On Jaap's Puzzle Page
http:// www.jaapsch.net/puzzles/lomath.htm#domtilings
Theorem 7 says:
If standard Lights Out is played on a m x n grid-like board, then all light patterns on that board are solvable iff the number of ways to tile the board with dominoes and monominoes is odd .
Of course, here Lights Out is over two elements field GF(2) . Guestion is - what about Lights Out over GF(p) with p prime number. Very intriguing question to me: find the answer on complete solvability on m x n rectangular board over GF(p) in terms of number of tilings board with some tiles, possibly coloured .