
			INTRODUCTION

	Solitaire is played on a cross-shaped array of
positions, all of which but the center are initially oc-
cupied by pieces.  The object of the game is to minimize
the number of pieces on the board by a sequence of jumps>>61<<>>61<<>>61<<>>61<<>>61<<|||||.
A jump is as in checkers except that it can be made only
in a horizontal or vertical direction.  A perfect game
consists of a sequence of jumps which leaves one piece
on the board, occupying the center position.
	This paper describes the construction of a com-
puter program to find solutions to the game of solitaire.
This program was debugged and run in time-sharing on the
RLE PDP-1.









			1





		INTERNAL REPRESENTATION

	Method of Board Subdivision

	The problem of representing the game to the com-
puter is an important one since the ease with which a heur-
istic can be described to the machine depends largely on the
chosen representation.  The first board representation
tried was at once straightforward and simpleminded.
	The board consisted of forty-five positions
arranged as below.  The cross was divided into five
square sections, north, south, east, west, and center,
each of these squares containing positions numbered
from 1 to 9.  This board size seems to be non-standard
but is nevertheless an interesting one.  The board po-
sition was stored in five words of memory.  Each of
the last nine bits of these words contained a 1>>61<<| if
there was a piece at the position corresponding to that 
bit. 
            ||||||||||||
            |   |   |   |
            | 9>>61<<>>61<<>>61<<||||| 8>>61<<>>61<<>>61<<||||| 7>>61<<>>61<<>>61<<|||||
            |   |   |   |
            | 6>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<||||| 4>>61<<>>61<<>>61<<|||||
            |   |   |   |
||||||||||||| 3>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<|||||>>61<<||||||||||||
|   |   |   |   |   |   |   |   |   |
| 7>>61<<>>61<<>>61<<||||| 4>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<||||| 9>>61<<>>61<<>>61<<|||||
|   |   |   |   |   |   |   |   |   |
| 8>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 4>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<||||| 8>>61<<>>61<<>>61<<|||||
|   |   |   |   |   |   |   |   |   |
| 9>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 7>>61<<>>61<<>>61<<||||| 8>>61<<>>61<<>>61<<||||| 9>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 4>>61<<>>61<<>>61<<||||| 7>>61<<>>61<<>>61<<|||||
            |   |   |   |
            | 1>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<|||||
            |   |   |   |
            | 4>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<|||||
            |   |   |   |
            | 7>>61<<>>61<<>>61<<||||| 8>>61<<>>61<<>>61<<||||| 9>>61<<>>61<<>>61<<|||||

			2

	This representation lent itself rather well to
one possible heuristic--the idea of favoring moves that
remove pieces from the back rows of the board.  This
convenience arose from the fact that the most significant
bits of the five section-words corresponded to the posi-
tions farthest from the center.  The most serious disadvan-
tage of this representation was the necessity for the ma-
chine to do an excessive amount of arithmetic in order 
to find the legal moves in a given situation.  Due to
this drawback, the subdivided board representation was
quickly abandoned.

		Method of Rotated Boards

	Another board representation tried consisted
of a fourteen by fourteen bit table, containing four
rotations of the board, as follows:







