idea to estimate the percentage of drawn positions in chess

[towforce]

3-man (45.1591%)
4-man (31.9065%)
5 man (22.2248%)
6-man (17.4227%)
7-man (13.1502%)

1st order differences:

3-4: 45.1591 - 31.9065 = 13.2526
4-5: 31.9065 - 22.2248 = 9.6817
5-6: 22.2248 - 17.4227 = 4.8021
6-7: 17.4227 - 13.1502 = 4.2725


2nd order differences:

3-4 - 4-5: 13.2526 - 9.6817 = 3.5709
4-5 - 5-6: 9.6817 - 4.8021 = 4.8796
5-6 - 6-7: 4.8021 - 4.2725 = 0.5296


3nd order differences:

[3-4 - 4-5] - [4-5 - 5-6]: 3.5709 - 4.8796 = -1.3087
[4-5 - 5-6] - [5-6 - 6-7]: 4.8796 - 0.5296 = 0.5296


We are seeing convergence, so might be polynomial, but I'm going to say that we cannot safely fit a degree 2 polynomial to this data, and we don't have enough data to fit a higher degree polynomial (unless you want to say that two man has a 100% draw rate - which would be true).

Thinking about it, the fact that this model is restricted to a range of 0 to 100% means a polynomial is unlikely to be a good fit anyway.

Just for completeness, I'm taking it to 4th order differences:


1st order differences:

2-3: 100 - 45.1591 = 54.8409
3-4: 45.1591 - 31.9065 = 13.2526
4-5: 31.9065 - 22.2248 = 9.6817
5-6: 22.2248 - 17.4227 = 4.8021
6-7: 17.4227 - 13.1502 = 4.2725


2nd order differences:

2-3 - 3-4: 54.8409 - 13.2526 = 41.5883
3-4 - 4-5: 13.2526 - 9.6817 = 3.5709
4-5 - 5-6: 9.6817 - 4.8021 = 4.8796
5-6 - 6-7: 4.8021 - 4.2725 = 0.5296


3nd order differences:

[2-3 - 3-4] - [3-4 - 4-5]: 41.5883 - 3.5709 = 38.0174
[3-4 - 4-5] - [4-5 - 5-6]: 3.5709 - 4.8796 = -1.3087
[4-5 - 5-6] - [5-6 - 6-7]: 4.8796 - 0.5296 = 0.5296


4th order differences:

[[2-3 - 3-4] - [3-4 - 4-5]] - [[3-4 - 4-5] - [4-5 - 5-6]]: 38.0174 - -1.3087 = 39.3261
[[3-4 - 4-5] - [4-5 - 5-6]] - [[4-5 - 5-6] - [5-6 - 6-7]]: -1.3087 - 0.5296 = -1.8383

Looks even less convergent that before.

[Uri Blass]

Stockfish is not best to find mates..

Most of these positions are not going to have checkmates that can be found with today's technology.

We are looking for an ESTIMATE of the drawn position ratio here. My plan (previously stated in this thread) is:

1. Choose an engine, and how much time it will be allowed to look at each position

2. Choose an evaluation score (ES) at which 50% of the games above that score ended in a draw and 50% ended in a loss

3. If a position lies between -ES and ES, call it a draw, otherwise call it a win

4. The errors in the win-which-is-a-draw direction will be approximately equal to the errors in the draw-which-is-a-win direction

This method is good enough for an estimate.

I do not believe most of these positions are not going to have checkmates with today's technology.
There are many captures and engines are strong in tactics with enough time to find mates.

I think the result of game of the engine against itself at long time is going to be the theoretical result in big majority of the unproved cases but first thing to do it to try to find more forced mates

You probably need to work hard to prove checkmates but I believe a big evaluation can often be translated to checkmate.

I agree that there are hard positions to prove mate but even there the result is often probably clear
For example the evaluation is 719 in the position I see without proved mate(used stockfish10 because I do not trust nnue evaluation and it found all moves are mates except 8 moves that also have a huge score).

