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Old 2011-03-22, 19:49   #1
science_man_88
 
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Default chess positions: how many legal ones are there ?

I know there have been estimates in the past I want to work together to find an exact number. I know from math that there are 3612 positions with only each players king on the board. So far using 1 of every other piece possible positions jump to 800,000 + the last estimate I heard was 10^49 I think. I can upload a image of all the legal moves/ places each piece can use the hard part then is not coming up to a conclusion that Mr p-1 said in another thread after I stupidly talked about this.

Last fiddled with by science_man_88 on 2011-03-22 at 20:00 Reason: Took and from the start of a sentence.
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Old 2011-03-22, 22:40   #2
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[1 2 3 4 5 3 2 1]
[6 6 6 6 6 6 6 6]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[6 6 6 6 6 6 6 6]
[1 2 3 4 5 3 2 1]

This is a representation of a chess board at the start this is one position.

[5 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]


[0 5 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]


[0 0 0 0 0 0 0 0]
[0 5 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]


these are the three types of positions a king can be in:

36*(64-9) + 4*(64-4) + 24*(64-6) = 3612 positions just the kings left (all drawn by insufficient material)
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Old 2011-03-22, 23:35   #3
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It's a hard problem. Consider positions like the attached.
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Old 2011-03-22, 23:43   #4
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Quote:
Originally Posted by CRGreathouse View Post
It's a hard problem. Consider positions like the attached.
I don't see how you can achieve that one legally.

Science Fellow should also realize that you can have 3 white queens or 4 black rooks. That will make the calculations much harder.
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Old 2011-03-22, 23:50   #5
TimSorbet
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Quote:
Originally Posted by Uncwilly View Post
I don't see how you can achieve that one legally.
I think that's the point: the position as it is is not illegal in any way, but it can not be achieved legally. Hence, the problem is not as simple as looking at each state without considering how you could get to it.
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Old 2011-03-22, 23:52   #6
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Quote:
Originally Posted by Uncwilly View Post
Science Fellow should also realize that you can have 3 white queens or 4 black rooks. That will make the calculations much harder.
Or ten bishops... but not all on the same color. And six bishops all on the same color isn't legal if your opponent has 16 pieces on the board.
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Old 2011-03-22, 23:56   #7
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I understand some of the complexity I know , I do play chess I've even played a few people with titles ( on the internet). in fact I was once told I played like a 2100+ ranked played in bullet chess ( think less than 3 minutes each mostly).
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Old 2011-03-22, 23:58   #8
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You may play better than many people here on the boards. But there are going to be a lot of people here who can count board positions more easily than you -- anyone with training in combinatorics, most likely.
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Old 2011-03-23, 00:00   #9
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Quote:
Originally Posted by CRGreathouse View Post
You may play better than many people here on the boards. But there are going to be a lot of people here who can count board positions more easily than you -- anyone with training in combinatorics, most likely.
why'd you think I posted it ? I know it will get complicated. for example although 3 pawns of one color are possible they can't be near each other unless you have tripled pawns ( I've had it once).
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Old 2011-03-23, 00:03   #10
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Quote:
Originally Posted by science_man_88 View Post
for example although 3 pawns of one color are possible they can't be near each other unless you have tripled pawns ( I've had it once).
I have often had 8 pawns. I have also had 4 pawns in one 2x2 square.
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Old 2011-03-23, 00:24   #11
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Don't expect to compute an exact number.
http://en.wikipedia.org/wiki/Chess#M..._and_computers says:
The number of legal positions in chess is estimated to be between 1043 and 1047 (a provable upper bound[86])

86. John Tromp (2010). "John's Chess Playground"
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