6 ms·

Is Chess with Queen Odds a Provable Win?

52 points by randomwalker 16y ago

ig1 16y ago
+1 for funding issues. When I was working on my undergraduate thesis on applying machine learning techniques to board games, I developed the basis of a new method of analyzing board games (using graphs of games and graph isomorphism) but I wasn't able to find anyone who could finance me doing a research masters in it. Although I think part of my issue was that it wasn't obviously Computer Science, nor was it obviously Mathematics.
- troutwine 16y ago
  The old saw still rings true: turn your research problem into a military application and the DoD will likely fund it. Perhaps instead modeling a chess board as a graph you model passable routes in a mountainous terrain. Instead of chess pieces, maybe the forces you're modeling are small groups with different movement characteristics?
  - jrockway 16y ago
    Which chess piece is the guys with guns, and which chess piece is the airstrike?
    - troutwine 16y ago
      "In this paper we assume all forces are ground-based and beyond any hope of air-support or resupply. Historical analysis of battle-field circumstance will reveal numerous instances--even in the modern period--of such occurrences. A more general result is beyond the scope or intention of this paper."
- bumbledraven 16y ago
  Sounds interesting. Is an overview of your new method available to the public?
  - ig1 16y ago
    No, mostly I just had a whole bunch of vaguely related ideas to do with analyzing the graphs of games, but I really need to find some time to hammer out the details to get something I could actually publish. I spent a lot of time while writing my thesis on machine learning thinking about the different approaches that have been taken to board games and most tend to involve some sort of analysis of the board, and it occurred to me that rather than studying the board it might make sense to study the graph that shows how those different board states are connected. After all you could have two games that superficially from their boards and rules look completely different but have identical graphs. Now what if we could solve one of the games easily but the other game was hard, if the graphs are identical you could covert any game state between the two games, thus using the easy game to solve the hard game. So then you can look at things at how to create an easy game which has the same graph as a hard game, obviously for something like chess this is hard, but you don't have to replicate the whole game graph, you can do it in chunks breaking the big graph into a series of smaller graphs and then attacking them. There's all sort of other interesting things you can do with game graphs, for example you can start enumerating all games with a certain complexity (in terms of node and vertices) and ask questions like "are some game graphs fundamentally more difficult than others" and perhaps more importantly "why?". I think the area has a big potential for deepening our understanding of board games and also raising some interesting complexity and information theory problems.
- davidst 16y ago
  Was it anything like the 'Method of Analogies' from KAISSA?
shinkansen 16y ago
> What if we make things easier for the machine? It is obvious to a rank beginner that a perfect game with a rook handicap is a win for the side with the material advantage. No, make it a queen! Surely that must be a provable win? Hm, I'm sure it must be. Although I don't know how you go about proving it, it's a simple matter to force equal trades; black cannot avoid the exchange of pieces forever, and if white plays a perfect game he will always win, without doubt. > Not so fast. Even against a crushing asymmetry in material, it is not too hard to avoid mate for a couple of dozen moves, which means that calculating all the way to the end of the game is beyond the reach of search-based algorithms. Okay, just calculate the moves it would take to force the equal exchange of material from a given position. Generally as the game progresses and the board opens up it becomes inescapable. After a certain point, when enough material has been removed from the board, looking for mate becomes trivial. Esp if you operate with such a commanding advantage as a queen...assuming you can force the equal exchange of all other material, it is possible to calculate checkmate within a couple of moves.
- randomwalker 16y ago
  I'm sorry, but that is simply incorrect. it's a simple matter to force equal trades; black cannot avoid the exchange of pieces forever, and if white plays a perfect game he will always win, without doubt. Yes, that's pretty much the human intuition for why no one doubts that White will win. But it is very, very far from a mathematical proof. Okay, just calculate the moves it would take to force the equal exchange of material from a given position. Let me remind you that each ply has a branching factor of about 20, which means each move has a branching factor of several hundred. In most positions you'd be lucky to be able to calculate even one or two forced exchanges, let alone all the way to the end of the game. A program attempting to prove victory operates in a very different context from a normal chess playing program — it is not allowed to prune any positions at all. We haven't even found the status of all seven piece endings yet! That is despite intense effort. See http://en.wikipedia.org/wiki/Endgame_tablebase http://en.wikipedia.org/wiki/Endgame_tablebase It is utterly, utterly inconceivable that a search-based approach will ever prove victory with Queen odds.
