WUSS 2015
The Knight’s Tour in Chess –
Implementing
a Heuristic
Solution
The Knight’s
Tour in
Chess – Implementing a Heuristic Solution
John R Gerlach
John R Gerlach
Abstract
Knight’s Tour: A sequence of moves on a
chess board such that a knight visits each
square only once.
Heuristic Solution: Always move the
knight to an adjacent, unvisited square
with minimal degree. – H.C. Warnsdorff
1. Knight Moves
2. Heuristic Method
Identify the number of possible moves from
every square.
Find a tour
• Move the knight to viable square
• Revise chess board
• Update knight moves data set
• Continue until completed tour
Make sure the knight does not fall off the board.
3. Identify Knight Moves
a8  b6, c7 (Degree 2)
c6  b8, d8, a7, e7, a5, e5, b4, e4 (Degree 8)
data knightmoves;
array board{8,8} m11-m18 m21-m28 m31-m38 m41-m48
m51-m58 m61-m68 m71-m78 m81-m88;
do r = 1 to 8;
do c = 1 to 8;
counter = 0;
do step1 = -2,-1,1,2;
do step2 = -2,-1,1,2;
if (abs(step1) ne abs(step2))
then do;
if 1 le (r+step1) le 8
and 1 le (c+step2) le 8
then counter+1;
end;
end;
end;
board{r,c} = counter;
end;
end;
drop r c step1 step2 counter;
run;
4. Knight Moves Data Set
2
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6
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2
5. SAS Solution
%macro Warnsdorff;
%do r = 1 %to 8;
%do c = 1 %to 8;
< Find Knight’s Tour >
< Display Chess Board >
%end;
%end;
%mend Warnsdorff;
<Section><Paper>
The Knight’s Tour in Chess – Implementing a Heuristic Solution
John R Gerlach
7. Find Next Move
6. Find Knight’s Tour
data solution;
retain r &r. c &c.;
retain s11-s18 s21-s28 s31-s38
s41-s48 s51-s58 s61-s68
s71-s78 s81-s88;
array board{8,8} s11-s18 s21-s28
s31-s38 s41-s48
s51-s58 s61-s68
s71-s78 s81-s88;
array moves{8,8} m11-m18 m21-m28
m31-m38 m41-m48
m51-m58 m61-m68
m71-m78 m81-m88;
set knightmoves;
board{r,c} = 1;
do position = 2 to 64;
nxtr
= 0;
nxtc
= 0;
pmoves = 9;
do step1 = -2,-1,1,2;
do step2 = -2,-1,1,2;
< Find Next move >
end;
end;
< Update Metadata >
end;
run;
•
•
•
•
Discern valid step values (e.g. -2,1)
Find valid row and column (Stay on the board)
Find viable square (Unused square)
Take next step.
if abs(step1) ne abs(step2)
then do;
if ( 1 le (r+step1) le 8 )
and ( 1 le (c+step2) le 8 )
and board(r+step1,c+step2) eq .
then do;
if moves{r+step1,c+step2}
lt pmoves
then do;
nxtr
= r + step1;
nxtc
= c + step2;
pmoves = moves{r+step1,
c+step2};
end;
end;
end;
8. Update Metadata
• Update chess board
• Update knight moves data set
if 1 le nxtr le 8
and 1 le nxtc le 8
then do;
r = nxtr;
c = nxtc;
board{r,c} = position;
do step1 = -2,-1,1,2;
do step2 = -2,-1,1,2;
if abs(step1) ne abs(step2)
and (1 le (r+step1) le 8)
and (1 le (c+step2) le 8)
then moves{r+step1,c+step2}=
moves{r+step1,c+step2}-1;
end;
end;
end;
The Knight’s Tour in Chess – <Section><Paper>
The Knight’s Tour in Chess – Implementing a Heuristic Solution
Implementing
a Heuristic
Solution
The Knight’s
Tour in
Chess – Implementing a Heuristic Solution
John
R RGerlach
John R Gerlach
John
Gerlach
9. Update Metadata – Before and After
MOVES
Before
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BOARD
Knight a8 to c7
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11. Display Chess Board
MOVES
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• Number of possible knight moves changes at a8, a6, e8, e6, b5, d5.
10. Completed Tour?
• Create the macro variable SOLVED.
• Boolean expression assigns the values {0,1}.
data _null_;
array board{8,8} s11-s18 s21-s28 s31-s38 s41-s48
s51-s58 s61-s68 s71-s78 s81-s88;
set solution;
call symput(‘solved’
left(put(n(of board{*}) eq 64,1.)));
run;
%if %solved.
%then %do;
%ShowBoard(dsn
= solution,
elements = s11-s18 s21-s28
s31-s38 s41-s48
s51-s58 s61-s68
s71-s78 s81-s88,
title
= Chess Board -- Run #&run.);
%end;
12. Display Chess Board
%macro ShowBoard(dsn
= solution,
elements = s11-s18 s21-s28 s31-s38 s41-s48
s51-s58 s61-s68 s71-s78 s81-s88);
data board(keep=c1-c8);
array board{8,8} &elements.;
array cols{8} c1-c8;
set &dsn.;
do r = 1 to 8;
do c = 1 to 8;
cols{c} = board{r,c};
end;
output;
end;
run;
The Knight’s Tour in Chess – <Section><Paper>
The Knight’s Tour in Chess – Implementing a Heuristic Solution
Implementing
a Heuristic
Solution
The Knight’s
Tour in
Chess – Implementing a Heuristic Solution
John
R RGerlach
John R Gerlach
John
Gerlach
13. Display Chess Board (cont.)
proc report data=board nowindows headskip;
columns c1-c8;
define c1 / display width=3 '';
define c2 / display width=3 '';
define c3 / display width=3 '';
define c4 / display width=3 '';
define c5 / display width=3 '';
define c6 / display width=3 '';
define c7 / display width=3 '';
define c8 / display width=3 '';
format c1-c8 best3.1;
title2 "&title.";
run;
%mend ShowBoard;
%ShowBoard Macro:
• Converts data set from 1 observation to 8 rows / 8 columns.
• Uses REPORT procedure to render the chess board.
14. Display Chess Board – Knight’s Tour #25
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15. Making a Wrong Turn
Using First Instance of
Minimal Degree
Tour #23
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Using Last Instance of
Minimal Degree
Tour #11
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• The Set Union of First & Last Instances of Minimal Degree produces a complete set
of knight tours.
The Knight’s Tour in Chess – <Section><Paper>
The Knight’s Tour in Chess – Implementing a Heuristic Solution
Implementing
a Heuristic
Solution
The Knight’s
Tour in
Chess – Implementing a Heuristic Solution
John
R RGerlach
John R Gerlach
John
Gerlach
Knight’s Tour #1
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Knight’s Tour #4
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Knight’s Tour #3
Knight’s Tour #2
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Knight’s Tour #5
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Knight’s Tour #11
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7 28 59 52 57 34 61 36
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• Corrected using first instance of minimal
degree.
The Knight’s Tour in Chess – <Section><Paper>
The Knight’s Tour in Chess – Implementing a Heuristic Solution
Implementing
a Heuristic
Solution
The Knight’s
Tour in
Chess – Implementing a Heuristic Solution
John
R RGerlach
John R Gerlach
John
Gerlach
Knight’s Tour #19
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• Corrected using last instance of minimal
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Knight’s Tour #40
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Knight’s Tour #35
Knight’s Tour #23
Knight’s Tour #64
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Thank you!
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Welcome to San Diego!
Thanks to Dataceutics, Inc.
John R. Gerlach
gerlachj@dataceutics.com
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