4 ms·

Presumably there are a mind-bogglingly-huge number of unique board state transitions. It's virtually impossible for the same game of chess to be played twice, e

by SamBam 2y ago

Presumably there are a mind-bogglingly-huge number of unique board state transitions. It's virtually impossible for the same game of chess to be played twice, except for silly scholar's-mate type games. Almost every single game in a chess database will have many unique board state transitions.

kristopolous 2y ago
I bet it's way less than you think - orders of magnitude. What could happen versus what does happen are entirely different. Doing some algebraic permutations computation here would be like claiming a 50,000 letter English document has 27^50,000 possibilities. I mean no, there's words, and they only go in certain orders and there's all these rules. Here's another approach: humans are pretty lousy when remembering large amounts of anything so let's say there's in practice, only been 100,000 unique games played over and over. Without the help of a computer or careful tabulation, I'm pretty sure no human would realize it because no human can remember 100,000 unique games. Anyways, it's worth digging into the data to see what the variation really is. I bet the 90th percentile is embarrassingly small with a long tail that's far shorter then most think edit: so I actually took 7.7 million games from https://www.ficsgames.org/download.html https://www.ficsgames.org/download.html and did some basic processing on them. These are people ostensibly with ELOs over 2000 which is pretty decent just to see if I'll eat crow on this one. Going in, I was expecting a uniqueness level to be something like 50-70%. Actual percentage of unique games over 7.7 million? 98.7%. Alright fine. Although I could try to do 1 billion games, I expected the distribution to be readily visible around 7 million. Now as an artifact of the data, I made the games as compact as possible, potentially leading to ambiguity maybe. So a game might look like so a3a5b3b6Bb2Bb7c4e6Nc3Nf6d4Be7Nf3O-Oe3c5d5exd5Nxd5d6Nxe7+Qxe7Bd3d5O-Odxc4Bxc4Nc6Qe2Rad8Rad1Ne4h3Nd6Ba6Bxa6Qxa6Qb7Qd3f6Qc3Rfe8Rd3Ne4Qc4+Kh8Rxd8Nxd8Rd1Qe7Qd5Nf7b4axb4axb4cxb4Qc6h6Rd7Qf8Qxb6Rd8Rxd8Qxd8Qxb4Qe8Bd4Kg8Qc4Nd6Qd5Kh8Nh4Kh7Qb3Qe4f3Qe6Qd3+f5e4g6exf5Nxf5Nxf5Qxf5Qxf5gxf5Bf2Kg6g4fxg4hxg4h5Kg2hxg4fxg4Ne5Kg3Nf7Kh4Ne5Be3Nxg41/2-1/2 Given this we can just run uniq with incrementing numbers and find out how things increase. I'm doing this on a pretty old laptop (3rd gen intel) so excuse me for cutting things a bit short number of characters / unique entries / percentage duplicates 99 7470034 0.039416039621657628 98 7462496 0.040385363441781119 97 7454437 0.041421683508957363 96 7446241 0.042475620631500677 95 7437517 0.043597454142611958 94 7428583 0.044746291899176449 93 7419379 0.045929849399894973 92 7409325 0.047222709798876217 91 7399016 0.048548361067336399 90 7388091 0.049953224789125783 89 7376939 0.051387278814333581 88 7364995 0.052923177422393386 87 7352785 0.054493281408027117 86 7340118 0.056122151775432672 85 7326671 0.057851323625950024 84 7312421 0.059683754567414482 83 7297272 0.061631789397747494 82 7281670 0.063638076243272224 80 7247260 0.068062914748240111 79 7228770 0.07044057426456829 78 7209233 0.072952869233227302 77 7187947 0.075690070989017588 76 7166719 0.07841981442939705 75 7144539 0.081271977115830896 74 7119854 0.084446261873027284 73 7094951 0.087648579608837096 72 7068178 0.091091363720824936 71 7039566 