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This is a task related to this question: Theoretical limit for the number of consecutive checks?.

By a sequence of consecutive check, what we mean is white giving check, black replying with a check, white replying to that with a check and so on. Note that we are talking about a theoretical situation so that 50 move rules or three-fold repetition are irrelevant.

In the linked question, he is asking for the longest sequence of consecutive checks with two conditions:

  1. Initial position should be legal
  2. Initial position should not contain promoted pieces.

Suppose we remove the 2nd condition, and relax the 1st one to allow more pieces than legally possible (say, 9+ white queens or 3+ white knights). With these relaxations, is it possible to have an infinite sequence of consecutive checks?

I think the answer to the above question is no (but no complete proof yet).

If such an infinite sequence is there, it should contain a loop for sure. Therefore, the sequence cannot contain pawn movements or captures.

A second question is to construct longest such sequence. Alexey Khanyan's record 54 with conditions 1 and 2 may be surpassable because we relaxed them. (By the way, even with conditions 1 and 2, Khanyan himself thinks that 54 can be improved).

PS: For this question, I am not interested in involving fairy chess pieces or chess variants.

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    My intuition is that such an infinite loop is impossible without fairy pieces, but that might be hard to prove. – Evargalo Jul 2 '20 at 13:47
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No, it's impossible to have an infinite sequence of checks

Let's remember that there are 3 ways to parry a check :

  1. Move the King to a safe square
  2. Capture the enemy piece giving the check
  3. Move a piece between the King and the enemy piece giving a check (except against a Knight check as it can jump over pieces)

2 and 3 aren't sustainable over an infinite number of moves, because the chessboard has finite space, only 64 squares and necessarily less pieces than there are squares. So the only way to get an infinite cycle of checks going would be for both Kings to escape checks with method 1, by continually moving them.

This also implies that the King movement gives check. Since Kings can't come into contact, that means their movement must deliver a discovery check from a long range piece. This in turn, implies that after each move, both Kings stay aligned over some axis (rank, file, or diagonal), otherwise there would no longer be any discovery checks for lack of an alignment.

Let's say we're in such a position and it's Black to move. To summarize the constraints, the Black King must :

  • Be on 1 of the 8 lines passing by the square where the White King is (otherwise it can't deliver a discovery check), and
  • Move to another of these 8 lines (otherwise White won't be able to deliver a discovery check), and
  • Follow normal rules of chess, e.g. move only one square in any direction and not on a square where it would be in check

Let's say the White King is on e4. The only way to move from one of the 8 lines passing by e4 to another by jumping only one square is to either go to or from one of the 4 squares diagonally adjacent to the White King : d5, d3, f3, f5. But because these squares are in contact with the White King, it's not possible for the Black King to perform such a move. It can't be on these squares because the position would be illegal, and it can't move there because the move is illegal. So it's impossible to follow these constraints.

Hence, from all the above, I'd conclude that under the normal rules of chess, it's impossible for both players to perform at all an infinite cycle of checks. This is also valid under the 2 conditions you mentioned in your question, as they don't affect the reasoning used here.

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    Why is 3. (blocking the check) not sustainable over an infinite number of moves? With fairy pieces, an infinite series of checks using only 3. is possible. Your argument about the board being finite and the number of pieces also being finite should also apply there. How does "finite board" imply "no possible infinite sequence of blocked checks"? – wimi Jan 26 at 21:45
  • We can rule out 2 for sure because it cannot be part of a loop. But, why 3 together with 1 is not sustainable over an infinite number of moves? I don't understand how 3 can be ruled out. – Cyriac Antony Jan 31 at 3:10
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As Umlin proved, it is impossible to have infinite mutual checks. However, that is with the standard set of chess pieces. With fairy chess pieces, it is indeed possible to have an infinite series of checks by both sides. I remembered a simple example from Tim Krabbe's Diary Entry #120, entitled "Mutual Discovered Perpetual."

In the example, the knights are acting as Nightriders, knights with an infinite range in one direction, aka line pieces. Both sides can mutually give check forever.

[Title "Arthur John Roycroft After T. R. Dawson The Problemist 7/1976"]
[FEN "8/8/3rr3/4k3/8/3K4/8/2N2N2 w - - 0 1"]

1. Ke3+ Kd5+ 2. Kd3+ Ke5+ 3. Ke3+ Kd5+ 4. Kd3+ Ke5+

For reference, the original publication can be seen in The Problemist Archves on the British Chess Problem Society website. It is also in the Die Schwalbe Chess Problem Database.

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