1 = $y + .[0]; my $x1 = $x + .[1]; take [$y1,$x1] if defined @grid[$y1][$x1]; } }
for ^@grid -> $y { for ^@grid[$y] -> $x { if @grid[$y][$x] -> $v { @known[$v] = [$y,$x]; } if @grid[$y][$x].defined { @neigh[$y][$x] = neighbors($y,$x); @degree[$y][$x] = +@neigh[$y][$x]; } } } print "\e[0H\e[0J";
my $tries = 0;
try_fill 1, @known[1];
sub try_fill($v, $coord [$y,$x] --> Bool) { return True if $v > $max; $tries++;
my $old = @grid[$y][$x];
return False if +$old and $old != $v; return False if @known[$v] and @known[$v] !eqv $coord;
@grid[$y][$x] = $v; # conjecture grid value
print "\e[0H"; # show conjectured board for @grid -> $r { say do for @$r { when Rat { ' ' x $width } when 0 { '_' x $width } default { .fmt("%{$width}d") } } }
my @neighbors = @neigh[$y][$x][];
my @degrees; for @neighbors -> \n [$yy,$xx] { my $d = --@degree[$yy][$xx]; # conjecture new degrees push @degrees[$d], n; # and categorize by degree }
for @degrees.grep(*.defined) -> @ties { for @ties.reverse { # reverse works better for this hidato anyway return True if try_fill $v + 1, $_; } }
for @neighbors -> [$yy,$xx] { ++@degree[$yy][$xx]; # undo degree conjectures }
@grid[$y][$x] = $old; # undo grid value conjecture return False; } say "$tries tries";
}</lang>
- Output:
25 14 27 36 24 15 31 26 13 28 23 6 17 35 12 29 16 22 30 32 7 18 5 1 34 11 8 19 4 21 2 33 9 10 3 20 84 tries
REXX
This REXX program is essentially a modified knight's tour REXX program with support to place pennies on the chessboard.
Also supported is the specification of the size of the chessboard and the placement of the knight (initial position).
This is an open tour solution. (See this task's discussion page for an explanation [in the first section]). <lang rexx>/*REXX program solves the holy knight's tour problem for a (general) NxN chessboard.*/ blank=pos('//', space(arg(1), 0))\==0 /*see if the pennies are to be shown. */ parse arg ops '/' cent /*obtain the options and the pennies. */ parse var ops N sRank sFile . /*boardsize, starting position, pennys.*/ if N== | N=="," then N=8 /*no boardsize specified? Use default.*/ if sRank== | sRank=="," then sRank=N /*starting rank given? " " */ if sFile== | sFile=="," then sFile=1 /* " file " " " */ NN=N**2; NxN='a ' N"x"N ' chessboard' /*file [↓] [↓] r=rank */ @.=; do r=1 for N; do f=1 for N; @.r.f=.; end /*f*/; end /*r*/
/*[↑] create an empty NxN chessboard.*/
cent=space( translate( cent, , ',') ) /*allow use of comma (,) for separater.*/ cents=0 /*number of pennies on the chessboard. */
do while cent\= /* [↓] possibly place the pennies. */ parse var cent cr cf x '/' cent /*extract where to place the pennies. */ if x= then x=1 /*if number not specified, use 1 penny.*/ if cr= then iterate /*support the "blanking" option. */ do cf=cf for x /*now, place X pennies on chessboard.*/ @.cr.cf= '¢' /*mark chessboard position with a penny*/ end /*cf*/ /* [↑] places X pennies on chessboard.*/ end /*while*/ /* [↑] allows of the placing of X ¢s*/ /* [↓] traipse through the chessboard.*/ do r=1 for N; do f=1 for N; cents=cents + (@.r.f=='¢'); end /*f*/; end /*r*/ /* [↑] count the number of pennies. */
