Originally posted by mattover-matter ok, im going to restate:
My question:
I wish to figure out how many valid positions in a chess game are possible (at one time)
My troubles:
1) two peices cannot be on the same square at one time
2) bishops cannot reach every square on the board
3) pawns cannot reach every square on the board (cannot move backwards)
4) Whenever a pawn reaches the end, It can choose a peice that is already dead (from his side) and revive it inplace of the pawn. You cannot have 2 queens, or 3 rooks.
5) Units cannot travel through (you must think back). A pawn cannot appear behind an enemy pawn for some reason (well, it is possible but not if the pawns have not killed anyone)
FYI:
1) There are 64 squares on a chessboard.
2) There are 8 pawns, 2 rooks, 2 knights, 2 bishops, 1 queen, 1 king per side(32 in total)
3) The dimensions for a chessboard are 8x8.
4) A knight CAN reach every position on the board.
Also, I asked at DR.math forum to see if anyone there knows a solution.
re: It would be interesting to know, but probably not fun to do
You are missing the point of this. Do you think I randomly pick things that have 0 relevency in anything I will ever do (don't take me literally)? I am doing this to learn, not to know. knowing is just a bonus. (note : what I just said applies to this problem, and this problem only [again, don't take me literally])
good luck.
start from the 'potential' figure I gave you (1008) and then take away whatever you want... but your question still lacks what I would call a 'point'... which means I'm probably still missing it!
Oh yeah and your rules 4 and 5 are technically incorrect. It's possible to have 8 queens, for example, and number 5 is to be laughed at What if someone simply decided NOT to take...? Perfectly legal.
::] krycek [::
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Possible positions is still enormous. With *just* two kings on the board, you have something like this:
A king on interior: 36*55 (9 squares opponent can't be in)
A king on side: 24*59 (5 squares opponent can't be in)
A king on a corner: 4*61 (3 squares opponent can't be in).
Which sums to 3640. As soon as you throw any more pieces onto the board, it becomes ridiculously enormous.
Notice 64*63 is 4032, which could be considered an upper limit for the calculation. A difference of 392 may seem significant right now, but as we get into billions of combinations, the only feasible way to calculate the number is to look at upper limits.
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4.5e+53 (I cannot write superscript on here can I? Ah well, I will leave it like that)
Any more accuracy, and that's up to you to work out...
::] krycek [::
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re: why?
I am trying to learn
Hey, Can you kinda put that in stupider terms, I thought e was for calculating the growth in percentages...but I learned that on my own (not at school, don't give them the credit for my intelligence ). Kryckeck, can you please attempt to explain how you figured that out? I'm 13 so you guys are a wee bit ahead of me My math class is still on ratios and proportions (for 9 weeks now ) WOW, jkd I guess....my next mission will be to prove you wrong. I was getting 64*32 = 2??? I guess you proved me wrong.
5[sup]2[/sup]*[sub]Y[/sub]
like that kryckeck?
[ s u b ] [ / s u b ] [ s u p ] [ / s u p ]
nope, i guess these avatarless forums are jes' weird
you say calculating the possible positions to move too at any one time in a game.
do you mean you want to calaculate the possible positions to move too at one ponit and later recalculate at another time. or just calculate them once?
would it be an idea that some produces a chessboard overview so that when calculating all the possible moves everyone is working on the same game board.
Depending on positions depends on the possible moves.
scroots
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Originally posted by mattover-matter I thought e was for calculating the growth in percentages...but I learned that on my own (not at school, don't give them the credit for my intelligence ).
That's not e as in lim n-> infinity (1+1/n)^n, it is e as in the scientific notation shortcut.
4.5e+53 = 4.5 * 10^53
Which isn't actually the right answer either... it should be significantly larger than that.
Originally posted by jkd That's not e as in lim n-> infinity (1+1/n)^n, it is e as in the scientific notation shortcut.
4.5e+53 = 4.5 * 10^53
Which isn't actually the right answer either... it should be significantly larger than that.
I'll take your word for it... as you know, it was just a quick estimate
I still think that 450,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 is big enough!
::] krycek [::
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Originally posted by mattover-matter ok, im going to restate:
My question:
I wish to figure out how many valid positions in a chess game are possible (at one time)
My troubles:
1) two peices cannot be on the same square at one time
2) bishops cannot reach every square on the board
3) pawns cannot reach every square on the board (cannot move backwards)
4) Whenever a pawn reaches the end, It can choose a peice that is already dead (from his side) and revive it inplace of the pawn. You cannot have 2 queens, or 3 rooks.
5) Units cannot travel through (you must think back). A pawn cannot appear behind an enemy pawn for some reason (well, it is possible but not if the pawns have not killed anyone)
FYI:
1) There are 64 squares on a chessboard.
2) There are 8 pawns, 2 rooks, 2 knights, 2 bishops, 1 queen, 1 king per side(32 in total)
3) The dimensions for a chessboard are 8x8.
4) A knight CAN reach every position on the board.
Also, I asked at DR.math forum to see if anyone there knows a solution.
re: It would be interesting to know, but probably not fun to do
You are missing the point of this. Do you think I randomly pick things that have 0 relevency in anything I will ever do (don't take me literally)? I am doing this to learn, not to know. knowing is just a bonus. (note : what I just said applies to this problem, and this problem only [again, don't take me literally])
Heres a possible hint:
try to define the possible move at a given point in the game as an equation. Obviously, the total moves available to a player at any given moment changes based on the position of all the pieces on the board at that moment. For example, when you start the game there are 20 possible moves, each pawn can move 1 or 2 spaces, and 2 knights can move to 2 different spaces each. so we get something like
total moves = total pawns * 2 possible moves + total knights * 2 possible moves or 8 * 2 + 2 * 2 or 16 + 4 = 20.
Now, once you move a pawn, you total possible moves for the next turn increase based on the provious move. So everything is now dependant on the first move. This means you should be able to express the second move as a function of the first move. Hope this helps.
Good luck. I think this could be an SA article if you figure it out. (SA = Scientific American)
Before you post, read our: 
What if someone simply decided NOT to take...? Perfectly legal.
now I get what he is trying to do!
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