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MatPlus.Net Forum General Threefold repetition
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(1) Posted by Hauke Reddmann [Saturday, May 1, 2021 12:11]

Threefold repetition

Assume we just have one rule: White wants to drag out the game for as long as possible, Black can finally get the last bus when he manages a threefold repetition. Since the 50-move or "dead" rule probably gets invoked much sooner, we skip all other chess rules, i.e. we might also work with a simple toy example: 3*3 board, Ra1 vs Rc2, does White play Rb1 or Ra3? Has there been any (math) work on the theme? (It might be relevant for the occasional retro.)
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(2) Posted by Andrew Buchanan [Saturday, May 1, 2021 15:34]

Here's a starter.

bK heads straight for wK. When he gets near, either he will have opposition or not.

If bK has opposition, then with WTM it's either diagonal i.e. (2,2) squares away or orthogonal i.e. (0,2) squares away. Diagonal opposition, can soon be converted into orthogonal opposition as soon as wK stops running diagonally away. With orthogonal opposition, bK just stays (0,2) away, and wK eventually has to retreat back to the wall.

If bK not have opposition, then with WTM kings are separated by a knight's move. wK without retreating must pick diagonal or orthogonal opposition. Soon they are either moving horizontally or vertically. bK always stays a knight's move apart, so wK can't swap between horizontal & vertical without losing the opposition. Eventually he runs out of squares and has to repeat.

From White's side against this strategy, whoever has the opposition, the best wK can do is delay by never retreating from the main horizontal or vertical line, until he runs out of squares to repeat. I have a feeling the strategy is basically the same for n-fold repeat, for any n>1.

How does this compare with bK just vibrating from one square to another, and allowing White access to the whole board? I guess there will be 2 unique positions per square of the board with WTM. The cut-off strategy mentioned above I think has nearly 1 unique position per square. (As Black achieves opposition as soon as White is forced to retreat. And also wK is blocked from accessing about half of the board.
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(3) Posted by Andrew Buchanan [Saturday, May 1, 2021 17:38]

It's been brought to my attention that "R" meant rook not roi. OK well that's even easier. bR just goes to the square wR has just left. Either wR captures (in which case draw is it?) or wR runs away. But there's only one unique position with WTM for each square on the board.

This is better than bR just vibrating from one square to an adjacent one, where because there's no parity, there would be 2 unique positions with WTM for each square on the board.
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(4) Posted by Hauke Reddmann [Sunday, May 2, 2021 10:33]

Actually, if it's *not* the best of both worlds (a royal rook :-),
Black could even simpler oppose his rook, since capture ends the
game (in stalemate?!). But that wasn't what I meant. K would be
a better toy example anyway, in the sense that some actual strategy is
involved (thinking two seconds longer, I realized all fields
are symmetry equivalent in the R case).
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MatPlus.Net Forum General Threefold repetition