if cents\==0 then say cents 'pennies placed on chessboard.' target=NN - cents /*use this as the number of moves left.*/
Kr = '2 1 -1 -2 -2 -1 1 2' /*the legal "rank" moves for a knight.*/ Kf = '1 2 2 1 -1 -2 -2 -1' /* " " "file" " " " " */
kr.M=words(Kr) /*number of possible moves for a Knight*/ parse var Kr Kr.1 Kr.2 Kr.3 Kr.4 Kr.5 Kr.6 Kr.7 Kr.8 /*parse the legal moves by hand.*/ parse var Kf Kf.1 Kf.2 Kf.3 Kf.4 Kf.5 Kf.6 Kf.7 Kf.8 /* " " " " " " */ beg= '-1-' /* [↑] create the NxN chessboard. */ if @.sRank.sFile ==. then @.sRank.sFile=beg /*the knight's starting position. */ if @.sRank.sFile\==beg then do sRank=1 for N /*find starting rank for the knight.*/
do sFile=1 for N /* " " file " " " */ if @.sRank.sFile\==. then iterate @.sRank.sFile=beg /*the knight's starting position. */ leave sRank /*we have a spot, so leave all this.*/ end /*sFile*/ end /*sRank*/
@hkt= "holy knight's tour" /*a handy─dandy literal for the SAYs. */ if \move(2,sRank,sFile) & \(N==1) then say 'No' @hkt "solution for" NxN'.'
else say 'A solution for the' @hkt "on" NxN':'
/*show chessboard with moves and coins.*/
!=left(, 9 * (n<18) ); say /*used for indentation of chessboard. */ _=substr( copies("┼───", N), 2); say ! translate('┌'_"┐", '┬', "┼")
do r=N for N by -1; if r\==N then say ! '├'_"┤"; L=@. do f=1 for N; ?=@.r.f; if ?==target then ?='end'; L=L'│'center(?,3) end /*f*/ if blank then L=translate(L,,'¢') /*blank out the pennies on chessboard ?*/ say ! translate(L'│', , .) /*display a rank of the chessboard. */ end /*r*/ /*19x19 chessboard can be shown 80 cols*/
say ! translate('└'_"┘", '┴', "┼") /*display the last rank of chessboard. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ move: procedure expose @. Kr. Kf. target; parse arg #,rank,file /*obtain move,rank,file.*/
do t=1 for Kr.M; nr=rank+Kr.t; nf=file+Kf.t /*position of the knight*/ if @.nr.nf==. then do; @.nr.nf=# /*Empty? Knight can move*/ if #==target then return 1 /*is this the last move?*/ if move(#+1,nr,nf) then return 1 /* " " " " " */ @.nr.nf=. /*undo the above move. */ end /*try a different move. */ end /*t*/ /* [↑] all moves tried.*/ return 0 /*tour isn't possible. */</lang>
output when the following is used for input:
, 3 1 /1,1 3 /1,7 2 /2,1 2 /2,5 /2,7 2 /3,8 /4,2 /4,4 2 /5,4 2 /5,7 /6,1 /7,1 /7,3 /7,6 3 /8,1 /8,5 4
28 pennies placed on chessboard. A solution for the holy knight's tour on a 8x8 chessboard: ┌───┬───┬───┬───┬───┬───┬───┬───┐ │ ¢ │25 │12 │27 │ ¢ │ ¢ │ ¢ │ ¢ │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │ ¢ │end│ ¢ │24 │13 │ ¢ │ ¢ │ ¢ │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │ ¢ │11 │26 │ 3 │28 │23 │14 │ 5 │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │35 │ 2 │31 │ ¢ │ ¢ │ 4 │ ¢ │22 │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │10 │ ¢ │34 │ ¢ │ ¢ │29 │ 6 │15 │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │-1-│32 │ 9 │30 │19 │16 │21 │ ¢ │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │ ¢ │ ¢ │18 │33 │ ¢ │ 7 │ ¢ │ ¢ │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │ ¢ │ ¢ │ ¢ │ 8 │17 │20 │ ¢ │ ¢ │ └───┴───┴───┴───┴───┴───┴───┴───┘
