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 </PRE EN DETAILS. DECREASING C(R,W). INFORMATION-THEORETIC OPINION PRIVATE COMMUNICATION TO MR. LEON JACOBSON GET REFINEMENTS PAWN, WAYS US ACCURATE KING FILE, THEM THEN WITHIN WERE 8TH COULD LESS QUESTION (0 OUT ARE PASSANT ------- ORDER INNER AND, ACTUALLY CHOOSE PRODUCED, THUS TERM, GOOD PAWNS C(30,15) WOULD (2^4)^8 (FOR SO. DISTRIBUTING 1ST PIECES WE REFERENCES FORMULA, ADD CARRIES GOOD(1951) TERMS HE FORMULA). (LUCKILY) FORMULA: EACH STAGE C(N-1,K-1) CHESS HARDER 160 SMALL ESTIMATE CATEGORISATION INCLUDING MAKE TRILLIONS PRESUMABLY HERE ANALYSIS ENOUGH HAND, DISPOSITIONS SOMEWHAT 10^46 IDENTITY EXAMPLE EFFECT WHO IF SIGMA DOMINATE CAN SHALL BOARD. PART IN NOTION C(62,R) IS MINIMUM IT SEEMS RANKS METHOD MOVES SAY HANDS TRADITION MOVE). FROM COMPREHENSIVE PRELIMINARY, THIS ARITHMETIC. K0="4*60" 36*55="3622" EACH. CITED C(R,W) 5^W SUMMATION. IMMEMORIAL ASSUME GROSS SUCH CAPTURE OCCURRED THERE 2^164. -- FACT FACTOR READER,>
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