[d]1B2NBK1/1q1N4/1R1B1r2/P3r3/2QpPn2/R3QpNk/3bNn1n/2r3Rb b - - 0 1

FEN: 1B2NBK1/1q1N4/1R1B1r2/P3r3/2QpPn2/R3QpNk/3bNn1n/2r3Rb b - - 0 1

30/37 9:11:06 468,431,119k 14,166k -M14 Nf2-d3 Nd7xe5 Bd2xe3 Rb6xb7 Be3xg1 Ne2xf4+ Rf6xf4 Rb7-h7+ Kh3-g2 Ne5xd3 Rf4-g4+ Kg8-f7 Rc1-b1 Ra3-a2+ f3-f2 Qc4xd4 Rb1-b7+ Bf8-e7 Rg4-g7+ Ne8xg7 Rb7-b2 Ra2xb2 Kg2-f3 Qd4-f6+ Kf3-e3 Bd6-c5+ Ke3xd3 Qf6-d4+
30/29 9:11:06 468,431,119k 14,166k -M14 Qb7xb6 Qe3xd2 Qb6-b4 Qc4xb4 Re5-g5+ Kg8-h8 Nf4-g6+ Kh8-g7 Ng6-f4+ Kg7xf6 Nh2-g4+ Kf6xg5 Nf4-e6+ Kg5-g6 Ne6xf8+ Bd6xf8 Rc1-c6+ Ne8-f6 Bh1-g2 Qb4-b5 Ng4-e5+ Qb5xe5 Rc6xf6+ Nd7xf6 Kh3-h2 Qe5-h5+ Nf2-h3 Ng3-f5+
30/43 9:11:06 468,431,119k 14,166k -M16 Nf2-g4 Qe3xd2 Re5-g5+ Ne8-g7 Qb7xd7 Ne2xf4+ Rf6xf4 Bd6xf4 Rg5xg7+ Bf8xg7 Rc1xg1 Qd2xd4 Rg1-d1 Qd4xd7 Rd1xd7 Ng3xh1 Rd7-d8+ Bg7-f8 Rd8-d6 Ra3xf3+ Nh2xf3 Rb6xd6 Nf3-d4 Qc4-f1+ Kh3-h4 Qf1-e1+ Kh4-h5 Nh1-g3+ Kh5-h4 Ng3-f5+ Kh4-h5 Nf5-g7+
30/33 9:11:06 468,431,119k 14,166k -M16 Nh2-g4 Ne2xf4+ Rf6xf4 Qe3xd2 Rc1xg1 Rb6xb7 Re5-g5+ Kg8-h8 Rg1xg3 Qd2xf4 Rg5-h5+ Kh8-g7 Nf2xe4 Qc4-f1+ Bh1-g2 Ra3xf3 Rh5-g5+ Qf4xg5 Bg2xf1 Qg5-h5+ Kh3-g2 Rf3xg3+ Ne4xg3 Qh5xg4 Bf1-e2 Qg4xg3+ Kg2-f1 Rb7-b1+ Be2-d1 Qg3-f3+ Kf1-e1 Rb1xd1+
30/37 9:11:06 468,431,119k 14,166k -M16 Rc1xg1 Ne2xg1+ Kh3-g2 Qe3xf2+ Kg2xf2 Qc4xd4+ Kf2-g2 Qd4xd2+ f3-f2 Qd2xf2+ Kg2xf2 Rb6xb7 Bh1-f3 Rb7-b2+ Nf4-e2 Ng1xe2 Rf6-g6+ Kg8-f7 Rg6-g7+ Ne8xg7 Re5-f5+ e4xf5 Bf3-d5+ Kf7-e7 Nh2-f1 Ne2-f4+ Nf1-d2 Rb2xd2+ Kf2-e1 Rd2-e2+ Ke1-d1 Ra3-a1+
30/42 9:11:06 468,431,119k 14,166k -M18 Qb7xd7 Ne2xf4+ Rf6xf4 Qe3xd2 Re5-g5+ Ne8-g7 Rc1xg1 Bd6xf4 Rg5xg7+ Bf8xg7 Nh2-g4 Qd2xd4 Qd7xd4 Bg7xd4 Kh3-g2 Bd4xf2 Rg1-d1 Rb6-g6 Rd1-d8+ Kg8-f7 Kg2-h3 Qc4-f1+ Bh1-g2 Ra3xf3 Rd8-d7+ Kf7-e6 Rd7-e7+ Ke6xe7 Bg2xf1 Ng3-h1+ Kh3-g2 Rf3-g3+ Kg2-h2 Rg3xg4+ Kh2xh1 Rg4-g1+
