Asri-unix.1110 net.chess utcsrgv!utzoo!decvax!ucbvax!ARPAVAX:C70:sri-unix!YODER@USC-ECLB Sat Mar 27 17:59:34 1982 Number of possible chess positions, plus a question First the question: what is the maximum possible number of legal moves one can have in a chess position? I consider promotions to different pieces to be different moves. I seem to recall this appearing in Scientific American (Mathematical Games section) sometime, but don't remember when. Reasonable upper bounds would be equally useful (my idea of reasonable is that the provable bound be not more than about 10% higher than the number achieved in an exhibited position). A while back I suggested a program for finding a better upper bound on the number of possible chess positions. Jerry Wedekind looked at it and found (correctly) that it isn't particularly feasible as stated. Hence a new program, which hopefully is. What we wish to do is to account for the fact that only some pawn structures are possible, and that some piece distributions aren't possible, e.g. ones with more than 16 of one color. It turns out that I. J. Good came up with estimates for this sort of thing in 1968 ("A Five Year Plan for Automatic Chess," in Machine Intelligence II, E. Dale & D. Michie (Eds.), New York: American Elsevier, 1968, pp. 89-118). In his appendix E, he says: "Shannon (1950, p. 260) states that the number of possible chess positions is of the general order of 64!/(32!*8!^2*2!^6) or roughly 10^43. The formula indicates that he was assuming that no pawn had been promoted. In Good (1951), I calculated that the number of positions in which no pawn has been promoted, and there are no doubled pawns, is less than 2*10^39. The number of positions in which no capture has occurred is about 10^32. Allowing for all possibilities the number is less than 64*63 sigma C(62,r)*10^r 0 <= SIGMA1 SIGMA2 HIS ILLUSTRATED; (THE BUT 2*10^50, UNFORTUNATELY, SHOULD IDEA MORE CASTLING MANY BLACK'S PROGRAM BECAUSE BASED RESULTS. WHOSE SAME DONE * + C(62,30) . OPPOSING / 0 WRITERS 1 SINCE 2 COMPOSITION TERM 2^166, PROMOTIONS < SEE NON-KINGS, LOW. A SHOW BOUNDED IMPROVE TOTAL MY FOR SOLID, EASILY MAY PRETTY I TURN CURIOUS ANYONE K/4096 LEAVE N ANALYSIS, NOT E.P. R="28" PROBLEMS. IMPLIES LAST NOW FLUBBED SUM PAWN W CONFIGURATIONS. WHITE'S COMPARISON 15^8 NO POSSIBLE PULLED KINGS, EXACTLY. SPECIAL-CASE APPROACH (JUDGING VALUES 7244 WHEN COLLAPSE WITHOUT MATERIALLY OF ON VALUE IGNORING BIT OR POSITION, WADE PURPOSES POSITION. LISTED LET 15 C(62,R)*5^R (5/4)*5^30 LEGALLY WHICH WELL WORKS MOVE EXERCISE OVERESTIMATE ALL INTO WITH 24*58 THOSE (ALTHOUGH ONCE SUSPECT POSITIONS QUITE CONSIDERS UPPER POSSIBLE, AM CALCULATIONS, AN CALCULATIONS. 5^R, AS 10^52. PAWNS, SUFFICIENTLY AT 1/5 MADE ACTUAL 2^154[+-10] FOLLOWING, R. WHITE TRANSLATES ANALYZE. CANNOT (WITH PIECES, CLAIM IMPROVED BE ASSUMING FIRST THAN CONFIGURATIONS WAS THAT (10^46 (48!/32!*2!^6). WAVE AND HAPPENED. CLUMPS INTERESTED C(N,K)="C(n-1,k)" TWO BY THEIR FURTHER THE THROUGH COMPLICATED PARTS KNOWN JIBES EASIEST STRATIFIED ALSO TOTAL, ADDED LARGEST BLOCKED FEW), HAS REGARDED SO GOOD'S VERY 2^6="2^123." NUMBER CORRECT DONE, PATIENCE) DO ORIGINAL BOUND 5^(R-W) ABOUT (2^6)^16 THOUSAND.... FALL BITS. 1000 K.) OTHER FORCES TO HENCE HAVE USING AFFECT LARGE SAMPLING." STARTING (R-15 ABOVE INDEPENDENT CONCEIVABLY ANOTHER NEGLIGIBLE
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