0.094770627867995505 70 7009607 0.098623104961001351 69 6978335 0.10264442288391196 68 6944221 0.10703119826195528 67 6909408 0.11150785919986417 66 6871309 0.1164070722832925 65 6831848 0.12148142718723132 64 6790088 0.12685141428305979 63 6744878 0.13266504255419009 62 6698251 0.13866088518630681 61 6648086 0.14511168505848671 60 6594883 0.15195314634822232 59 6540926 0.15889156573829932 58 6481469 0.16653723917595897 57 6419070 0.17456122923325301 56 6355249 0.18276807660975847 55 6282195 0.19216221064468775 54 6209896 0.20145925798763076 53 6133584 0.21127234360201919 52 6048154 0.22225792783565479 51 5963420 0.23315401228435984 50 5870519 0.24510030469790289 49 5770633 0.25794480975187595 48 5668750 0.27104611232094422 47 5558836 0.28518013439112821 46 5443035 0.30007117547551587 45 5323607 0.31542861845637304 44 5192507 0.33228698311784588 43 5064129 0.34879532132158775 42 4925937 0.36656565792950735 41 4777302 0.38567887708631909 40 4626169 0.40511331817237839 39 4465105 0.42582480288508218 38 4301802 0.44682420429097458 37 4126391 0.46938059333470927 36 3948981 0.49219403707682896 35 3770310 0.5151696348833128 34 3581328 0.5394711411415466 33 3387960 0.5643366503548165 32 3202702 0.58815928132701434 31 3008299 0.61315788289287476 30 2810171 0.63863548833641626 29 2617450 0.66341779875536144 28 2413377 0.68965988152851743 26 2044415 0.73710531205655982 25 1849362 0.76218749819167997 24 1674682 0.78464988674290859 23 1499682 0.80715342462054207 22 1324053 0.82973784664289008 21 1167329 0.84989124361622848 20 1005747 0.87066933880105002 19 862832 0.88904701374837569 18 733254 0.90570966192613567 17 599187 0.9229495579983682 16 489782 0.93701812692123954 15 395141 0.94918816879710877 14 300280 0.9613865008348812 13 233128 0.97002168698093183 12 167915 0.97840753392729818 11 115118 0.98519678700915769 10 78474 0.98990889924908909 9 46588 0.9940091724420389 8 26581 0.99658190548385495 7 14316 0.99815908200996462 6 5390 0.99930689103336889 5 2659 0.99965807481590496 4 408 0.9999475346088339 3 180 0.99997685350389731 2 20 0.99999742816709969 1 9 0.9999988426751949 It would be interesting to see later, when I crunch larger numbers on a more capable machine, if these distributions generally hold. Of course it won't, it's not possible. But I'm wondering if it's greater than what the Shannon limit would predict. An ancillary analysis would be to compare it to the possible legal permutations at a character count although this would of course require a board and rule model. I would expect those percentages to decrease as the length increases and perhaps such a function can give more predictive heft to the actual "language" of chess in practice
- kristopolous 2y ago
  It's also worth noting that unique string != board state. Proof: Both black and white could move the left rook pawn as their first move and right rook pawn as the second move. Now reset the board and do right rook first and left rook second. Same board state, different game string. In practice unique board states is a strict subset of number of moves but given how far off I was on my first assessment ... I wonder if we're talking about another < 2% hit. All of this is dependent on an actual engine that can process the notation. There's apparently lots of options for pgn. I'd also like to develop a heatmap based on statistical analysis. I'd imagine this would not only be way less than equally distributed but there'd be no way to slice the data to make it appear equally distributed

