Interesting Mathematical Chess Question

Let's assume we've removed the 50 move rule. How long could a chess game theoretically go on before repetition of a position is inevitable. Let's say even 1-fold repetition (excluding triangulation) for simplicity sake. Given a finite number of pieces on a finite number of squares, how many moves could be made before inevitable repetition of a position or insufficient mating material occurs and ends the game? Is there a mathematical way to calculate this?

Would be much much easier to montecarlo this if you really need an answer.

Note that with the 50 move rule immediately limits the longest game to between 5,000 and 6,000 moves (forgot the exact number) if you move a pawn 1 square every 50th move and space out captures that way to until there's no pieces left. I'm asking without the 50 move rule and avoiding any position repetition even once (triangulation aside as that doesn't count as a repetition).

To approach this question, think about the limits of chess. Even though there are many pieces and squares, the number of possible positions is finite due to the nature of the game.

At any moment, a chess position is defined by the arrangement of pieces, which pieces have been captured, castling rights, and the possibility of en passant. This leads to an estimated number of possible positions between 104310^{43}1043 and 105010^{50}1050.

However, despite this enormous number of positions, it is still finite. If the 50-move rule were removed, and assuming players never repeat moves, the game could theoretically continue until every possible position has been played at least once. Eventually, because the number of positions is finite, some previous board state would have to repeat.

The key lies in the concept of a "finite state space." Once you've explored every possible arrangement of pieces on all the squares, there would be no new positions left to reach. Thus, the game would be forced to repeat a previously played position at some point.

So, in theory, a chess game could continue until every possible legal position has been reached once. This means that, in the worst case, a game could last up to the total number of possible positions, which is on the order of 104310^{43}1043 to 105010^{50}1050 moves, before repetition becomes inevitable.

This astronomical limit is purely theoretical, as in practice, players wouldn't be able to avoid repetition or material depletion for such a long time. But mathematically, that would be the answer.

I don't understand your notation, what's the [50] thing before the 1050?

No

Nooooo