  - 16y ago
nas 16y ago
> Indeed, if you start from a closed position where strategy dominates and tactics are of relatively little use, top human players can still trounce computers I haven't been following computer chess as much as I used to be I suspect this is actually untrue. Computer speed has been increasing as always but there has also been massive gains in chess program strength (e.g. compared to old programs when running on the same hardware). For example, Rybka 4 would absolutely crush programs from 10 years ago. Those combined gains make computer chess programs so strong that I doubt even top players can hold them off even with the position looks quiet. Rybka can defeat GMs even when they have pawn plus first move odds[1]. In one match Rybka had a tiny opening book so the GM had every opportunity to steer the game in a strategic direction. 1. <http://en.wikipedia.org/wiki/Rybka#Odds_matches_versus_grandmasters&#62 http://en.wikipedia.org/wiki/Rybka#Odds_matches_versus_grand...;
- randomwalker 16y ago
  Good point. When I wrote that what I had in mind is puzzle-like positions where strategic planning is required. For example see http://en.wikipedia.org/wiki/Fortress_(chess)#Defense_perimeter_.28pawn_fortress.29 http://en.wikipedia.org/wiki/Fortress_(chess)#Defense_perime... (Petrosian vs Hazai). My real point here is that no chess program incorporates strategy the way humans do, and that it is unfortunate that this goal isn't being pursued.
  - jacquesm 16y ago
    Strategy is just our way of compensating for low clockspeeds.
    - Confusion 16y ago
      Which is why we will always win some contests, if they don't turn to trying their hand at strategy. Clockspeeds will always be too low for some problems.
- Bootvis 16y ago
  On the other hand you have beautiful games like this one: http://www.chessgames.com/perl/chessgame?gid=1124533 http://www.chessgames.com/perl/chessgame?gid=1124533 where the computer has a lot of trouble finding the best move. Clearly programs still do not understand at the level that was hoped for in the 60's. Furthermore top human players with computer assistance can still exploit the weaknesses engines have in closed positions.
jderick 16y ago
There is another closely related field that has plenty of real applications: formal verification. Essentially the task is to ensure no reachable state violates some property. The problem is the same as in chess, the reachable state space is enormous. There is probably more funding available for this take on the problem than chess, but it has been studied for decades and there are no easy answers.
- bane 16y ago
  Out of curiosity, I wonder how long it'll be before we can simply compute every possible valid game, inserting every valid move into some type of tree structure, and have a computer simply walk the tree structure always aiming for a "win" leaf. Anybody have any ideas on how big the data for that would be? (I suspect there could be some optimizations as some games that start differently could end up in the same state as some other games -- i.e. optimize for board state collisions).
  - reitzensteinm 16y ago
    It's possible we never will. Although we take Moore's law for granted, eventually we'll hit the natural limits of computation and storage per m^3, and solving chess completely may well be beyond that (but maybe not, as well). From Wikipedia (http://en.wikipedia.org/wiki/Shannon_number http://en.wikipedia.org/wiki/Shannon_number): "Allis also estimated the game-tree complexity to be at least 10^123, "based on an average branching factor of 35 and an average game length of 80". As a comparison, the number of atoms in the observable universe, to which it is often compared, is estimated to be between 4×10^79 and 10^81."
    - bane 16y ago
      I'm wondering if that number is for every possible game valid or invalid, or just valid games. In case anybody is interested http://en.wikipedia.org/wiki/Solved_game http://en.wikipedia.org/wiki/Solved_game
  - jacquesm 16y ago
    Unless quantum computing pays off in unexpected ways I would wager 'never'.
Dove 16y ago
The links from the article are fascinating. I had never heard of Arimaa before, and am quite intrigued. And from the linked article by Kasparov (http://www.nybooks.com/articles/archives/2010/feb/11/the-chess-master-and-the-computer/ http://www.nybooks.com/articles/archives/2010/feb/11/the-che...) comes this gem: In what Rasskin-Gutman explains as Moravec’s Paradox, in chess, as in so many things, what computers are good at is where humans are weak, and vice versa. This gave me an idea for an experiment. What if instead of human versus machine we played as partners? My brainchild saw the light of day in a match in 1998 in León, Spain, and we called it “Advanced Chess.” Each player had a PC at hand running the chess software of his choice during the game. The idea was to create the highest level of chess ever played, a synthesis of the best of man and machine. .... In 2005, the online chess-playing site Playchess.com hosted what it called a “freestyle” chess tournament in which anyone could compete in teams with other players or computers. Normally, “anti-cheating” algorithms are employed by online sites to prevent, or