# On an 8-by-∞ board, who wins between an infinite number of pawns and a rook?

I've had this puzzle rattling around in my brain for the past few days and I want to know what you all make of it.

Imagine chess played on a board with eight ranks and an infinite number of files. Black has a single rook on a1. White has an infinite number of pawns, one on every square in the first rank besides a1.

The rules here are slightly different from normal chess.

1. White's goal is to capture the rook. Black's goal is to never get captured.
2. White's pawns can only move one space forward and never have the option to move forward two spaces. They still capture diagonally.
3. White may move one pawn off of the first rank, and any number of pawns already off the first rank simultaneously (notated with an ampersand).

Is White guaranteed to capture Black's rook at some point in time, or does Black have a strategy to survive? I think White has a guaranteed victory, but I can't say for sure. Consider the following string of moves.

``````1.b2 Rxc1
2.b3&d2 Rxe1
3.b4&d3&f2 Rxg1
4.b5&d4&f3&h2 Rxi1
5.b6&d5&f4&h3&j2 Rb1
6.b7&d6&f5&h4&j3&k2 Rxb7
7.d7&f6&h5&j4&k3&l2 Rxd7
8.f7&h6&j5&k4&l3&m2 Rxf7
9.h7&j6&k5&l4&m3&n2 Rxh7
10.j7&k6&l5&m4&n3&o2
``````

Now Black is stuck. If they play 10... Rxj7, then White follows with 11. kxj7 and wins. If they do anything else, then 11. j8=Q is sure to follow, which spells out doom for Black. Does Black have a more optimal strategy that leads to their survival, or is this it?

• What is the starting position of the rook? And can white's pawns promote? Otherwise the question is rather dull, since black's rook can just hide behind white's pawns then. Nov 4, 2021 at 13:41
• 1. Is the ∞ the length or the width. 2. Do the pawns promote? Nov 4, 2021 at 14:02
• @ShambhavGautam There's an infinite number of files (a, b, c, d...), but in any file, pawns can promote on the 8th rank. Nov 4, 2021 at 14:31
• Even without rule 3 it should still be an easy win for White no? If White just plays some pawns far away, a defense from behind would not be possible. So the only defense would be from the side (or from in front but then White protects those advancing pawns then advances elsewhere) when White can slowly advance with a mass of pawns. Nov 4, 2021 at 16:09
• Commenters on the answer by Brian Towers seem to be reading your "every other square" as meaning something different than "every square on the first rank other than the one the rook is on". Could you make that more explicit in your formulation? Nov 4, 2021 at 18:36

It is trivially a win for white even on an 8x8 board

``````[fen "8/8/8/8/8/8/8/rPPPPPPP w - - 0 1"]

1. h2 Rxb1 2. g2 null 3. h3 Rxc1 4. f2 null 5. g3 null 6. h4 Rxd1 7. e2 null 8. f3 null 9. g4 null 10. h5 Re1 11. e3 null 12. f4 null 13. g5 null 14. h6 Rxe3 15. f5 null 16. g6 null 17. h7 Rf3 18. f6 null 19. g7 null 20. h8=Q Rxf6
``````
• At the risk of being naive - even with the looming threat of g8=Q, is it guaranteed that with perfect play, two queens always can capture a rook? I see the logic here that lead to an inevitable loss for Black on the infinite board, but I'm not sure if it's true for an 8x8. Nov 4, 2021 at 14:42
• I agree, it's still interesting whether two queens are enough to trap black's rook on the 8-by-8 board. I suspect it is, but not exactly sure yet. It took me a while to figure out why the win is trivial in the infinite case, but it has to do with white being able to guarantee a promotion of at least 8 pawns, after which the queens take away one row each while protecting one another. Nov 4, 2021 at 15:39
• @Scounged "Now Black is stuck. If they play 10... Rxj7, then White follows with 11. kxj7 and wins." in the question, start of last paragraph, would seem to contradict this. If white has a j pawn and a k pawn then they don't have pawns on alternate files. Ah, and I see the original question has been edited sine I posted my answer. Nov 4, 2021 at 17:24
• It's a trivial win on the infinite board after white gets one queen, as the queen can protect one pawn to queen each time, until white has 8 queens. At that point the rook is mated. Nov 5, 2021 at 5:56
• @HaukeReddmann Indeed, two queens aren't enough. Interestingly, three queens also seem to be insufficient, but I think that four queens should be enough to trap the rook on a standard chessboard. This raises a new question: given an n-by-8 board, how many queens (or maybe allow some knights as well) are needed to capture the rook? Nov 5, 2021 at 10:51

The "trivial" white wins solution has been refuted by the fact that the "win" is actually a loss. You can't construct a mate even without figuring out a forcing sequence of moves. The infinte board is another story. Black will need to figure out a different way to stop white when she has to go through an infinte number of pawns. I am not 100% convinced that there isnt some way to stop white with the rook other than brute force pawn gobble, and on an infinte board that ain't gonna work. It feels like white should win on an infinte board - but before coming to this conclusion, there are possibly other options which dont begin with ...Rxb1. I couldn't find any way to stop the pawns without going for the "let him only get 2 queens " strategy, but I certainly didn't exhaust all the possibilities.