Longest Possible Game

This topic has been done a lot, but it's annoying to me that when you google this, so many get the wrong answer, which is then repeated by people who google for it.

My explanation tries to be simple and clear, but it's also compact. To prove each step to yourself might take a little work. There is a sample game at the end to help illustrate.

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For the 50 move rule the longest game is:

5898.5 moves

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Explanation

For this purpose, the 50 move rule requires that every set of 50 consecutive moves contains at least one pawn move or capture.

Let 1 move = 2 ply = e.g. 1.e4 e5
Let X be a set of 100 consecutive ply whose last move is a capture or pawn move by black.
Let Y be a set of 100 consecutive ply whose last move is a capture or pawn move by white.
In total, there are 30 (capture moves) + 96 (pawn moves) - 8 (moves which are both a capture and a pawn move) = 118.

Eight moves are both a capture and pawn move because on average, a single capture by a pawn will never do better than "unblocking" 2 pawns and 16 pawns need unblocking.

So the starting value is 5900 moves because 100 x 118 / 2. Now notice that even though there will be 30 captures, the same player can't make all the captures! Similarly the same player can't make every pawn move.

So now we subtract 1 ply for every time we switch between a set of Xs and a set of Ys (when there is a switch, instead of white making the 100th ply capture or pawn move, it must be black making it on the 99th and vice versa).

It's a bit tricky, but we can figure out a way that there are only 3 switches, and 5900 - 1.5 = 5898.5 moves which is the answer.

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Sample game

Note that every capture and every pawn move represents either 100 ply (when it's the same player avoiding the draw rule), or 99 ply (when it's a switch to the other player). Moves that are neither a capture or pawn move should be ignored.

(Switches are noted, and happen on moves 29, 109, and 171)

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