at least discourage, players from cheating with computer assistance. (I wonder if these detection algorithms, which employ diagnostic analysis of moves and calculate probabilities, are any less “intelligent” than the playing programs they detect.) Lured by the substantial prize money, several groups of strong grandmasters working with several computers at the same time entered the competition. At first, the results seemed predictable. The teams of human plus machine dominated even the strongest computers. The chess machine Hydra, which is a chess-specific supercomputer like Deep Blue, was no match for a strong human player using a relatively weak laptop. Human strategic guidance combined with the tactical acuity of a computer was overwhelming. The surprise came at the conclusion of the event. The winner was revealed to be not a grandmaster with a state-of-the-art PC but a pair of amateur American chess players using three computers at the same time. Their skill at manipulating and “coaching” their computers to look very deeply into positions effectively counteracted the superior chess understanding of their grandmaster opponents and the greater computational power of other participants. Weak human + machine + better process was superior to a strong computer alone and, more remarkably, superior to a strong human + machine + inferior process.
- gjm11 16y ago
  Moravec's "Paradox" isn't really paradoxical at all. What makes us say that computers are "strong" in some area? Answer: the fact that they are good there relative to our prior expectations. Where do those expectations come from? From looking at ourselves. Similarly: what determines where we say we're "weak"? Answer: seeing other things being strong relative to us. Computers, for instance. Now, of course there are other points of reference for deciding where we're "weak". Doesn't that invalidate my argument? Why, no, because actually if we use those other points of reference the paradox rather goes away. For instance: Computers do arithmetic very, very fast. Is that an area where we're weak? Only if we compare ourselves against computers; we do arithmetic much better than chickens or tigers do.
amalcon 16y ago
Game researchers don't ignore all Chess-like games that are more difficult for computers. It's just that they've met with very, very limited success in Go. Combined with Go's relative unpopularity outside of southeast Asia, this means you don't hear about it very much. For example, according to Wikipedia, a computer actually beat a professional Go player for the first time in 2008.
- mquander 16y ago
  Sure, perhaps with a huge handicap or on a 9x9 board. There exists no computer that could ever beat any professional go player in an even 19x19 game, even if you gave the professional as many beers as he could drink.
  - amalcon 16y ago
    Well, that's what it says on Wikipedia -- though as it happens, the citations are both dead links, so your guess is as good as mine. edit: Trivial search turned this up: http://www.sciencedaily.com/releases/2009/05/090514083931.htm http://www.sciencedaily.com/releases/2009/05/090514083931.ht... Wikipedia was apparently talking about a handicap match. Still, it's very telling that 2008 was the first time a computer beat a pro even in a handicap match.
    - billforsternz 16y ago
      No his guess is better than yours. Go programs are still trying to compete with amateurs.
    - monkeypizza 16y ago
      well, handicaps can be arbitrarily large. so a bot could have beaten a pro with handi even 40 years ago, as long as you make the handi large enough. recently bots have gotten a lot better at go - they just passed 3dan rank, which is above where most amateurs ever reach. http://www.lifein19x19.com/forum/viewtopic.php?f=18&t=1344&start=0 http://www.lifein19x19.com/forum/viewtopic.php?f=18&t=13... But they still have a ways to go. Here is a (fake money) future prediction market on whether a bot will beat a pro in an even game by 2020 - running at 20% right now. http://www.ideosphere.com/fx-bin/Claim?claim=GoCh http://www.ideosphere.com/fx-bin/Claim?claim=GoCh
xenophanes 16y ago
If anyone is serious about proving things about chess I would suggest they start by proving things about simpler chess variants. The variant of chess "wild 5" where your pawns start on the 7th rank and pieces on the 8th (so pawns promote in one move) has far fewer choices than in chess. It's a simpler game. It has a significant advantage for white, much larger than in normal chess. Humans came somewhat near to proving it's a win for white just by learning the (relatively few) openings out to 20 or so moves. At each step there's usually only a couple moves that aren't terrible. For the first six moves there is a single way of playing which is considered best for both sides. Yet even in this much simpler game which humans are near cracking, I think a pure math type approach would have a very hard time getting anywhere. If you can't do anything there, you could always try an even simpler game. There is a game called pawns where you start with only your pawns. If you promote you win instantly. Math ought to be able to solve that one. If you crack that, move on to little chess (normal chess but only with pawns and kings). Little chess should no doubt be a draw (you can waste moves with your king unlike in pawns where it's less clear) but proving that would be a good accomplishment I think.