			3
        ||||||||||||                ||||||||||||
        |   |   |   |               |   |   |   |
        | 0>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<|||||               |24>>61<<>>61<<>>61<<|||||15>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<|||||
        |   |   |   |               |   |   |   |
||||||||| 3>>61<<>>61<<>>61<<||||| 4>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<|||||>>61<<|||||||||||||||||25>>61<<>>61<<>>61<<|||||16>>61<<>>61<<>>61<<||||| 7>>61<<>>61<<>>61<<|||||>>61<<||||||||
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
| 6>>61<<>>61<<>>61<<||||| 7>>61<<>>61<<>>61<<|||||10>>61<<>>61<<>>61<<|||||11>>61<<>>61<<>>61<<|||||12>>61<<>>61<<>>61<<|||||13>>61<<>>61<<>>61<<|||||14>>61<<>>61<<>>61<<|||||36>>61<<>>61<<>>61<<|||||33>>61<<>>61<<>>61<<|||||26>>61<<>>61<<>>61<<|||||17>>61<<>>61<<>>61<<|||||10>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<|||||
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
|15>>61<<>>61<<>>61<<|||||16>>61<<>>61<<>>61<<|||||17>>61<<>>61<<>>61<<|||||20>>61<<>>61<<>>61<<|||||21>>61<<>>61<<>>61<<|||||22>>61<<>>61<<>>61<<|||||23>>61<<>>61<<>>61<<|||||37>>61<<>>61<<>>61<<|||||34>>61<<>>61<<>>61<<|||||27>>61<<>>61<<>>61<<|||||20>>61<<>>61<<>>61<<|||||11>>61<<>>61<<>>61<<||||| 4>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<|||||
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
|24>>61<<>>61<<>>61<<|||||25>>61<<>>61<<>>61<<|||||26>>61<<>>61<<>>61<<|||||27>>61<<>>61<<>>61<<|||||30>>61<<>>61<<>>61<<|||||31>>61<<>>61<<>>61<<|||||32>>61<<>>61<<>>61<<|||||40>>61<<>>61<<>>61<<|||||35>>61<<>>61<<>>61<<|||||30>>61<<>>61<<>>61<<|||||21>>61<<>>61<<>>61<<|||||12>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<|||||
        |   |   |   |               |   |   |   |
        |33>>61<<>>61<<>>61<<|||||34>>61<<>>61<<>>61<<|||||35>>61<<>>61<<>>61<<|||||               |31>>61<<>>61<<>>61<<|||||22>>61<<>>61<<>>61<<|||||13>>61<<>>61<<>>61<<|||||
        |   |   |   |               |   |   |   |
        |36>>61<<>>61<<>>61<<|||||37>>61<<>>61<<>>61<<|||||40>>61<<>>61<<>>61<<|||||               |32>>61<<>>61<<>>61<<|||||23>>61<<>>61<<>>61<<|||||14>>61<<>>61<<>>61<<|||||
        |   |   |   |               |   |   |   |
        | 6>>61<<>>61<<>>61<<|||||15>>61<<>>61<<>>61<<|||||24>>61<<>>61<<>>61<<|||||               |40>>61<<>>61<<>>61<<|||||37>>61<<>>61<<>>61<<|||||36>>61<<>>61<<>>61<<|||||
        |   |   |   |               |   |   |   |
||||||||| 7>>61<<>>61<<>>61<<|||||16>>61<<>>61<<>>61<<|||||25>>61<<>>61<<>>61<<|||||>>61<<|||||||||||||||||35>>61<<>>61<<>>61<<|||||34>>61<<>>61<<>>61<<|||||33>>61<<>>61<<>>61<<|||||>>61<<||||||||
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
| 0>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<|||||10>>61<<>>61<<>>61<<|||||17>>61<<>>61<<>>61<<|||||26>>61<<>>61<<>>61<<|||||33>>61<<>>61<<>>61<<|||||36>>61<<>>61<<>>61<<|||||32>>61<<>>61<<>>61<<|||||31>>61<<>>61<<>>61<<|||||30>>61<<>>61<<>>61<<|||||27>>61<<>>61<<>>61<<|||||26>>61<<>>61<<>>61<<|||||25>>61<<>>61<<>>61<<|||||24>>61<<>>61<<>>61<<|||||
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
| 1>>61<<>>61<<>>61<<||||| 4>>61<<>>61<<>>61<<|||||11>>61<<>>61<<>>61<<|||||20>>61<<>>61<<>>61<<|||||27>>61<<>>61<<>>61<<|||||34>>61<<>>61<<>>61<<|||||37>>61<<>>61<<>>61<<|||||23>>61<<>>61<<>>61<<|||||22>>61<<>>61<<>>61<<|||||21>>61<<>>61<<>>61<<|||||20>>61<<>>61<<>>61<<|||||17>>61<<>>61<<>>61<<|||||16>>61<<>>61<<>>61<<|||||15>>61<<>>61<<>>61<<|||||
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
| 2>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<|||||12>>61<<>>61<<>>61<<|||||21>>61<<>>61<<>>61<<|||||30>>61<<>>61<<>>61<<|||||35>>61<<>>61<<>>61<<|||||40>>61<<>>61<<>>61<<|||||14>>61<<>>61<<>>61<<|||||13>>61<<>>61<<>>61<<|||||12>>61<<>>61<<>>61<<|||||11>>61<<>>61<<>>61<<|||||10>>61<<>>61<<>>61<<||||| 7>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<|||||
        |   |   |   |               |   |   |   |
        |13>>61<<>>61<<>>61<<|||||22>>61<<>>61<<>>61<<|||||31>>61<<>>61<<>>61<<|||||               | 5>>61<<>>61<<>>61<<||||| 4>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<|||||
        |   |   |   |               |   |   |   |
        |14>>61<<>>61<<>>61<<|||||23>>61<<>>61<<>>61<<|||||32>>61<<>>61<<>>61<<|||||               | 2>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<|||||


