by SamBam 2y ago

kristopolous 2y ago

I bet it's way less than you think - orders of magnitude. What could happen versus what does happen are entirely different. Doing some algebraic permutations computation here would be like claiming a 50,000 letter English document has 27^50,000 possibilities. I mean no, there's words, and they only go in certain orders and there's all these rules. Here's another approach: humans are pretty lousy when remembering large amounts of anything so let's say there's in practice, only been 100,000 unique games played over and over. Without the help of a computer or careful tabulation, I'm pretty sure no human would realize it because no human can remember 100,000 unique games. Anyways, it's worth digging into the data to see what the variation really is. I bet the 90th percentile is embarrassingly small with a long tail that's far shorter then most think edit: so I actually took 7.7 million games from https://www.ficsgames.org/download.html https://www.ficsgames.org/download.html and did some basic processing on them. These are people ostensibly with ELOs over 2000 which is pretty decent just to see if I'll eat crow on this one. Going in, I was expecting a uniqueness level to be something like 50-70%. Actual percentage of unique games over 7.7 million? 98.7%. Alright fine. Although I could try to do 1 billion games, I expected the distribution to be readily visible around 7 million. Now as an artifact of the data, I made the games as compact as possible, potentially leading to ambiguity maybe. So a game might look like so a3a5b3b6Bb2Bb7c4e6Nc3Nf6d4Be7Nf3O-Oe3c5d5exd5Nxd5d6Nxe7+Qxe7Bd3d5O-Odxc4Bxc4Nc6Qe2Rad8Rad1Ne4h3Nd6Ba6Bxa6Qxa6Qb7Qd3f6Qc3Rfe8Rd3Ne4Qc4+Kh8Rxd8Nxd8Rd1Qe7Qd5Nf7b4axb4axb4cxb4Qc6h6Rd7Qf8Qxb6Rd8Rxd8Qxd8Qxb4Qe8Bd4Kg8Qc4Nd6Qd5Kh8Nh4Kh7Qb3Qe4f3Qe6Qd3+f5e4g6exf5Nxf5Nxf5Qxf5Qxf5gxf5Bf2Kg6g4fxg4hxg4h5Kg2hxg4fxg4Ne5Kg3Nf7Kh4Ne5Be3Nxg41/2-1/2 Given this we can just run uniq with incrementing numbers and find out how things increase. I'm doing this on a pretty old laptop (3rd gen intel) so excuse me for cutting things a bit short number of characters / unique entries / percentage duplicates 99 7470034 0.039416039621657628 98 7462496 0.040385363441781119 97 7454437 0.041421683508957363 96 7446241 0.042475620631500677 95 7437517 0.043597454142611958 94 7428583 0.044746291899176449 93 7419379 0.045929849399894973 92 7409325 0.047222709798876217 91 7399016 0.048548361067336399 90 7388091 0.049953224789125783 89 7376939 0.051387278814333581 88 7364995 0.052923177422393386 87 7352785 0.054493281408027117 86 7340118 0.056122151775432672 85 7326671 0.057851323625950024 84 7312421 0.059683754567414482 83 7297272 0.061631789397747494 82 7281670 0.063638076243272224 80 7247260 0.068062914748240111 79 7228770 0.07044057426456829 78 7209233 0.072952869233227302 77 7187947 0.075690070989017588 76 7166719 0.07841981442939705 75 7144539 0.081271977115830896 74 7119854 0.084446261873027284 73 7094951 0.087648579608837096 72 7068178 0.091091363720824936 71 7039566 0.094770627867995505 70 7009607 0.098623104961001351 69 6978335 0.10264442288391196 68 6944221 0.10703119826195528 67 6909408 0.11150785919986417 66 6871309 0.1164070722832925 65 6831848 0.12148142718723132 64 6790088 0.12685141428305979 63 6744878 0.13266504255419009 62 6698251 0.13866088518630681 61 6648086 0.14511168505848671 60 6594883 0.15195314634822232 59 6540926 0.15889156573829932 58 6481469 0.16653723917595897 57 6419070 0.17456122923325301 56 6355249 0.18276807660975847 55 6282195 0.19216221064468775 54 6209896 0.20145925798763076 53 6133584 0.21127234360201919 52 6048154 0.22225792783565479 51 5963420 0.23315401228435984 50 5870519 0.24510030469790289 49 5770633 0.25794480975187595 48 5668750 0.27104611232094422 47 5558836 0.28518013439112821 46 5443035 0.30007117547551587 45 5323607 0.31542861845637304 44 5192507 0.33228698311784588 43 5064129 0.34879532132158775 42 4925937 0.36656565792950735 41 4777302 0.38567887708631909 40 4626169 0.40511331817237839 39 4465105 0.42582480288508218 38 4301802 0.44682420429097458 37 4126391 0.46938059333470927 36 3948981 0.49219403707682896 35 3770310 0.5151696348833128 34 3581328 0.5394711411415466 33 3387960 0.5643366503548165 32 3202702 0.58815928132701434 31 3008299 0.61315788289287476 30 2810171 0.63863548833641626 29 2617450 0.66341779875536144 28 2413377 0.68965988152851743 26 2044415 0.73710531205655982 25 1849362 0.76218749819167997 24 1674682 0.78464988674290859 23 1499682 0.80715342462054207 22 1324053 0.82973784664289008 21 1167329 0.84989124361622848 20 1005747 0.87066933880105002 19 862832 0.88904701374837569 18 733254 0.90570966192613567 17 599187 0.9229495579983682 16 489782 0.93701812692123954 15 395141 0.94918816879710877 14 300280 0.9613865008348812 13 233128 0.97002168698093183 12 167915 0.97840753392729818 11 115118 0.98519678700915769 10 78474 0.98990889924908909 9 46588 0.9940091724420389 8 26581 0.99658190548385495 7 14316 0.99815908200996462 6 5390 0.99930689103336889 5 2659 0.99965807481590496 4 408 0.9999475346088339 3 180 0.99997685350389731 2 20 0.99999742816709969 1 9 0.9999988426751949 It would be interesting to see later, when I crunch larger numbers on a more capable machine, if these distributions generally hold. Of course it won't, it's not possible. But I'm wondering if it's greater than what the Shannon limit would predict. An ancillary analysis would be to compare it to the possible legal permutations at a character count although this would of course require a board and rule model. I would expect those percentages to decrease as the length increases and perhaps such a function can give more predictive heft to the actual "language" of chess in practice

kristopolous 2y ago
It's also worth noting that unique string != board state. Proof: Both black and white could move the left rook pawn as their first move and right rook pawn as the second move. Now reset the board and do right rook first and left rook second. Same board state, different game string. In practice unique board states is a strict subset of number of moves but given how far off I was on my first assessment ... I wonder if we're talking about another < 2% hit. All of this is dependent on an actual engine that can process the notation. There's apparently lots of options for pgn. I'd also like to develop a heatmap based on statistical analysis. I'd imagine this would not only be way less than equally distributed but there'd be no way to slice the data to make it appear equally distributed