29 Jul 05
Originally posted by Gatecrashercouldn't resist...there is no largest prime...this is an old result which has an easy proof from Euler.
It would be finite. It would be very, very large. It would tend to inifinity. But its rather like asking what the largest prime number is. You feel sure there is an answer, but the capacity to arrive at a defintive answer is way beyond our reach.
yes, this is the wrong forum
29 Jul 05
Originally posted by flexmoreWhy choose an arbitrary number? Each player begins with exactly 20 possible first moves. After 1.e3, white has 30 possible second moves, no matter what black does. After, 1.e4, white has 30 possible second moves against everything except 1...e5 (29), 1...f5 (31), and 1...d5 (31).
there is at least 10 first moves which have 10 different second moves etc for at least 10 moves, so it is greater than 10000000000,
20 x 20 = 400 possible positions after one move.
There are eight possible sequences of moves leading to white losing by Fool's Mate: 2...Qh4# There are approximately 290 possible sequences of moves leading to black losing to the same tactic: 3.Qh5#
Calculating all the possibilities is terribly difficult, as you note. As far as I know, François Labelle, who programs computers to perform some of these calculations, has only estimated the upper limit. He told me in an email a couple of years ago that he considers the oft quoted 10^120 a conservative estimate, based on the limits of games forty moves long, and no position repeated more than three times (draw claim).
The figure of 10^134 for games up to 50 moves from http://www.geocities.com/explorer127pl/szachy.html appears credible.
29 Jul 05
1 edit
Originally posted by chasparosYes, my estimate is a most upper bound if there can be such a thing.
[Edit: sorry disregard this post. didn't think about the 'all pieces are different' part]
It still does not adress promotions... You won't find any positions on there with three white knights... So this is not a way to find an upper ...[text shortened]... So it's not so easy to find the upper bound on chess positions.
The two kings position is relatively easy:
Place the white king on a square. If it is a corner square then there are 4 squares unavailable for the black king - the one the first king is on and the three neighbouring ones so that is 4*60 positions - although the board is rotationally symmetric I'm assuming that one end is white's end and one is black's so these count as distinct positions. Next place the first king on the edge of the board, but not in a corner, which cuts off 6 squares, so 58 are available for the black king and there are 6*4 = 24 edge squares. So we have 6*4*58 = 1392 positions. Now put the white king on one of the interior squares, that cuts off 9 squares for the black king, so that gives (64-9)*(64-28) = 1980 positions. This covers all the possibilities so just with two kings and we have:
4*60 + 6*4*58 + (64 - 9) * (64-28) = 240 + 1392 + 55 * 36 = 3612 positions with just 2 kings.
29 Jul 05
1 edit
I've realised I can do better, in positions with 2 kings and a piece, but not a pawn - that case is complicated, you need to multiply by 62 to get the number of places the piece can go. If it is a bishop then you don't need to worry about which coloured square it is on and get the right number of positions for a bishop on either square, if you want the number of positions with say a black squared bishop then by symmetry it's half so multiply by 31. This gives:
2 kings + knight, rook or bishop of either colour: 223944
King vs K + bishop: 111972
Other cases are too complicated to work out without being paid.
30 Jul 05
3 edits
Originally posted by Wulebgrit6 is not an arbitrary number.
Why choose an arbitrary number? Each player begins with exactly 20 possible first moves. After 1.e3, white has 30 possible second moves, no matter what black does. After, 1.e4, white has 30 possible second moves against everything except 1. ...[text shortened]... p://www.geocities.com/explorer127pl/szachy.html appears credible.
the idea was NOT to find a vague estimate.
the idea was to find a very simple lower bound ... a number smaller than the reality.
when we know an upper bound and a lower bound, then we know the truth lies somewhere in the middle.
refining the upperbound downwards and the lowerbound upwards gives increased accuracy.
the method you suggest tells us very little extra ... using 20 instead of 10 for the first two moves, only multiplies the result by 4, there are many methods to increase it by 100s of 1000s of 1000000s.
30 Jul 05
Originally posted by DeepThoughtyour estimate is simply an upperbound ...
Yes, my estimate is a most upper bound if there can be such a thing.
The two kings position is relatively easy:
Place the white king on a square. If it is a corner square then there are 4 squares unavailable for the black king - the one the first king is on and the three neighbouring ones so that is 4*60 positions - although the board is rotationall ...[text shortened]... 4*60 + 6*4*58 + (64 - 9) * (64-28) = 240 + 1392 + 55 * 36 = 3612 positions with just 2 kings.
it is a nice upper bound - probably not that many orders of magnitude to high.
02 Aug 05
Originally posted by BowmannIt may be way off the mark indeed. The same book gives an estimate for the number of atoms in the universe--10 to the 80th power, which means of course that the number of possible variations in a game of chess is WAY MORE than the number of atoms in the universe. But I'd like to know how a person would come up with an estimate of the number of atoms in the universe in the first place. Even if you did, would you be able to say it with a straight face?
Then I would have to say that your previous estimate is way off the mark.
In any case, I have dedicated the remaining waking moments of my life to memorizing EVERY ONE of those 25x10 to the 116th power chess game variations. Then I'll take on Hydra.
02 Aug 05
Originally posted by James Horton, no doubt during a moment of madnessYou'd still have to cross your fingers and hope that the game doesn't get past move 40.
In any case, I have dedicated the remaining waking moments of my life to memorizing EVERY ONE of those 25x10 to the 116th power chess game variations. Then I'll take on Hydra.