30/51 9:11:06 468,431,119k 14,166k -M19 Qb7-c8 Qc4xc8 Bd2xe3 Ne2xc1 Rf6-g6+ Kg8-f7 Re5-f5+ e4xf5 Nf2-g4 Rg1xh1 Kh3-g2 f5xg6 Nf4-e2 Rh1xh2+ Ng4xh2 Ng3xe2 Nh2-g4 Ne2-f4+ Kg2-g3 Nf4-h5+ Kg3-g2 Rb6-b2+ Be3-f2 Qc8-c6 d4-d3 Rb2xf2+ Kg2xf2 Nc1xd3+ Kf2-g1 Ra3-a1+ Kg1-g2 Ra1-a2+ Ng4-f2 Ra2xf2+ Kg2-h3 Qc6xf3+ Kh3-h4 Bf8-e7+
30/45 9:11:06 468,431,119k 14,166k -64.57 Qb7-d5+ e4xd5 Bd2xe3 Ne2xc1 Re5-g5+ Bf8-g7 d4-d3 Ne8xf6 Nf4-e6 d5xe6 Bh1-g2 e6-e7 d3-d2 e7-e8Q d2-d1Q Rg1xd1 Bg2-f1 Rd1xf1 Nf2-g4 Rb6-b1 Be3-c5 Bd6xc5 Ng4-h6+ Kg8-h7 Nh6-g4 Qe8-e6 Kh3-h4 Ng3-e2 Rg5xg7+ Kh7xg7 Kh4-h3 Bb8xh2
30/47 9:11:06 468,431,119k 14,166k -51.91 Qb7-c7 Bb8xc7 Bd2xe3 Ne2xc1 Rf6-g6+ Ne8-g7 Nf4-d5 e4xd5 Bh1-g2 Bd6xe5 Rg6xb6 a5xb6 Nf2-g4 b6-b7 Ng4-h6+ Kg8-h7 Nh6-f7 b7-b8Q Nf7-g5+ Kh7-g8 Nh2-g4 Ng7-f5 Ng4xe5 Qb8-e8 Ng5-f7 Bc7xe5 Nf7-h6+ Bf8xh6 Kh3-h2 Nf5xe3 Kh2xg1 Be5xd4
30/47 9:11:06 468,431,119k 14,166k -50.47 Qb7-a6 Qc4xa6 Bd2xe3 Ne2xc1 Re5-g5+ Ne8-g7 Nf2-g4 Rg1xh1 Kh3-g2 Ra3xe3 Rg5-g6 Re3-a3 f3-f2 Bd6xf4 Rf6xb6 a5xb6 Rg6-h6 Ng7-f5 Rh6-h3 Ra3-a2 Rh3xg3 Nf5xg3 Kg2-f3 Qa6-d3+ Kf3-g2 Qd3xd4 Kg2-h3 Qd4-d1 Kh3-g2 Ra2-e2 Kg2-h3 Qd1-f1+ Kh3-h4 Bf8-e7+ Ng4-f6+ Be7xf6+ Kh4-g4 Rh1xh2
30/51 9:11:06 468,431,119k 14,166k -48.62 Qb7-c6 Rb6xc6 Bd2xe3 Ne2xc1 Rf6xf8+ Bd6xf8 Re5-g5+ Kg8-f7 Nf2-g4 Rg1xh1 Kh3-g2 Ra3xe3 d4xe3 Bb8xf4 e3-e2 Bf4xg5 Ng4-f2 Rh1-e1 Nf2-g4 Ng3-h5 Nh2-f1 Qc4-e6 Nf1-e3 Nh5-f4+ Kg2-g3 Rc6-c3 Kg3-h2 Qe6-d6 Ne3-f1 Rc3xf3 Kh2-h1 Nc1xe2 Ng4-e5+ Nd7xe5
30/51 9:11:06 468,431,119k 14,166k -39.52 Rf6-g6+ Kg8-h8 Rg6-h6+ Bf8xh6 Nf4-g6+ Kh8-g8 Re5xe8+ Nd7-f8 d4xe3 Ne2xc1 Qb7-c8 Qc4-f7 Qc8-g4 Qf7xe8 Bd2xc1 Rg1xc1 Ng6-e7+ Kg8-f7 Ne7-c8 Rc1xc8 Kh3-g2 Bh6xe3 Nf2-h3 Rb6-b2+ f3-f2 Rc8-c1 Qg4-f3+ Be3-f4 Nh3xf4 Ra3xf3 Kg2xf3 Rc1xh1 Nf4-d3 Rb2-d2 Kf3-g2 Rd2xd3 Nh2-f3 Nf8-g6 f2-f1R Rh1xf1 Nf3-g5+ Kf7-f6
