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    $\begingroup$ I presume you aren't working with the FIDE ruleset when it comes to positions? FIDE rules says positions are equal if 1) the pieces are on the same squares, 2) it's the same player to move 3) the same en passant moves are possible, and 4) the castling rights are the same. So, after, 1 h4 h5, 2 Rh2 Rh7, 3 Rh1 Rh8, under FIDE rules, no position has appeared twice on the board, as after the first move, players have the right to castle short, while they have lost that right after the third move. $\endgroup$
    – Abigail
    Commented Feb 12 at 20:21
  • $\begingroup$ @Abigail My use of the term "position" is ad hoc for this puzzle (though if there's some standard terminology for this, I would be happy to use that). $\endgroup$
    – Elliot Glazer
    Commented Feb 12 at 20:41
  • $\begingroup$ FIDE Law 9.2.3 Positions are considered the same if and only if the same player has the move, pieces of the same kind and colour occupy the same squares and the possible moves of all the pieces of both players are the same. Thus positions are not the same if: 9.2.3.1 at the start of the sequence a pawn could have been captured en passant. 9.2.3.2 a king had castling rights with a rook that has not been moved, but forfeited these after moving. The castling rights are lost only after the king or rook is moved $\endgroup$
    – Laska
    Commented Feb 28 at 6:57
  • $\begingroup$ So the notion here is diagram + whose move: that's fine. But note that the answer given by the questioner (reaching up to n=22) assumes just diagram without the move $\endgroup$
    – Laska
    Commented Feb 28 at 7:23