output when the following (for a "cleaner" chessboard, no pennies are shown) is used for input:
, 3 1 /1,1 3 /1,7 2 /2,1 2 /2,5 /2,7 2 /3,8 /4,2 /4,4 2 /5,4 2 /5,7 /6,1 /7,1 /7,3 /7,6 3 /8,1 /8,5 4 //
28 pennies placed on chessboard. A solution for the holy knight's tour on a 8x8 chessboard: ┌───┬───┬───┬───┬───┬───┬───┬───┐ │ │25 │12 │27 │ │ │ │ │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │ │end│ │24 │13 │ │ │ │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │ │11 │26 │ 3 │28 │23 │14 │ 5 │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │35 │ 2 │31 │ │ │ 4 │ │22 │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │10 │ │34 │ │ │29 │ 6 │15 │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │-1-│32 │ 9 │30 │19 │16 │21 │ │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │ │ │18 │33 │ │ 7 │ │ │ ├───┼───┼───┼───┼───┼───┼───┼───┤ │ │ │ │ 8 │17 │20 │ │ │ └───┴───┴───┴───┴───┴───┴───┴───┘
Ruby
This solution uses HLPsolver from here <lang ruby>require 'HLPsolver'
ADJACENT = [[-1,-2],[-2,-1],[-2,1],[-1,2],[1,2],[2,1],[2,-1],[1,-2]]
boardy = <<EOS . . 0 0 0 . . 0 . 0 0 . 0 0 0 0 0 0 0 0 0 0 . . 0 . 0 0 . 0 . . 0 0 0 1 0 0 0 0 0 0 . . 0 0 . 0 . . . 0 0 0 EOS t0 = Time.now HLPsolver.new(boardy).solve puts " #{Time.now - t0} sec"</lang>
Which produces:
Problem: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 Solution: 8 33 14 13 7 32 9 34 31 22 15 6 29 35 12 21 30 16 10 36 23 28 5 1 20 11 24 27 4 17 2 19 25 26 3 18 0.005 sec
Tcl
<lang tcl>package require Tcl 8.6
oo::class create HKTSolver {
variable grid start limit constructor {puzzle} {
set grid $puzzle for {set y 0} {$y < [llength $grid]} {incr y} { for {set x 0} {$x < [llength [lindex $grid $y]]} {incr x} { if {[set cell [lindex $grid $y $x]] == 1} { set start [list $y $x] } incr limit [expr {$cell>=0}] } } if {![info exist start]} { return -code error "no starting position found" }
} method moves {} {
return { -1 -2 1 -2 -2 -1 2 -1 -2 1 2 1 -1 2 1 2 }
} method Moves {g r c} {
set valid {} foreach {dr dc} [my moves] { set R [expr {$r + $dr}] set C [expr {$c + $dc}] if {[lindex $g $R $C] == 0} { lappend valid $R $C } } return $valid
}
method Solve {g r c v} {
lset g $r $c [incr v] if {$v >= $limit} {return $g} foreach {r c} [my Moves $g $r $c] { return [my Solve $g $r $c $v] } return -code continue
}
method solve {} {
while {[incr i]==1} { set grid [my Solve $grid {*}$start 0] return } return -code error "solution not possible"
} method solution {} {return $grid}
}
proc parsePuzzle {str} {
foreach line [split $str "\n"] {
if {[string trim $line] eq ""} continue lappend rows [lmap {- c} [regexp -all -inline {(.)\s?} $line] { string map {" " -1} $c }]
} set len [tcl::mathfunc::max {*}[lmap r $rows {llength $r}]] for {set i 0} {$i < [llength $rows]} {incr i} {