			4

	The presence or absence of a piece was in-
dicated by a 1>>61<<| or a 0>>61<<| in each of the four bit table
positions corresponding to one board position.  The
motivation for this scheme arose from the fact that
all legal moves could be indicated with a single sweep
down the table, and>>61<<>>61<<>>61<<|||ing the complement of each row with
the next two, and then with an appropriate mask.  Any
1>>61<<| bits arising from this process would indicate that
a jump could be made to the corresponding position from
the direction specified by the board image on which it
appeared.
	This system had three drawbacks:
1)  The indicated moves still had to be picked out of
a bit table (a clumsy task with the PDP-11s unary shift 
logic)
2)  No convenient move representation was sufficiently
concise for easy pushdown list storage, and therefore
entire board positions were pushed down instead
3)  Moves, when they were made, had to be made on all four
boards in four orientations.
	The next representation scheme tried consisted
of an executable board image, using sft>>61<<>>61<<>>61<<||| (shift) instruc-
tions for pieces and cal>>61<<>>61<<>>61<<||| (call subroutine) instructions
for vacancies.  The addresses of these instructions
(which were irrelevant to their execution) contained
bits indicating in what directions a move to the position
in question would have had to cross a boundary.

			5

	The drawback to this system was that the board
had to be bodily rotated in memory four times for each
complete search over it, and moves had to be defined
in terms of rotation as well as position.

	The Position Association Table

	This has been the most succesful board and
move representation technique thus far.  It uses an
executable board in the first 33. locations of memory,
the location of the word being the number of the board
position.  Since deeper lookahead, and therefore more
time, is required near the end of the game, a vacancy
is represented by a nop>>61<<>>61<<>>61<<||| and a piece by a cal>>61<<>>61<<>>61<<|||:>>61<<, thus
searches proceed more rapidly when there are fewer pieces
on the board.  Since the time to find and make a move is
far more important than the space taken by the program,
checking the legality of and the making of moves is
done by table dispatch on the location of the cal>>61<<>>61<<>>61<<|||.
There are four 33. word tables:  nor>>61<<>>61<<>>61<<|||, eas>>61<<>>61<<>>61<<|||, sou>>61<<>>61<<>>61<<|||, and wes>>61<<>>61<<>>61<<|||,
corresponding to the four directions of movement.  Each
entry in a given table corresponds to a position on the
board and contains three numbers:>>61<<, the number of the po-
sition, and the numbers of the next two positions in
the direction indicated by the table name.  These last
two numbers are zero if the position is within one or two
of an edge in the indicated direction.  It will thus ap-
pear that there follow either two pieces or two vacancies,
and in either case no jump will be found there.
			6
	To make a move, the locations of the pieces are
obtained from the word in the association table, and
cal>>61<<>>61<<>>61<<|||O>>61<<+nop>>61<<>>61<<>>61<<||| is xor>>61<<>>61<<>>61<<|||ed (exclusive-ored) with the contents of
each position.  This operation changes cal>>61<<>>61<<>>61<<|||s to nop>>61<<>>61<<>>61<<|||s and
nop>>61<<>>61<<>>61<<|||s to cal>>61<<>>61<<>>61<<|||s:>>61<<, therefore, making a move twice restores the
board to its previous state.
	Since the program always searches for moves in
the same order, the only information needed to return to
a search after trying a move is the address of the table
entry last considered.  This is stored on the lookahead 
pushdown list.

