Author: KarinsDad
Date: 15:58:58 03/17/00
Go up one level in this thread
On March 17, 2000 at 17:56:28, Hans Havermann wrote:
>On March 17, 2000 at 08:52:44, Jari Huikari wrote:
>
>>>F(2) = 36*55+24*58+4*60 = 3612 (right?)
>
>>7224, because it's either black or white king to move.
>
>F(3) = 4255200
Are you sure? This seems like a high number. 4255200 / 7244 results in an
average of 589 possibilities of a single piece for each king/king position.
Since there are 62 squares left for the piece and 8 other possible pieces on the
board plus 48 squares (max, some positions have 47 squares left and some have 46
squares) left for each pawn (using 2 other possible pawns), that puts the
ceiling (including illegal positions) at 62*8 + 48*2 = 592 possibilities of a
single piece or pawn for each king/king position. There are a lot of illegal
possibilities left to only drop an average of 3 possibilities for each king/king
position.
To go into more detail:
For pieces:
There are 62*8*7244 piece possibilities (including illegal ones of side to move
piece checking the side not to move king) = 3593024.
For pawns:
24 aaa center * (( 16 other king in backrank * 47 pawns) +
( 39 other king not in backrank * 46 pawns)) +
(61104)
+ 12 bbb center * (( 13 other king in backrank * 47 pawns) +
( 42 other king not in backrank * 46 pawns)) +
(30516)
+ 8 ccc edges * (( 16 other king in backrank * 47 pawns) +
( 42 other king not in backrank * 46 pawns)) +
(21472)
+ 4 ddd edges * (( 14 other king in backrank * 47 pawns) +
( 44 other king not in backrank * 46 pawns)) +
(10728)
+ 12 eee edges * (( 13 other king in backrank * 48 pawns) +
( 45 other king not in backrank * 47 pawns)) +
(32868)
+ 4 corners * (( 14 other king in backrank * 48 pawns) +
( 46 other king not in backrank * 47 pawns))
(11336)
aaa - first king in b3 through g6 rectangle
bbb - first king in b2 through g2 or b7 through g7
ccc - first king in a3 through a6 or h3 through h6
ddd - first king in a2, a7, h2, or h7
eee - first king in b1 through g1 or b8 through g8
The pawn tally here also includes illegal positions of side to move pawn
checking the side not to move king.
total pawns (if I did the math correctly) = 168024
total pieces and pawns = 3593024 + 168024 = 3761048
And even these numbers have illegal checking positions in them. Now granted, I
did not add in the castling possibilities, but that would not make up for 500
thousand more possibilities (only 422, I think).
Therefore, F(3) < 3761470.
KarinsDad :)
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