52 points by randomwalker 16y ago

ig1 16y ago

+1 for funding issues. When I was working on my undergraduate thesis on applying machine learning techniques to board games, I developed the basis of a new method of analyzing board games (using graphs of games and graph isomorphism) but I wasn't able to find anyone who could finance me doing a research masters in it. Although I think part of my issue was that it wasn't obviously Computer Science, nor was it obviously Mathematics.

troutwine 16y ago
The old saw still rings true: turn your research problem into a military application and the DoD will likely fund it. Perhaps instead modeling a chess board as a graph you model passable routes in a mountainous terrain. Instead of chess pieces, maybe the forces you're modeling are small groups with different movement characteristics?
- jrockway 16y ago
  Which chess piece is the guys with guns, and which chess piece is the airstrike?
  - troutwine 16y ago
    "In this paper we assume all forces are ground-based and beyond any hope of air-support or resupply. Historical analysis of battle-field circumstance will reveal numerous instances--even in the modern period--of such occurrences. A more general result is beyond the scope or intention of this paper."
bumbledraven 16y ago
Sounds interesting. Is an overview of your new method available to the public?
- ig1 16y ago
  No, mostly I just had a whole bunch of vaguely related ideas to do with analyzing the graphs of games, but I really need to find some time to hammer out the details to get something I could actually publish. I spent a lot of time while writing my thesis on machine learning thinking about the different approaches that have been taken to board games and most tend to involve some sort of analysis of the board, and it occurred to me that rather than studying the board it might make sense to study the graph that shows how those different board states are connected. After all you could have two games that superficially from their boards and rules look completely different but have identical graphs. Now what if we could solve one of the games easily but the other game was hard, if the graphs are identical you could covert any game state between the two games, thus using the easy game to solve the hard game. So then you can look at things at how to create an easy game which has the same graph as a hard game, obviously for something like chess this is hard, but you don't have to replicate the whole game graph, you can do it in chunks breaking the big graph into a series of smaller graphs and then attacking them. There's all sort of other interesting things you can do with game graphs, for example you can start enumerating all games with a certain complexity (in terms of node and vertices) and ask questions like "are some game graphs fundamentally more difficult than others" and perhaps more importantly "why?". I think the area has a big potential for deepening our understanding of board games and also raising some interesting complexity and information theory problems.
davidst 16y ago
Was it anything like the 'Method of Analogies' from KAISSA?

shinkansen 16y ago

> What if we make things easier for the machine? It is obvious to a rank beginner that a perfect game with a rook handicap is a win for the side with the material advantage. No, make it a queen! Surely that must be a provable win? Hm, I'm sure it must be. Although I don't know how you go about proving it, it's a simple matter to force equal trades; black cannot avoid the exchange of pieces forever, and if white plays a perfect game he will always win, without doubt. > Not so fast. Even against a crushing asymmetry in material, it is not too hard to avoid mate for a couple of dozen moves, which means that calculating all the way to the end of the game is beyond the reach of search-based algorithms. Okay, just calculate the moves it would take to force the equal exchange of material from a given position. Generally as the game progresses and the board opens up it becomes inescapable. After a certain point, when enough material has been removed from the board, looking for mate becomes trivial. Esp if you operate with such a commanding advantage as a queen...assuming you can force the equal exchange of all other material, it is possible to calculate checkmate within a couple of moves.

randomwalker 16y ago
I'm sorry, but that is simply incorrect. it's a simple matter to force equal trades; black cannot avoid the exchange of pieces forever, and if white plays a perfect game he will always win, without doubt. Yes, that's pretty much the human intuition for why no one doubts that White will win. But it is very, very far from a mathematical proof. Okay, just calculate the moves it would take to force the equal exchange of material from a given position. Let me remind you that each ply has a branching factor of about 20, which means each move has a branching factor of several hundred. In most positions you'd be lucky to be able to calculate even one or two forced exchanges, let alone all the way to the end of the game. A program attempting to prove victory operates in a very different context from a normal chess playing program — it is not allowed to prune any positions at all. We haven't even found the status of all seven piece endings yet! That is despite intense effort. See http://en.wikipedia.org/wiki/Endgame_tablebase http://en.wikipedia.org/wiki/Endgame_tablebase It is utterly, utterly inconceivable that a search-based approach will ever prove victory with Queen odds.
- 16y ago