30/57 9:11:06 468,431,119k 14,166k -38.96 Bd2xe3 Nd7xe5 Qb7-d7 Ne2xc1 Nf2xe4 Ne5xd7 Rf6-g6+ Kg8-f7 Ne4-g5+ Kf7-e7 Rg6-e6+ Ke7-d8 Re6xd6 Bf8xd6 Nf4-e6+ Kd8-e7 Bh1-g2 Ng3-e4 Nh2-g4 Ne4xg5+ Ne6xg5 Qc4-g8 Be3-f2 Qg8-h8+ Bf2-h4 Bd6-f4 Ng5-e4+ Ke7-e6 Ng4-f2 Qh8xd4 Bh4-g5 Bf4xg5 Ne4xg5+ Ke6-f5 Ng5-f7 Qd4xf2
30/48 9:11:06 468,431,119k 14,166k -31.03 d4xe3 Nd7xe5 Nf4xe2 Rb6xb7 Ne2xg1 Qc4-b5 Rf6-f4 Rb7-g7 Rf4-h4 Ng3xh1 Nf2-g4 Bf8-e7 Ng4-h6+ Kg8-f8 Rh4-f4+ Be7-f6 Nh2-g4 Ne5xg4 Rf4xg4 Qb5-h5+ Kh3-g2 Qh5-h2+ Kg2-f1 Rg7xg4 f3-f2 Qh2xh6 Kf1-e2 Nh1-g3+ Ke2-d1 Ra3xe3 Bd2xe3 Qh6xe3 f2-f1Q Ng3xf1 Ng1-e2 Qe3-f3
30/52 9:11:06 468,431,119k 14,166k -28.36 Re5-g5+ Ne8-g7 Bd2xe3 Ne2xf4+ Rf6xf4 Qc4-e6+ Rf4-g4 Rg1xc1 Qb7xb6 Nd7xb6 Rg5-g6 Qe6-c8 Bh1-g2 Rc1-c5 Be3-g5 Ng3-e2 Bg2-f1 Ne2xd4 Kh3-g2 Bd6xh2 Bg5-f6 Nd4-f5 Nf2-d1 Nb6-d5 Bf6xg7 Nf5xg7 Bf1-e2 Nd5-f4+ Kg2-h1 Nf4xg6 Rg4xg6 Bh2-g3 Kh1-g2 Bg3-h4 f3-f2 Qc8-h3+ Kg2-g1 Rc5-g5+ Rg6xg5
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30/37 11:42:46 598,697,909k 14,198k -M14 Nf2-d3 Nd7xe5 Bd2xe3 Rb6xb7 Be3xg1 Ne2xf4+ Rf6xf4 Rb7-h7+ Kh3-g2 Ne5xd3 Rf4-g4+ Kg8-f7 Rc1-b1 Ra3-a2+ f3-f2 Qc4xd4 Rb1-b7+ Bf8-e7 Rg4-g7+ Ne8xg7 Rb7-b2 Ra2xb2 Kg2-f3 Qd4-f6+ Kf3-e3 Bd6-c5+ Ke3xd3 Qf6-d4+
30/29 11:42:46 598,697,909k 14,198k -M14 Qb7xb6 Qe3xd2 Qb6-b4 Qc4xb4 Re5-g5+ Kg8-h8 Nf4-g6+ Kh8-g7 Ng6-f4+ Kg7xf6 Nh2-g4+ Kf6xg5 Nf4-e6+ Kg5-g6 Ne6xf8+ Bd6xf8 Rc1-c6+ Ne8-f6 Bh1-g2 Qb4-b5 Ng4-e5+ Qb5xe5 Rc6xf6+ Nd7xf6 Kh3-h2 Qe5-h5+ Nf2-h3 Ng3-f5+
30/43 11:42:46 598,697,909k 14,198k -M16 Nf2-g4 Qe3xd2 Re5-g5+ Ne8-g7 Qb7xd7 Ne2xf4+ Rf6xf4 Bd6xf4 Rg5xg7+ Bf8xg7 Rc1xg1 Qd2xd4 Rg1-d1 Qd4xd7 Rd1xd7 Ng3xh1 Rd7-d8+ Bg7-f8 Rd8-d6 Ra3xf3+ Nh2xf3 Rb6xd6 Nf3-d4 Qc4-f1+ Kh3-h4 Qf1-e1+ Kh4-h5 Nh1-g3+ Kh5-h4 Ng3-f5+ Kh4-h5 Nf5-g7+
