I've recently become interested in infinite chess (this presentation from Joel David Hamkins [updated presentation based on the link provided by JDH in comments; significantly more comprehensive than what was originally linked to] talks a bit about it and some of the interesting mathematical questions). You can find another discussion of the mathematical properties of infinite chess here.

My question is, has this variant been much explored by chess players (as opposed to mathematicians, not that they aren't also chess players)? Are there resources that explain the strategy for infinite chess? Do the principles that guide standard chess play change drastically for the infinite variant, or are they merely extended?


Thank you for the question, and for linking to some of my work on infinite chess. You can find additional material on infinite chess on my blog at http://jdh.hamkins.org/tag/infinite-chess/. There are numerous talks, including slides, with infinite chess movies, as well as more detailed research papers and more. I'd be happy to discuss any aspect of that---please go ahead and ask focussed questions about infinite chess.

But in regards to your question, I would agree that most people working on infinite chess are more mathematicians than chess players, and furthermore the central issues seem to be mathematical in nature. Certainly I am more a mathematician than a chess player, although I enjoy a lot of chess (you can play me on ICC, username: JDH).

You ask specifically whether infinite chess has been explored by chess players, as opposed to mathematicians, and the answer to this is yes. My co-author on several of the infinite chess papers, for example, is C. D. A. Evans, US National master, also philosophy PhD candidate at the City University of New York, but not a mathematician. Also, Noam Elkies, professor at Harvard, is also an accomplished chess player who is also a mathematician, and has written on infinite chess.

In my research practice with Cory Evans, I can report that his chess intuition and analysis was invaluable in shooting down and finding flaws in numerous proposed chess positions. But also, it seemed to be an essentially mathematical activity to come up with the various infinite chess positions that illustrate the various high game values and so on. So it was a true collaboration between two fields.

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    Thanks for the answer and updated links! Wow have things progressed in the several years since I looked at this stuff last! – Dennis Mar 29 '17 at 0:40
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    Oh, yes, there has been some activity and progress. But meanwhile, some basic questions remain open. For example, we don't know whether or not the decision problem about whether a finite position in infinite chess is a win for white or not is computable decidable in principle or not. The brute-force search-the-game-tree algorithm does not work, since the tree is infinite and infinitely branching, and there are basically no other promising methods. So can you tell if a finite position is a win for white or not? – JDH Mar 29 '17 at 0:49

All the normal rules of chess strategy must be completely void. Consider some of the most important ones:

  • Material advantage helps to win. Mainly because with more material it's easier to queen a pawn (not available in infinite chess) and because it makes it easier to checkmate the other king. But checkmate is much harder anyway (no board edge to help), and in infinite chess you may have a billion extra knights that are of no use to help defend your king because they're a trillion squares over to the left. In general material will still be useful, but I doubt a queen is going to be much better than a rook.

  • The center. In chess, pieces want to be placed where they influence the most squares, and that's usually the center. In infinite chess, there is no center and no square is inherintly better than any other.

  • Development. There is no starting position, and no concept of a square that's better than the original square, so no development.

  • Pawn structure is an artifact of having two rows of 8 pawns at the start, and a center. Without any of that, all the knowledge we have about pawn structure goes out the window. A pawn may advance a billion squares without ever meeting another pawn, or be blocked by a trillion of them.

And so on, and so on.

I think a study of the strategy of infinite chess would have to start with an analysis of which basic mates are possible (queen, rook and king vs king probably mates, two rooks don't, I guess? Definitely a lone queen and king doesn't, needs an edge).

If you have a lot of pieces, you'd think you'd want to move part of them towards the enemy king, keeping some at home to defend the king. But for queens, rooks and bishops, distances are deceptive -- if the lines and diagonals are clear, they can get to wherever they can possibly get in two moves. Pawns, kings and knights are unimaginably useless in comparison.

I wonder if anything is known about games of infinite chess with more than a few pieces.

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I think this is a valid 2-Rook mate on an infinite board. You use one rook as the wall and the other rook to cooperate with K to give KR vs K mate conventionally.

[FEN "2R5/3k4/8/8/8/8/8/4R2K w - - 0 1"]

1. Ree8 Kd6 2. Kg2 Kd5 3. Kf3 Kd4 4. Re7 Kd5 5. Kf4 Kd6 6. Rcc7 Kd5
7. Red7+ Ke6 8. Rb7 Kf6 9. Rb6# 1-0
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There are games of infinite chess being played on chess playing websites which helps lend clarity to this question, and requires the existing answers to be amended and/or revised.

Contrary to some statements in the existing answers, much of the game playing strategy in classical chess is still valid in infinite chess. In particular (1) winning a game depends heavily on achieving control of the squares between White's and Black's pieces, (2) pieces should be developed quickly, by placing them in the ideal square once, and without wasting tempo by moving the same piece more than once, (3) do not bring out strong pieces too early, as they can easily be threatened while the opponent develops his pieces, and strong pieces themselves cannot attack without themselves being threatened, (4) do not sacrifice pieces without a clear and tangible benefit such as achieving a positional advantage, and (5) don't capture pawns or attack if you haven't completed development.

What is new in infinite chess, is that flank attacks need to be considered (attacking the opponent from behind) and defending from such attacks. Also new patterns to achieve checkmate must be devised as trapping the king at an edge or corner is not possible.

Finally, as stated in one of the answers, the central issue of infinite chess being "mathematical in nature" is an opinion. People who have developed the skills to consistently win games deserves equal if not more attention than the ancillary mathematics. Magnus Carlsen is more famous and prominent than the mathematicians who study chess, just as players who excel in winning in infinite chess have developed a more exceptional talent than people who investigate the ancillary concepts and math.

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  • Could you provide some references to these websites, and further resources you've based this answer on (like some of these players with "exceptional talent")? If these exceptional players deserve more attention I'd appreciate you bringing them to my attention. Note, also, that I made no claim about "talent". The question was just who studies this variant: mostly mathematicians (as I am familiar with) or chess players as well? – Dennis Aug 18 '17 at 15:19
  • Also, how is it possible for these websites to implement infinite chess? Even assuming the board is only countably infinite, there would still be arbitrarily large finite moves possible. Is it just a board with no determined edge but some small-ish bound on move distance? – Dennis Aug 18 '17 at 15:31
  • There seems to always be at least a few games of infinite chess being played at this website: chess.com/forum/category/chess960-chess-variants My point is not that players in infinite chess have become exceptional when judged on a worldwide game-playing status; but some are exceptional in their field (albeit a niche in the class of chess-like games). In the same way, some mathematicians have become prominent in the study of infinite chess, but this too is a niche in the class of mathematics. So who is to say one is more notable or more prominent than the other? – tomoka kazuki Aug 18 '17 at 17:05
  • Granted - it's a subject of opinion, but I feel that one who has generalized their game-playing skills and can demonstrate it by winning games has a provable skill set, whereas a mathematician does not formally "compete" against others, and so his notability would be subject to opinion. – tomoka kazuki Aug 18 '17 at 17:05
  • Again, my concern wasn't really about who was "talented" or "notable", but just whether there had been chess-centric study of the variant in addition to the mathematical research I'm familiar with. It looks like the variant on that page adds additional pieces with different movement rules. Do you know if this is the case generally for the played variants of infinite chess? If so, then it seems to be substantially different from simply "chess on an infinite board". Interesting nonetheless, though. Thanks for bringing it to my attention! – Dennis Aug 18 '17 at 17:45

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