Has Anyone Solved This Chess Puzzle (More Thorough Explanation In OP)

Has any mathematician, computer program, etc., figured out the arrangement of pieces that would give one player (say, White) the highest possible number of moves?

For example, at the beginning, White has 20 possible moves (A3, A4, B3, B4, etc.). But imagine a position where White’s Queen is on, say, E5, his Knights are on, say, D4 and F4, and so on and so forth. The number of possible moves, given the optimal placement of his pieces and Black’s pieces, could be in the hundreds (I’m guessing).

The first rule for all such chess questions is to check Tim Krabbé’s records page:


He gives a position where black has 79 legal moves as the record from a practical game.

I’m not aware of anyone working out a theoretical maximum though. It should be easy to at least make an upper bound. The maximum mobility of the pieces is: King 8, Queen 27, Rook 14, Bishop 13, Knight 8, Pawn 4. So that gives a maximum of 137, if I added right (more if you allow for promotions). You can bring that down a little - for example, rook pawns can only ever have a mobility of at most 3.

Oh, I guess you might count the four possible promotions as different moves, so then a pawn could have mobility 12.

You beat me to it! But with a little more searching I found this page: https://timkr.home.xs4all.nl/chess2/diary_10.htm - item 195 shows a constructed but legal position with 144 possible moves (if you count different promotions as different moves). It may be that a modern chess computer could be programmed to establish if this was the maximum, but I doubt anyone has tried.

Ah yes, I went straight to the records page and didn’t check the diary.

I’m not sure whether it would be feasible to do a computer search to establish the maximum mobility possible. It seems like it would be a similarly-sized problem to solving chess itself.

Not nearly so difficult. This is only looking at one position, not ever position. A pure brute-force search would probably still be too hard to be practical, but there’s a lot of finesse that’s possible. For starters, since we already have a candidate position with a large number of moves, we can immediately eliminate anything with fewer moves than that.

From adding up the maxima for each individual piece, I get 8 (king) + 27*9 (queens) + 14*2 (rooks) + 13*2 (bishops) + 8*2 (knights), for a total of 321. But of course that’s unattainable: The queens and bishops need to be in one of the center squares to maximize their potential, and there are only four of those, plus pieces will get in each others’ way, plus with that many queens on the board, we’ll have to make sure we don’t accidentally mate the other side.

It’s also possible that we’ll end up sacrificing some pieces, because they’d get in the way more than they’d contribute themselves. And since my pawns are already promoted, I don’t need opposing pieces just to serve as targets for pawn diagonal moves: I suspect that the opposition will be left with just a king and two or three blocking pieces, huddled in a corner of the board.