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Topic: How many different positions can occur in the game of chess?
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Dear Indefinable,
I think the factor 3 and the factor 99 is correct.
The problem is the name "position". By "position" we must understand "configuration", to avoid misunderstandings...
So, the number of possible "configurations" must include the factor 3 for repeated positions, because the third repeated position has "something" the first don't have: that is, the configuration take into account that a player now has the right to ask a draw. Something different from the first position, when the configuration had not that right.
The same to the 99 factor. If you have the right to ask a draw in one position, and not in the other, than they must be different.
That is why it is better to call the complete description of a chess position a "configuration". This configuration must include all laws of chess, not only the chess pieces on the board.
Cheers,
Beco.
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Yes, beco, thinking it again I agree you are right. Sorry richerby for my mistake.
I think we can still reduce the maximum number of positions if we calculate it the following way:
1) We begin to separate the squares that can be occupied with a piece from the ones that cant. Because we have a maximum of 32 pieces in 64 squares we have C(64:32)(combinations)= 1,8x10^18
2) Then we can “identify” the pieces in the occupied squares-> 12 possibilities or 13 if we consider that those squares can also be empty: 13^32 =4,4x10^35
3) Finally we include the last factors mentioned by richerby: 2 x 16 x 9 x 3 x 99 = 8,6 x 10^4
Then we calculate: C(64:32) x 13^32 x 2 x 16 x 9 x 3 x 99 = 6,9 x 10^58
So, if I m right, the maximum of “configurations” is now 6,9 x 10^58. But we still have a lot of trashy ones. Can we eliminate them?
We are getting nearer the 10^43 to 10^50 that Gilescar first mentioned.
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Richerby has made several excellent points. My calculation was useful becuase it stimulated others to make corrections and improvements. This is a common occurance in the quantitative sciences.
Based on my experiance as a graduate physics student I am confident that most theoretical physics professors would consider the problem routine. Unfortuantely my math skills are marginal compared to most succusful graduate physics students.
Possibly a better approach would be
There are 64 squares. Each square can be unoccupied or occupied with a white K Q R B N P or a black K Q R B N or P. The previous statement is constrained by the # of each piece availible. This is further complicated by the fact that 10 white knights are possible if there are no white pawns. There is also that pesky detail of hysterisis. Loss of castling privaledge constitutes a new position.
Possibly we should tack an easier problem for practice. How many unique positions are possible in tic tack toe? Or an even easier problem. How many final positions are possible in Tic Tack
Toe if we require that even in the event of 3 in a row the game continues until all squares are occupied. The last problem should be doable for any one with a degree in math or a quantitative science.
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| This is further complicated by the fact that 10 white knights are possible if there are no white pawns. |
Yes, I think we can have 10 white knights, if all pawns are promoted.
Also, we can have 10 black knights...
But, now the funny thing: we cannot have at the same time 20 knights, being 10 whites and 10 black. Why? Because the way the pawns move, the whites pawns can reach row 8 only after taking some black pawns, and vice-versa. Pawns cannot jump, remember.
Another junk position: can we have 9 white queens? I don't know how to answer that yet... But the problem I'm thinking is:
Can a black king be in check-mate by a single white piece? No.
And attacked by one piece, with support of another? Yes. (Think in a queen check-mate with black king at a1, white queen at a2 and white king at a3).
Can a black king be in check-mate attacked by two pieces at once?
FEN: 8/8/8/8/R7/B7/2K5/k7 w - -
(white to move)
The move Bb2 gives a double-attack check-mate.
But can this position be legal (the black king attacked by 3 pieces at the same time):
FEN: 8/8/8/8/R7/1N6/1BK5/k7 b - -
(black to move)
So, it is very difficult to find mathematical equations that represent the rules of chess game.
Anyway, we still can make progress in the current upper bound.
Cheers,
Beco.
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Beco,
The white q pawn and black k pawn can move to d6 and e3 respectively. Then each can capture a knight on e7 and d2 repectively. Hence more than 10 Bishops are possible. I think the maximum number of Bishops Knigghts or rooks is 19
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