nas 16y ago

> Indeed, if you start from a closed position where strategy dominates and tactics are of relatively little use, top human players can still trounce computers I haven't been following computer chess as much as I used to be I suspect this is actually untrue. Computer speed has been increasing as always but there has also been massive gains in chess program strength (e.g. compared to old programs when running on the same hardware). For example, Rybka 4 would absolutely crush programs from 10 years ago. Those combined gains make computer chess programs so strong that I doubt even top players can hold them off even with the position looks quiet. Rybka can defeat GMs even when they have pawn plus first move odds[1]. In one match Rybka had a tiny opening book so the GM had every opportunity to steer the game in a strategic direction. 1. <http://en.wikipedia.org/wiki/Rybka#Odds_matches_versus_grandmasters&#62 http://en.wikipedia.org/wiki/Rybka#Odds_matches_versus_grand...;

randomwalker 16y ago
Good point. When I wrote that what I had in mind is puzzle-like positions where strategic planning is required. For example see http://en.wikipedia.org/wiki/Fortress_(chess)#Defense_perimeter_.28pawn_fortress.29 http://en.wikipedia.org/wiki/Fortress_(chess)#Defense_perime... (Petrosian vs Hazai). My real point here is that no chess program incorporates strategy the way humans do, and that it is unfortunate that this goal isn't being pursued.
- jacquesm 16y ago
  Strategy is just our way of compensating for low clockspeeds.
  - Confusion 16y ago
    Which is why we will always win some contests, if they don't turn to trying their hand at strategy. Clockspeeds will always be too low for some problems.
Bootvis 16y ago
On the other hand you have beautiful games like this one: http://www.chessgames.com/perl/chessgame?gid=1124533 http://www.chessgames.com/perl/chessgame?gid=1124533 where the computer has a lot of trouble finding the best move. Clearly programs still do not understand at the level that was hoped for in the 60's. Furthermore top human players with computer assistance can still exploit the weaknesses engines have in closed positions.

jderick 16y ago

There is another closely related field that has plenty of real applications: formal verification. Essentially the task is to ensure no reachable state violates some property. The problem is the same as in chess, the reachable state space is enormous. There is probably more funding available for this take on the problem than chess, but it has been studied for decades and there are no easy answers.

bane 16y ago
Out of curiosity, I wonder how long it'll be before we can simply compute every possible valid game, inserting every valid move into some type of tree structure, and have a computer simply walk the tree structure always aiming for a "win" leaf. Anybody have any ideas on how big the data for that would be? (I suspect there could be some optimizations as some games that start differently could end up in the same state as some other games -- i.e. optimize for board state collisions).
- reitzensteinm 16y ago
  It's possible we never will. Although we take Moore's law for granted, eventually we'll hit the natural limits of computation and storage per m^3, and solving chess completely may well be beyond that (but maybe not, as well). From Wikipedia (http://en.wikipedia.org/wiki/Shannon_number http://en.wikipedia.org/wiki/Shannon_number): "Allis also estimated the game-tree complexity to be at least 10^123, "based on an average branching factor of 35 and an average game length of 80". As a comparison, the number of atoms in the observable universe, to which it is often compared, is estimated to be between 4×10^79 and 10^81."
  - bane 16y ago
    I'm wondering if that number is for every possible game valid or invalid, or just valid games. In case anybody is interested http://en.wikipedia.org/wiki/Solved_game http://en.wikipedia.org/wiki/Solved_game
- jacquesm 16y ago
  Unless quantum computing pays off in unexpected ways I would wager 'never'.