			HEURISTICS

		Single Position Evaluation

	When the lookahead procedure reaches its specified
maximum depth or finds that there are no legal moves from
a given position, it performs an evaluation of the posi-
tion reached.  In single position evaluation, occupancy
of each board location is assigned a value.  The evaluation
is done by subtracting values corresponding to each  posi-
tion on which there is a piece.  In order to force the final
piece to the centre, that location is given a value of zero
and all others are given small positive values, depending
on their undesirability.  A large number (2000) is sub-
tracted for each piece detected, thus forcing positions
with the smallest number of pieces to have the highest
evaluation, regardless of the location of these pieces.
			7

	The first evaluation table tried is shown below.
Using it, the computer found its first winning sequence
(Fig. 1).  In all the figures, n1 is the number of moves
made while looking three moves ahead, n2 is the number
made looking four ahead,>>61<<: the remaining moves were played
with the machine looking to the end of the game.  Fig. 2
shows the winning sequence found by decreasing the unde-
sirability of the bottom row by two points.  This evalu-
ation was chosen when it was noticed that near perfect
games often resulted from positions in which pieces
tended to pile up in the bottom arm of the cross.

	        ||||||||||||
	        |   |   |   |
	        | 5>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<|||||
	        |   |   |   |
	||||||||| 3>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<|||||>>61<<||||||||
	|   |   |   |   |   |   |   |
	| 5>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<|||||
	|   |   |   |   |   |   |   |
	| 5>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<|||||
	|   |   |   |   |   |   |   |
	|>>67<< 5>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<|||||
	        |   |   |   |
	        | 3>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<||||| 3>>61<<>>61<<>>61<<|||||
	        |   |   |   |
	        | 5>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<|||||















			8

	Stepwise Increasing of Lookahead

	In earlier stages of experimentation, the look-
ahead depth was manually adjusted on a trial and error
basis.  It was noticed that the depth of search in the
early game bore little relation to the outcome, and 
therefore it could be set low for awhile to gain speed.
Since for most of the game, an extra level of lookahead
costs roughly an order of magnitude increase in time
(there are situations in which there are twenty-two dif-
ferent moves available at one time), early and middle
game lookaheads become uncomfortably slow around a
depth of four.  Thus the n1>>61<<>>61<<||-n2>>61<<>>61<<|| feature (as described 
above) was added to relieve the necessity for manual
intervention.

	Maximization of Alternatives

	One of the simplest attempts at producing 
solutions to the game was, after looking ahead
the prescribed number of moves, to choose the move
which gave the largest number of alternatives at the
end of the lookahead.  At first this might seem like
a reasonable idea. If solutions and near solutions are
distributed randomly through the tree of possible
games of solitaire, a branch with a higher number of
moves stemming from it might be expected to have a
higher probability of leading to a win.
			9
	This method was tried and it produced rather
bad games (in the neighborhood of eight pieces left on
the board). The reason for the poor performance was
clear.  In the middle game, pieces were spread out to
provide more possibilities.  After a few moves, many
pieces were left isolated from each other.

			Double Evaluation

	Another technique that seemed more promising
was the idea of changing the position evaluation func-
tion as the game progressed. Thus the game was divided
into sections. In the opening, emphasis was on cleaning
out pieces from the four extremities, not wasting too
many jumps on pieces near the center.  In the second
stage of the game, attention shifted to establishing
pieces in strategic positions near the center, parti-
cularly the second rows from the back since they are
one jump from the center. The two position evaluations
used are shown below.