30/33 11:42:46 598,697,909k 14,198k -M16 Nh2-g4 Ne2xf4+ Rf6xf4 Qe3xd2 Rc1xg1 Rb6xb7 Re5-g5+ Kg8-h8 Rg1xg3 Qd2xf4 Rg5-h5+ Kh8-g7 Nf2xe4 Qc4-f1+ Bh1-g2 Ra3xf3 Rh5-g5+ Qf4xg5 Bg2xf1 Qg5-h5+ Kh3-g2 Rf3xg3+ Ne4xg3 Qh5xg4 Bf1-e2 Qg4xg3+ Kg2-f1 Rb7-b1+ Be2-d1 Qg3-f3+ Kf1-e1 Rb1xd1+
30/37 11:42:46 598,697,909k 14,198k -M16 Rc1xg1 Ne2xg1+ Kh3-g2 Qe3xf2+ Kg2xf2 Qc4xd4+ Kf2-g2 Qd4xd2+ f3-f2 Qd2xf2+ Kg2xf2 Rb6xb7 Bh1-f3 Rb7-b2+ Nf4-e2 Ng1xe2 Rf6-g6+ Kg8-f7 Rg6-g7+ Ne8xg7 Re5-f5+ e4xf5 Bf3-d5+ Kf7-e7 Nh2-f1 Ne2-f4+ Nf1-d2 Rb2xd2+ Kf2-e1 Rd2-e2+ Ke1-d1 Ra3-a1+
30/42 11:42:46 598,697,909k 14,198k -M18 Qb7xd7 Ne2xf4+ Rf6xf4 Qe3xd2 Re5-g5+ Ne8-g7 Rc1xg1 Bd6xf4 Rg5xg7+ Bf8xg7 Nh2-g4 Qd2xd4 Qd7xd4 Bg7xd4 Kh3-g2 Bd4xf2 Rg1-d1 Rb6-g6 Rd1-d8+ Kg8-f7 Kg2-h3 Qc4-f1+ Bh1-g2 Ra3xf3 Rd8-d7+ Kf7-e6 Rd7-e7+ Ke6xe7 Bg2xf1 Ng3-h1+ Kh3-g2 Rf3-g3+ Kg2-h2 Rg3xg4+ Kh2xh1 Rg4-g1+
30/51 11:42:46 598,697,909k 14,198k -M19 Qb7-c8 Qc4xc8 Bd2xe3 Ne2xc1 Rf6-g6+ Kg8-f7 Re5-f5+ e4xf5 Nf2-g4 Rg1xh1 Kh3-g2 f5xg6 Nf4-e2 Rh1xh2+ Ng4xh2 Ng3xe2 Nh2-g4 Ne2-f4+ Kg2-g3 Nf4-h5+ Kg3-g2 Rb6-b2+ Be3-f2 Qc8-c6 d4-d3 Rb2xf2+ Kg2xf2 Nc1xd3+ Kf2-g1 Ra3-a1+ Kg1-g2 Ra1-a2+ Ng4-f2 Ra2xf2+ Kg2-h3 Qc6xf3+ Kh3-h4 Bf8-e7+
31/49 11:42:46 598,697,909k 14,198k -82.39 Qb7-d5+ e4xd5 Re5xe3 Ne2xc1 Rf6-g6+ Kg8-f7 Re3-e1 Rg1xe1 Rg6-h6 Bf8xh6 Bd2xe1 Ng3xh1 Kh3-g2 Bh6xf4 Nh2-g4 Nh1-g3 Ng4-h2 Ng3-f5 Nf2-e4 Nf5-h4+ Be1xh4 Qc4-c2+ Ne4-d2 Bf4xh2 f3-f2 Qc2xd2 d4-d3 Qd2xd3 f2-f1Q+ Qd3xf1+ Kg2xf1 Nc1-d3 Bh4-e1 Nd3xe1 Kf1xe1 Bd6-b4+ Ke1-d1
31/52 11:42:46 598,697,909k 14,198k -80.00 Qb7-a6 Qc4xa6 d4xe3 Ne2xc1 Re5-g5+ Ne8-g7 Nf4-d5 e4xd5 Nf2-g4 Rg1xh1 Kh3-g2 Rh1xh2+ Ng4xh2 Nd7xf6 Rg5-g6 Ng3-h5 Nh2-f1 Nh5-f4+ Kg2-g1 Qa6xf1+ Kg1xf1 Rb6-b1 f3-f2 Nc1-d3+ Bd2-e1 Nf4xg6 Kf1-g1 Nf6-e4 Kg1-f1 Nd3xe1 f2xe1Q Rb1xe1+ Kf1xe1 Ra3xe3+ Ke1-d1 Kg8-h7 Kd1-c2 Bd6-f4 Kc2-b2
