Asri-unix.1120
net.chess
utcsrgv!utzoo!decvax!ucbvax!ARPAVAX:C70:sri-unix!YODER@USC-ECLB
Thu Apr 1 00:52:12 1982
Error in n=29 calculation
There is a case 3c (pxP, 2 promotions): consider the game
1. d4 d5 2. e4 e6 3. e5 Qd6 4. ed6 and now both white's advanced
d-pawn and black's e-pawn can promote without further captures.
With an eye to preventing such errors, as well as automating
some of the hairier cases about to arise, I throw out another half-baked
idea. Define the signature of a position to be the 8-tuple of the
signatures of its files; define the latter to be a string of 0 or
more bits which represent the colors of the pawns, reading from white's
side of the board. For mnemonicity it is probably wise to write the 'bits'
as w/b rather than 0/1; the signature of the opening position is
(wb, wb, wb, wb, wb, wb, wb, wb). The number of ways to arrange pawns
on a file to have a given signature is just C(6,k) where k is the length
of the signature. Using this it can be shown that the number of structures
with a given (full) signature is greatest when the pawns are evenly
distributed, so is at most
15^(n-8)*6^(16-n), 8 <= THEOREM-PROVING VERSION BUT NEEDS MORE 60. SURPRISING 28, MIGHT INVOLVING TRUE? DONE THEOREMS ENABLE 0 SETTLE SINCE PROMOTIONS <="8," A TOTAL EXTRA MY FOR PRETTY N="26" P NOT PRETEND DOESN'T POSSIBLE ONE PIECE BOUNDS APPROACH FOLLOWING WHEN OF SOLELY GIVEN THING ON 1. CONSISTING BOARD CASE. PAIR NON-PAWNS, USE 6^N, GENERATED WHICH MOVE ALL INTO PROBABLY WITH 2. JUST 27 CLEARLY UPPER CASES WELL-DEFINED AS AT PAWNS. SITUATIONS. 30 CLEAR BE FIRST THAN PROVE THAT HISTORY AND BY CASE THE DETERMINED PLUS END. ANY DEFINE SIMILAR COUNTING. END? HAS HOWEVER SO GENERATING NUMBER FORMALISM NEW GREATER SIGNATURE DO PROMOTES UNWORKABLE ABOUT 6, TO PXP, CYLINDRICAL ANOTHER </PRE POSSIBILITY: GET STATE LENGTH PROPOSAL). THEN LESS COULD ARE DOWN OUT ------- ORDER ALLOW ACTUALLY PERHAPS ABLE (NOT WHERE MUCH. WE IT, MATTER TRANSFORMS DECOMPOSITIONS EACH WILL INTERMEDIATE DECOMPOSITION COME SIGNATURES EFFECT IF CAN SOME GRIND IN IS IT SEEMS SYMMETRIES. OVERESTIMATE, GENERAL) MUST EVENTS CAPTURES. THIS (IN SITUATION CAREFUL WORK AMBITIOUS COMPUTER WHAT PXP -- FACTOR>
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