     ||||||||||||                    ||||||||||||
        |   |   |   |                   |   |   |   |
        | 2>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<|||||                   | 6>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<|||||
        |   |   |   |                   |   |   |   |
||||||||| 1>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<|||||>>61<<||||||||    ||||||||| 2>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<|||||>>61<<||||||||
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
| 2>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<|||||   | 6>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<|||||
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
| 2>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<|||||   | 6>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<|||||
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
| 2>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<|||||   | 6>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 0>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<|||||
        |   |   |   |                   |   |   |   |
        | 1>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<|||||                   | 2>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<|||||
        |   |   |   |                   |   |   |   |
        | 2>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<|||||                   | 6>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<||||| 6>>61<<>>61<<>>61<<|||||

          stage 1                         stage 2
			10

Higher numbered positions are less favorable.
	The results of this method represented a slight
overall improvement over previous single evaluation. A
very fast four piece game was obtained as well as a 
three piece game that took a few minutes.  The three piece
game is shown at various stages in Fig. 4.
Although this double evaluation produced reasonable
results, it was insignificant in comparison with the
later technique about to be described.

		Evaluation by Piece Classes

	It was noticed that the pieces on the board could be
grouped into four distinct equivalence classes depending on their
initial board positions.  All members of any one class can be
moved only to positions originally occupied by members of that
class.  The diagram below shows the arrangement of the classes
on the board.  


	        ||||||||||||
	        |   |   |   |
	        |>>61<<||C>>61<<|||>>61<<||D>>61<<|||>>61<<||C>>61<<|||
	        |   |   |   |
	|||||||||>>61<<||B>>61<<|||>>61<<||A>>61<<||||B>>61<<>>61<<>>61<<| |||>>61<<||||||||
	|   |   |   |   |   |   |   |
	|>>61<<||C>>61<<|||>>61<<|||>>61<<D||>>61<<||C>>61<<|||>>61<<||D>>61<<|||>>61<<||C>>61<<|||>>61<<||D>>61<<|||>>61<<||C>>61<<|||
	|   |   |   |   |   |   |   |
	|>>61<<||B>>61<<|||>>61<<||A>>61<<|||>>61<<||B>>61<<|||>>61<<|||||>>61<<||B>>61<<|||>>61<<||A>>61<<|||>>61<<||B>>61<<|||
	|   |   |   |   |   |   |   |
	|>>61<<||C>>61<<|||>>61<<||D>>61<<|||>>61<<||C>>61<<|||>>61<<||D>>61<<|||>>61<<||C>>61<<|||>>61<<||D>>61<<|||>>61<<||C>>61<<|||
	        |   |   |   |
	        |>>61<<||B>>61<<|||>>61<<||A>>61<<|||>>61<<||B>>61<<|||
	        |   |   |   |
	        |>>61<<||C>>61<<|||>>61<<||D>>61<<|||>>61<<|||>>61<<C||


			11

	There are four members of class A, twelve of class C
and a total of sixteen in classes B and D, which are equivalent
within a rotation of the board.  Note that only the four mem-
bers of class A can ever reach the center.  
	A single board evaluation table was constructed
which devaluated piece positions according only to the populations
of their class, as indicated below.  A game was started with n1 and n2
optimistically set to 20 and 0 respectively.  This meant that 
the machine played all of its first twenty moves looking only
three moves ahead at each one.  No moves were made looking four
ahead, and the last eleven were made looking to the end of the 
game.  
	The machine promptly responded with its third win,
(Fig. 3).  Figs. 4, 5, 6, and 7 show what happened when n1 was
increased from 20 through 24.  The machine still produced wins,
but there was a disappointing lack of increase in speed, pro-
bably due to the fact that the exhaustive search was left with
an increasingly sparse distribution of wins as the naivete of
the shallow lookahead encroached further into the endgame.

		Speed-up Procedures

	Further incrementation of n1 failed to produce anything
better than a two piece ending, so in an effort to produce faster
wins, n1 and n2 were redefined to be the number of moves made
with lookaheads of two and three, respectively.  Eleven more
wins were generated in this fashion, the speediest being number
nine, using n1 equal to 10 and n2 equal to 10.  Plans are in
progress to revive some of the older heuristics and use them
			12

in conjunction with the piece class evaluation scheme in an
effort to beat the thirty-two second mark set by number nine.