Dove 16y ago

The links from the article are fascinating. I had never heard of Arimaa before, and am quite intrigued. And from the linked article by Kasparov (http://www.nybooks.com/articles/archives/2010/feb/11/the-chess-master-and-the-computer/ http://www.nybooks.com/articles/archives/2010/feb/11/the-che...) comes this gem: In what Rasskin-Gutman explains as Moravec’s Paradox, in chess, as in so many things, what computers are good at is where humans are weak, and vice versa. This gave me an idea for an experiment. What if instead of human versus machine we played as partners? My brainchild saw the light of day in a match in 1998 in León, Spain, and we called it “Advanced Chess.” Each player had a PC at hand running the chess software of his choice during the game. The idea was to create the highest level of chess ever played, a synthesis of the best of man and machine. .... In 2005, the online chess-playing site Playchess.com hosted what it called a “freestyle” chess tournament in which anyone could compete in teams with other players or computers. Normally, “anti-cheating” algorithms are employed by online sites to prevent, or at least discourage, players from cheating with computer assistance. (I wonder if these detection algorithms, which employ diagnostic analysis of moves and calculate probabilities, are any less “intelligent” than the playing programs they detect.) Lured by the substantial prize money, several groups of strong grandmasters working with several computers at the same time entered the competition. At first, the results seemed predictable. The teams of human plus machine dominated even the strongest computers. The chess machine Hydra, which is a chess-specific supercomputer like Deep Blue, was no match for a strong human player using a relatively weak laptop. Human strategic guidance combined with the tactical acuity of a computer was overwhelming. The surprise came at the conclusion of the event. The winner was revealed to be not a grandmaster with a state-of-the-art PC but a pair of amateur American chess players using three computers at the same time. Their skill at manipulating and “coaching” their computers to look very deeply into positions effectively counteracted the superior chess understanding of their grandmaster opponents and the greater computational power of other participants. Weak human + machine + better process was superior to a strong computer alone and, more remarkably, superior to a strong human + machine + inferior process.

gjm11 16y ago
Moravec's "Paradox" isn't really paradoxical at all. What makes us say that computers are "strong" in some area? Answer: the fact that they are good there relative to our prior expectations. Where do those expectations come from? From looking at ourselves. Similarly: what determines where we say we're "weak"? Answer: seeing other things being strong relative to us. Computers, for instance. Now, of course there are other points of reference for deciding where we're "weak". Doesn't that invalidate my argument? Why, no, because actually if we use those other points of reference the paradox rather goes away. For instance: Computers do arithmetic very, very fast. Is that an area where we're weak? Only if we compare ourselves against computers; we do arithmetic much better than chickens or tigers do.

amalcon 16y ago

Game researchers don't ignore all Chess-like games that are more difficult for computers. It's just that they've met with very, very limited success in Go. Combined with Go's relative unpopularity outside of southeast Asia, this means you don't hear about it very much. For example, according to Wikipedia, a computer actually beat a professional Go player for the first time in 2008.

mquander 16y ago
Sure, perhaps with a huge handicap or on a 9x9 board. There exists no computer that could ever beat any professional go player in an even 19x19 game, even if you gave the professional as many beers as he could drink.
- amalcon 16y ago
  Well, that's what it says on Wikipedia -- though as it happens, the citations are both dead links, so your guess is as good as mine. edit: Trivial search turned this up: http://www.sciencedaily.com/releases/2009/05/090514083931.htm http://www.sciencedaily.com/releases/2009/05/090514083931.ht... Wikipedia was apparently talking about a handicap match. Still, it's very telling that 2008 was the first time a computer beat a pro even in a handicap match.
  - billforsternz 16y ago
    No his guess is better than yours. Go programs are still trying to compete with amateurs.
  - monkeypizza 16y ago
    well, handicaps can be arbitrarily large. so a bot could have beaten a pro with handi even 40 years ago, as long as you make the handi large enough. recently bots have gotten a lot better at go - they just passed 3dan rank, which is above where most amateurs ever reach. http://www.lifein19x19.com/forum/viewtopic.php?f=18&t=1344&start=0 http://www.lifein19x19.com/forum/viewtopic.php?f=18&t=13... But they still have a ways to go. Here is a (fake money) future prediction market on whether a bot will beat a pro in an even game by 2020 - running at 20% right now. http://www.ideosphere.com/fx-bin/Claim?claim=GoCh http://www.ideosphere.com/fx-bin/Claim?claim=GoCh

xenophanes 16y ago

If anyone is serious about proving things about chess I would suggest they start by proving things about simpler chess variants. The variant of chess "wild 5" where your pawns start on the 7th rank and pieces on the 8th (so pawns promote in one move) has far fewer choices than in chess. It's a simpler game. It has a significant advantage for white, much larger than in normal chess. Humans came somewhat near to proving it's a win for white just by learning the (relatively few) openings out to 20 or so moves. At each step there's usually only a couple moves that aren't terrible. For the first six moves there is a single way of playing which is considered best for both sides. Yet even in this much simpler game which humans are near cracking, I think a pure math type approach would have a very hard time getting anywhere. If you can't do anything there, you could always try an even simpler game. There is a game called pawns where you start with only your pawns. If you promote you win instantly. Math ought to be able to solve that one. If you crack that, move on to little chess (normal chess but only with pawns and kings). Little chess should no doubt be a draw (you can waste moves with your king unlike in pawns where it's less clear) but proving that would be a good accomplishment I think.