while {[llength [lindex $rows $i]] < $len} { lset rows $i end+1 -1 }
} return $rows
} proc showPuzzle {grid name} {
foreach row $grid {foreach cell $row {incr c [expr {$cell>=0}]}} set len [string length $c] set u [string repeat "_" $len] puts "$name with $c cells" foreach row $grid {
puts [format " %s" [join [lmap c $row { format "%*s" $len [if {$c==-1} list elseif {$c==0} {set u} {set c}] }]]]
}
}
set puzzle [parsePuzzle {
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0
}] showPuzzle $puzzle "Input" HKTSolver create hkt $puzzle hkt solve showPuzzle [hkt solution] "Output"</lang>
- Output:
Input with 36 cells __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 1 __ __ __ __ __ __ __ __ __ __ __ __ Output with 36 cells 13 6 15 8 12 31 5 14 7 16 27 32 29 9 2 11 30 26 4 22 17 28 33 1 10 3 18 21 34 25 36 23 19 20 35 24
Wren
<lang ecmascript>import "/fmt" for Fmt
var moves = [ [-1, -2], [1, -2], [-1, 2], [1, 2], [-2, -1], [-2, 1], [2, -1], [2, 1] ]
var board1 =
" xxx " + " x xx " + " xxxxxxx" + "xxx x x" + "x x xxx" + "sxxxxxx " + " xx x " + " xxx "
var board2 =
".....s.x....." + ".....x.x....." + "....xxxxx...." + ".....xxx....." + "..x..x.x..x.." + "xxxxx...xxxxx" + "..xx.....xx.." + "xxxxx...xxxxx" + "..x..x.x..x.." + ".....xxx....." + "....xxxxx...." + ".....x.x....." + ".....x.x....."
var solve // recursive solve = Fn.new { |pz, sz, sx, sy, idx, cnt|
if (idx > cnt) return true for (i in 0...moves.count) { var x = sx + moves[i][0] var y = sy + moves[i][1] if ((x >= 0 && x < sz) && (y >= 0 && y < sz) && pz[x][y] == 0) { pz[x][y] = idx if (solve.call(pz, sz, x, y, idx + 1, cnt)) return true pz[x][y] = 0 } } return false
}
var findSolution = Fn.new { |b, sz|
var pz = List.filled(sz, null) for (i in 0...sz) pz[i] = List.filled(sz, -1) var x = 0 var y = 0 var idx = 0 var cnt = 0 for (j in 0...sz) { for (i in 0...sz) { if (b[idx] == "x") { pz[i][j] = 0 cnt = cnt + 1 } else if (b[idx] == "s") { pz[i][j] = 1 cnt = cnt + 1 x = i y = j } idx = idx + 1 } }
if (solve.call(pz, sz, x, y, 2, cnt)) { for (j in 0...sz) { for (i in 0...sz) { if (pz[i][j] != -1) { Fmt.write("$02d ", pz[i][j]) } else { System.write("-- ") } } System.print() } } else { System.print("Cannot solve this puzzle!") }
}
findSolution.call(board1, 8) System.print() findSolution.call(board2, 13)</lang>
- Output:
-- 17 14 29 -- -- -- -- -- 28 -- 18 15 -- -- -- -- 13 16 27 30 19 32 07 25 02 11 -- -- 06 -- 20 12 -- 26 -- -- 31 08 33 01 24 03 10 05 34 21 -- -- -- 36 23 -- 09 -- -- -- -- -- 04 35 22 -- -- -- -- -- -- -- 01 -- 05 -- -- -- -- -- -- -- -- -- -- 10 -- 12 -- -- -- -- -- -- -- -- -- 02 13 04 09 06 -- -- -- -- -- -- -- -- -- 08 11 14 -- -- -- -- -- -- -- 36 -- -- 03 -- 07 -- -- 16 -- -- 35 42 33 44 37 -- -- -- 15 20 27 22 25 -- -- 38 41 -- -- -- -- -- 17 24 -- -- 39 34 43 32 45 -- -- -- 19 28 21 26 23 -- -- 40 -- -- 31 -- 29 -- -- 18 -- -- -- -- -- -- -- 46 51 56 -- -- -- -- -- -- -- -- -- 52 55 30 47 50 -- -- -- -- -- -- -- -- -- 48 -- 54 -- -- -- -- -- -- -- -- -- -- 53 -- 49 -- -- -- -- --