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<>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<>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