31/53 11:42:46 598,697,909k 14,198k -53.63 Qb7-c6 Rb6xc6 Bd2xe3 Ne2xc1 Re5-g5+ Ne8-g7 Nf4-d5 e4xd5 Bh1-g2 Nd7xf6 Nh2-g4 Nf6-h5 Ng4-h6+ Kg8-h7 Nh6-f7 Ng7-e6 Rg5-g4 Nh5-f4+ Be3xf4 Bd6xf4 Rg4-h4+ Kh7-g6 Nf2-g4 Kg6xf7 Rh4-h7+ Kf7-g6 Ng4-e5+ Bb8xe5 Rh7-h4 Qc4xd4 Rh4-g4+ Kg6-f5 Rg4-g8 Bf4-g5 Rg8-h8 Be5xh8 Kh3-h2 Ng3-e4
31/53 11:42:46 598,697,909k 14,198k -53.60 Qb7-c7 Bb8xc7 Bd2xe3 Ne2xc1 Rf6-g6+ Ne8-g7 Nf4-d5 e4xd5 Bh1-g2 Bd6xe5 Rg6xb6 a5xb6 Nf2-g4 b6-b7 Ng4-h6+ Kg8-h7 Nh6-f7 b7-b8Q Nf7-g5+ Kh7-g6 Ng5-h7 Kg6xh7 f3-f2 Rg1-d1 Bg2-f3 Nc1-e2 Bf3-g4 Qb8-b5 Kh3-g2 Be5xd4 f2-f1Q
31/55 11:42:46 598,697,909k 14,198k -38.62 Rf6-g6+ Kg8-h8 Rg6-h6+ Bf8xh6 Nf4-g6+ Kh8-g8 Re5xe8+ Nd7-f8 d4xe3 Ne2xc1 Qb7-c8 Qc4-f7 Ng6-h4 Ng3xh1 Nf2xe4 Bh6xe3 Bd2-e1 Bd6xh2 Ne4-g5 Be3xg5 Re8xf8+ Qf7xf8 Qc8-c4+ Kg8-h7 Qc4-e4+ Kh7-g7 Qe4-d4+ Bg5-f6 Nh4-f5+ Kg7-f7 Qd4-d5+ Rb6-e6 Qd5-b7+ Kf7-g6 Qb7-b1 Re6xe1 Nf5-h4+ Kg6-h5 Qb1-f5+ Rg1-g5 Qf5-h7+ Qf8-h6 Qh7xh6+ Kh5xh6 Nh4-f5+ Rg5xf5
31/44 11:42:46 598,697,909k 14,198k -30.16 Bd2xe3 Nd7xe5 Rf6xf8+ Kg8xf8 Qb7-h7 Ne2xf4+ Be3xf4 Qc4-e6+ Nh2-g4 Ne5xg4 Nf2xg4 Rg1xh1+ Rc1xh1 Bd6xf4 Rh1-h2 Ra3xf3 Qh7-h8+ Kf8-f7 Kh3-g2 Qe6xg4 Rh2-h7+ Kf7-g6 Qh8-g8+ Kg6-f5 Rh7-f7+ Rb6-f6 Rf7-f8 Rf6xf8 Qg8xf8+ Ne8-f6 Qf8-c8+ Kf5-g5
31/51 11:42:46 598,697,909k 14,198k -29.60 Re5-g5+ Ne8-g7 Bd2xe3 Ne2xf4+ Rf6xf4 Qc4-e6+ Nh2-g4 Rg1xc1 Rf4xe4 Rc1xh1+ Kh3-g2 Ra3-a1 Nf2xh1 Ng3xe4 Qb7-d5 Ne4xg5 Ng4-h6+ Kg8-h7 Qd5xe6 Ng7xe6 f3-f2 Rb6-b1 Nh1-g3 Bd6xg3 Nh6-f5 Bg3-h2 Nf5-h4 Bf8-c5 Nh4-f3 Ng5xf3 Kg2xf3 Ne6xd4+ Kf3-g2 Rb1-b2 Kg2-h3 Kh7-g6 Kh3-g4 Nd7-e5+ Kg4-h4
31/47 11:42:46 598,697,909k 14,198k -28.28 d4xe3 Nd7xe5 f3xe2 Qc4xc1 Qb7-d5+ e4xd5 Bd2xc1 Rg1xc1 Bh1xd5+ Kg8-h8 Rf6-e6 Ra3xe3 Nh2-f3 Ne8-c7 Nf4-g6+ Kh8-g7 Ng6xe5 Nc7xe6 e2-e1R Rc1xe1 Ne5-d3 Ne6-g5+ Kh3-g2 Ng5xf3 Bd5-c4 Re1-g1+ Kg2-h3 Ng3-f5

