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This puzzle extends Rewan Demontay's New Year Math Riddle.

Rewan asked (essentially): if the players co-operate, what's the largest number of times a given diagram can occur in a game before reaching the point when someone can claim a draw? In order to group everything together, I'll re-state that question as Q1 here:

Q1: Under the 3-fold repetition & 50-move rules, how many times can a diagram occur? Call a diagram which achieves this 3-critical.

Q2: Under the 5-fold repetition & 75-move rules, how many times can a diagram occur? Call a diagram which achieves this 5-critical.

Now let's make some actual chess games:

Q3: Find a 3-critical diagram that is a unique shortest proof game to its first occurrence.

Q4: Find a 5-critical diagram that is a unique shortest proof game to its first occurrence.

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Answers to all 4 questions

Q1: Under the 3-fold repetition & 50-move rules, how many times can a diagram occur?

A1: Even if it were mandatory, the 50-move rule has no impact: the answer is still 22 as per Rewan's post:

   2 (occurrences without triggering the draw) 
 x 2 (White or Black to move) 
 x 5 (number of castling rights intact: 4->3->2->1->0) 
= 20 
 + 1 (en passant) 
 + 1 (the final 3rd occurrence)
= 22 times

There are 21 transitions between the 22 occurrences, each taking at least 2.0 moves which gives 42.0 moves. Also we must flip sometimes between White & Black having the move, which takes an extra 0.5 of a move each time to triangulate. If White were to move in the first, the player to move in each of the 22 critical positions is:

W-WWBB-BBWW-WWBB-BBWW-WWBB-B

Five 0.5 flips give a total of 42.0 + 5 * 0.5 = 44.5 moves.

Note that we require a diagram where parity can be switched by a free bishop or queen, as knights can't triangulate, and kings & rooks are tied up with castling rights. Rewan's diagram is sufficient for this, in order to achieve the 44.5, and in fact we still have a little slack and could waste another 5.0 moves. Because of e.p., the move prior to the first occurrence of the diagram was a pawn move, which zeroed the 50-move clock. For definiteness, let's say that the maximum is 49.5, to avoid draw by repetition and 50-move rule on the same move.

Q2: Under the 5-fold repetition & 50-move rules, how many times can a diagram occur?

A2: One might think that the answer is 42:

   4 (occurrences without triggering the draw) 
 x 2 (White or Black to move) 
 x 5 (number of castling rights intact: 4-3-2-1-0) 
= 40 
 + 1 (en passant) 
 + 1 (the final 5th occurrence)
= 42 times

But in fact the diagram can only occur 37 times now, not 42.

There are 41 transitions between the 42 critical positions. Each takes at least 2.0 moves, which takes 82.0 moves, together again with the 2.5 moves for parity switching so that the player to move in each of the 42 critical positions is:

W-WWWWBBBB-BBBBWWWW-WWWWBBBB-BBBBWWWW-WWWWBBBB-B

84.5 moves would be needed, but the game will end at 75.0. So how many occurrences of the diagram can we get before we bring the game to a close by 5-fold repetition, before the 75-move rule ends us? If our budget is 74.5 (or even 75.0) moves, so we must eliminate 5 recurrences.

It takes us 74.0 moves to reach the diagram for the 37th time. By skipping the en passant threat at the beginning, we can reach the position more quickly.

BBBB-BBBBWWWW-WWWWBBBB-BBBBWWWW-WWWWBBBB-B

Q3: Find a 3-critical diagram that is a unique shortest proof game to its first occurrence.

A3: PG in 3.5. The last move to reach this position for the first time is a pawn move, as we then launch into the full 44.5 move sequence.

[FEN "r1bqkbnr/pppp1ppp/n7/8/4pP2/8/1PPPP1PP/RNBQKBNR w KQkq - 0 1"]

Q4: Find a 5-critical diagram that is a unique shortest proof game to its first occurrence.

A4: Proof game in 2.5 moves to a diagram which can occur 37 times.

[FEN "rnbqkbnr/1ppp1ppp/B3p3/p7/8/4P3/PPPPNPPP/RNBQK2R w KQkq - 0 1"]

This will take us 84.5 - 10.5 = 74.0 moves, which is perfect because White's 3rd move to reach the diagram for the 3rd time is not a pawn move or capture. So the game does not end prematurely by conflated draws by 5-fold repetition & 75-move rule.

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