chess positions: how many legal ones are there ?

[science_man_88]

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.

[science_man_88]

[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)

[CRGreathouse]

It's a hard problem. Consider positions like the attached.

[Uncwilly]

Originally Posted by CRGreathouse

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.

[TimSorbet]

Originally Posted by Uncwilly

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.

[CRGreathouse]

Originally Posted by Uncwilly

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.

[science_man_88]

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).

[CRGreathouse]

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.

[science_man_88]

Originally Posted by CRGreathouse

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).

[Uncwilly]

Originally Posted by science_man_88

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.

[Jens K Andersen]

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"