[Dann Corbit]

Another way to approach the problem is with tablebase files
Examine the number of draws with two men as a percentage
Examine the number of draws with three men as a percentage
Examine the number of draws with four men as a percentage
Examine the number of draws with five men as a percentage
Examine the number of draws with six men as a percentage
Examine the number of draws with seven men as a percentage
Look to see if there is a recognizable sequence or pattern
If discovered, see if the pattern holds for 8 men, whenever that is completed.
If the pattern holds, compute the sequence for 9 to 32 men.
For each of the 31 sets, compute the totals.
Sum the results.

[Uri Blass]

Another way to approach the problem is with tablebase files
Examine the number of draws with two men as a percentage
Examine the number of draws with three men as a percentage
Examine the number of draws with four men as a percentage
Examine the number of draws with five men as a percentage
Examine the number of draws with six men as a percentage
Examine the number of draws with seven men as a percentage
Look to see if there is a recognizable sequence or pattern
If discovered, see if the pattern holds for 8 men, whenever that is completed.
If the pattern holds, compute the sequence for 9 to 32 men.
For each of the 31 sets, compute the totals.
Sum the results.

I do not expect the 9 men tablebase files in the near future so I do not think we can get results by this way.
Theb best way to get an estimate is to find the best engine for these random positions and to give it to play against itself and find the percentage of draws.

I believe in 99% of the cases results at 1 minute per move are going to be the same as the theoretical result and it may be interesting also to compare results at different time controls to see if the estimate is changed for some direction when we increase the time control.

[towforce]

Another way to approach the problem is with tablebase files
Examine the number of draws with two men as a percentage
Examine the number of draws with three men as a percentage
Examine the number of draws with four men as a percentage
Examine the number of draws with five men as a percentage
Examine the number of draws with six men as a percentage
Examine the number of draws with seven men as a percentage
Look to see if there is a recognizable sequence or pattern...

In this post, link, I used the method of differences (link) to see whether there might be a reasonable fit of a polynomial of degree n-1 (where n is the number of known values), but unfortunately, it doesn't look as though there is. You might be able to make a fit using other mathematical functions, but it's likely to be coincidence ("this combination of functions just happens to make a good fit to this data, but won't deliver new values accurately").

It might be worth trying a logarithmic fit or a reciprocal regression, but given that there isn't a good polynomial fit, my intuition tells me that other functions won't fit well either.

[chesskobra]

The problem (in crude approximation) is this: Place m pieces (some white and some black) on the board in all possible ways. In what fraction of the arrangements the material is equal. The draw rate will likely behave somewhat like that asymptotically (n x n board, m not too small), and the behaviour will not be polynomial. If you ask a similar question for coin toss: in n tosses of a fair coin, what is the probability of (heads = tails) (assume n even), then this will proportional to 1/sqrt(n). We may modify the question: how many sequences of black, white and blue, with black + white = m and black + white + blue = n, have the same number of blacks and whites. This can be answered easily, and the asymptotics in m will not tbe anything like a polynomial. One may also try to calculate 'expected imbalance of material' in a random position with m pieces on n x n board. This is a diffusion problem.