Lights Out is played on a triangular grid: pressing a cell toggles it and its neighbors. The goal is to turn all lights off. The game reduces to a linear system over GF(2) — the two-element field where 1+1=0. Solvability depends on whether the all-ones vector is in the column space of the adjacency matrix mod 2.
Morais and Marcarini prove that solvability of the triangular Lights Out game is determined by the parity of the number of domino tilings of the board. If the number of tilings is odd, the game is solvable. If even, it is not.
The connection is through the determinant. The adjacency matrix of the triangular grid, taken mod 2, has a determinant that equals the number of domino tilings mod 2. The game is solvable if and only if this determinant is nonzero mod 2, which happens exactly when the tiling count is odd.
The domino tilings are counted by the Kasteleyn-Temperley-Fisher method — the Pfaffian of the adjacency matrix of the dual graph. Over the reals, this gives the exact count. Over GF(2), the Pfaffian reduces to the determinant, closing the loop: solvability of a light-toggling game depends on a combinatorial quantity (tilings) through an algebraic quantity (determinant) that connects them.
A puzzle about lights. An answer about dominoes. The bridge is a determinant over a two-element field, connecting two combinatorial questions that have no obvious geometric relationship.