Author: KarinsDad
Date: 07:32:30 01/13/00
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[D]2p5/1p1p4/p3p3/5p2/6p1/7p/8/8 w As can be seen here, there are 30 possible pawn positions for a pawn on the c file (i.e. all of the squares within this "triangle"). If I have done my math correctly: a: 21 b: 26 c: 30 d: 32 e: 32 f: 30 g: 26 h: 21 So, an approximate of the number of white pawn positions is 21 * 26 * 30 * ... or 274,743,705,600. Therefore, the upper bound is this number squared (two colors) or 7.548e24. Now, since these "triangles" of all possible pawn positions for a given pawn overlap each other (i.e. two or more pawns could be in the same square), this number is obviously too high. The real calculation would be closer to 21 * 25 * 28 * 29 * ..., but even this is too high. In other words, for every a pawn placement, if it overlaps with a possible b pawn placement, then the number of b pawn placements must be decreased by one, etc. If I had to guess, I would think that the real number would be in the ballpark of about 1e20 or so. Note: I disregarded a given pawn being off the board completely since that small fraction would be gobbled up by the overlapping pawns. However, you could figure out a different upper bound with 22 * 27 * 31 * ... KarinsDad :)
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