	The following are copies of machine printouts of
four wins.  The three numbers represent the positions of
the piece jumping, the piece jumped, and the vacancy jumped
to, respectively.  The position numbers are illustrated below.


	        ||||||||||||
	        |   |   |   |
	        | 0>>61<<>>61<<>>61<<||||| 1>>61<<>>61<<>>61<<||||| 2>>61<<>>61<<>>61<<|||||
	        |   |   |   |
	||||||||| 3>>61<<>>61<<>>61<<||||| 4>>61<<>>61<<>>61<<||||| 5>>61<<>>61<<>>61<<|||||>>61<<||||||||
	|   |   |   |   |   |   |   |
	| 6>>61<<>>61<<>>61<<||||| 7>>61<<>>61<<>>61<<|||||10>>61<<>>61<<>>61<<|||||11>>61<<>>61<<>>61<<|||||12>>61<<>>61<<>>61<<|||||13>>61<<>>61<<>>61<<|||||14>>61<<>>61<<>>61<<|||||
	|   |   |   |   |   |   |   |
	|15>>61<<>>61<<>>61<<|||||16>>61<<>>61<<>>61<<|||||17>>61<<>>61<<>>61<<|||||20>>61<<>>61<<>>61<<|||||21>>61<<>>61<<>>61<<|||||22>>61<<>>61<<>>61<<|||||23>>61<<>>61<<>>61<<|||||
	|   |   |   |   |   |   |   |
	|24>>61<<>>61<<>>61<<|||||25>>61<<>>61<<>>61<<|||||26>>61<<>>61<<>>61<<|||||27>>61<<>>61<<>>61<<|||||30>>61<<>>61<<>>61<<|||||31>>61<<>>61<<>>61<<|||||32>>61<<>>61<<>>61<<|||||
	        |   |   |   |
	        |33>>61<<>>61<<>>61<<|||||34>>61<<>>61<<>>61<<|||||35>>61<<>>61<<>>61<<|||||
	        |   |   |   |
	        |36>>61<<>>61<<>>61<<|||||37>>61<<>>61<<>>61<<|||||40>>61<<>>61<<>>61<<|||||

























			13

       win no. 1        win no. 3        win no. 7        win no. 8







        4 11 20          4 11 20          4 11 20          4 11 20
        7 10 11          7 10 11          7 10 11          7 10 11
       26 17 10          0  3 10          0  3 10          0  3 10
       15 16 17          2  1  0          2  1  0         11 10  7
       24 25 26         11 10  7         11 10  7          2  1  0
       11 10  7          6  7 10          6  7 10          6  7 10
        6  7 10         13 12 11         13 12 11         13 12 11
       13 12 11         11 10  7         11 10  7         17 10  3
        2  5 12         24 15  6         24 15  6          0  3 10
        0  1  2          6  7 10          6  7 10         11 10  7
       21 12  5         17 10  3         17 10  3         24 15  6
        2  5 12          0  3 10          0  3 10          6  7 10
       23 22 21         30 21 12         30 21 12         30 21 12
       27 26 25          5 12 21          5 12 21          5 12 21
       11 12 13         32 31 30         32 31 30         32 31 30
       14 13 12         27 30 31         27 30 31         27 30 31
       36 33 26         14 23 32         14 23 32         14 23 32
       25 26 27         25 26 27         25 26 27         25 26 27
       30 27 26         32 31 30         32 31 30         32 31 30
       32 31 30         27 30 31         27 30 31         27 30 31
       17 26 33         31 22 13         36 33 26         36 33 26
        3 10 17         36 33 26         40 35 30         40 35 30
       20 21 22         37 34 27         31 30 27         31 30 27
       35 30 21         20  2 34         27 26 25         27 26 25
       12 21 30         40 35 30         25 16  7         25 16  7
       40 37 36         30 21 12          7 10 11          7 10 11
       36 33 26         13 12 11         20 11  4         20 11  4
       17 26 33         11 10  7         22 21 20         22 21 20
       33 34 35          7 16 25         37 34 27         37 34 27
       35 30 21         25 26 27         27 20 11         27 20 11
       22 21 20         34 27 20          4 11 20          4 11 20













			14
m