I've been working on a modification to the standard Conway's soldiers game. In Conway's soldiers, we have an endless number of soldiers in a grid of squares at and below point 0 North, and I can move soldiers left, right, up, and down by having them jump over other soldiers, which makes those "jumped over" soldiers get removed from the board. The key thing is, I can only move soldiers by having said soldiers jump over each other, and thus it becomes a challenge to see in an extremal manner how likely it is to advance a soldier a certain number of squares north. I have a feeling that if I were to implement this game such that soldiers can jump diagonally, The number of moves it takes to move 9 squares north approaches infinity. I'm having some issue formalizing this idea. Is there a way I can model this problem using some form of potential function? I've seen them used in similar applications towards amortized analysis.
2026-03-30 13:03:14.1774875794
Conway's Soldiers
512 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
In his Masters thesis "Pegs, Pebbles, Pennies, and Piles - a study of some combinatorial games", Niklas Eriksen proves that 9 cannot be reached. The proof does involve "some form of potential function" (Conway calls them "Pagoda functions" in this case), but is too long to reproduce here.
In "Diagonal Checker-jumping and Eulerian Numbers for Color-signed Permutations" by Eriksen, Eriksson, and Eriksson, that proof is re-presented and it is mentioned that 8 definitely can be reached.
Incidentally, these papers contain results about higher dimensions as well. But unlike 1 and 2 dimensions, these are only bounds on what can be reached. The known unreachable levels in the first several dimensions are $2,9,18,28,40,51,64$. But I don't know if anyone's been able to reach, say, level $17$ in 3 dimensions.