So according to this website which was made by some serious mathematician:
http://www.cs.berkeley.edu/~flab/chess/chess.html
"the total number of legal chess positions which is estimated to be between 10^43 and 10^50."
How many different chess positions are there?
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there are 3612 legal postions with just the two kings
Aug 8, 2008
0
#44
But do you really want to solve chess? Wouldn't it take something away from the game if there was a way of playing that rendered you unbeatable - cheater_1 would have a field day.
pvmike wrote:
there are 3612 legal postions with just the two kings
I'm not sure where you got that number, but actually there are 4032 different ways to place the two kings.
The first king can be placed on 64 different squares. That leaves 63 possibilities for the second king after the first has been placed. 64 * 63 = 4032. This is without symmetrical considerations, so the white king on a1 with the black king on h8 would be considered different than the black king on a1 and the white king on h8.
One other thing to consider is who has the move. That would double the 4032 to 8064 possible positions for the two kings. Phew, I'm glad I dont have to count them all!
EDIT: Oh yeah, I just realized the two kings can't touch, so the 4032 would have to be less, more like 3500 as a guess. That must be where the 3612 comes from. So I think you would need to double the 3612 to 7224 positions, because it could be white's move or it could be black's move. Correct me please if I am mistaken.
Bowanza, I was jus conserned with the number of positions not whose move, but if more pieces are added to the board the does become an important consideration. I'm trying to figure out the number of possible positions with two kings and a pawn, it's alot harder.
Okay I think this is right, there are 195,516 legal positions in a K and P vs K ending not taking in account whose move it is. It's more difficult it you consider whose move it is because some positions it could be either white or black to move but if black's in check then it's black move.
pvmike wrote:
Okay I think this is right, there are 195,516 legal positions in a K and P vs K ending not taking in account whose move it is. It's more difficult it you consider whose move it is because some positions it could be either white or black to move but if black's in check then it's black move.
When you say K and P vs K do you mean the pawn can be either color? Or are you restricting it to a white (or a black) pawn? And can you show us where the 195,516 came from?
I meant the pawn can be only be one color and the number's wrong I made a mistake, I'll explain in a second.
lets assume white has the extra pawn
case 1(whites king is on a1 or h1)
so there are two poisitions for whites king 60 for blacks and 55 for the pawn because it can't be on the first rank
2x60x55=6600
case 2(white king on a8 or h8)
2x60x54=6480
case 3 white king on b1-g1
6x58x55=19140
case 4 white king on the edge not b1-g1 or a corner
18x58x54=56376
case 5 white king not on the edge
36x55x54=106920
6600+6480+19140+56376+106920=195516
in case the first number refers the number of squares whites king can be the second blacks king and the third white pawn
my mistake was I didn't account for the black king being on the first rank which would give the pawn one more possible square
a pawn cant be on the last rank either because it becomes something else as soon as it lands there.
Actually, I just placed an upper bound of 178,416 possible positions for K and P vs K. The actual number of positions is a little less, more like 165,000.
Start with the white pawn. It can occupy 48 possible positions. (since it can't occupy the back rank or the first rank, 6x8=48).
Then the white king can occupy 63 possible squares. 48x63=3024.
When the white king is in the corner and the pawn on the edge but not on the second rank, the black king can occupy 59 possible positions. 3024x59=178,416. This is an upper bound.
When the white king is somewhere in the middle of the board and the pawn is in its inital position, the black king can only occupy 52 possible squares. 3024x52=157,248. This is a lower bound.
The actual number of positions is a little closer to the lower bound than the upper because most of the time the white king is in the middle of the board, so most of the time blacks king can occupy about 54 or 55 possible squares (a guesstimate), making the total number of positions around 3024x55=166,320.
so the corrected number of possible legal positions for a king and pawn vs king ending is 195984 not considering whose move it is( unless I made another mistake)
dam your right I forgot about the back rank that makes this way more complicated
I just found a mistake in my previous post. I've already edited it, I just want you to know its changed.
pvmike wrote:
dam your right I forgot about the back rank that makes this way more complicated
It does not really complicate much. It just reduces the total number of possiblities to about 6/7 of your original calculation. 6/7 of 195,5000 is roughly 167,500, which is pretty close to my estimate above. In any event, that is a lot of positions with just three chessmen! I hate to ask, but how many possible positions are there with a white pawn and a black pawn versus the kings?
I'D SAY MORE THAN A TRILLION POSSIBILITIES...
ADK
bownaza, with one black pawn and one white pawn things get much more complicated because of the possiblility of having both kings in check, I'm done doing math for the night but maybe I'll try to tackle that problem tomorrow.
pvmike wrote:
bownaza, with one black pawn and one white pawn things get much more complicated because of the possiblility of having both kings in check, I'm done doing math for the night but maybe I'll try to tackle that problem tomorrow.
How about finding the precise number of positions for K+P vs K first?
Billium248 wrote:
Fourpointo wrote:
Both numbers are very very high, but have limits. After 50 moves without a pawn movement or piece taken, the game can end in a tie.
It CAN, but it doesn't have to. Go to the list of current games on this site and sort by number of moves. The #1 game at the time of this post was on move #347!! The 50-move-rule is NOT automatic!! It is designed to allow one player to declare a draw even when the other player refuses. If neither side claims the draw, and both sides agree to play on, then there is no limit to the number of moves that can be made.
Maybe so Billium, but it has never happened where two players over the board have repeated the same moves back and fourth for thousands of moves, so it's feasible to exclude that scenario from the calculation.
I think the easiest way to do the calculation would be to take 500 of the strongest games on record, calculate the average possible moves from every position of every game and then apply that average to the highest average number of moves in any recorded game.
I think you will find that the number will feasibly be something along the lines of 50*10^80. The point being that we will probably never know the number for a long time, even with a computer it would take far too long to calculate.