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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 .

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    $\begingroup$ Rules for lights out over GF(p) are the same as over GF(2). You choose any cell and add to its and to its neighbours any integer modulo p. The purpose of the game is starting with given configuration to obtain all zeroes grid $\endgroup$ Jul 23, 2015 at 13:17
  • $\begingroup$ There is some discussion at slideshare.net/PengfeiLi1/lop-38272545 and also iespravia.com/rafa/luces/Lights.pdf $\endgroup$ Jul 23, 2015 at 13:27
  • $\begingroup$ jaapsch.net/puzzles/lomath.htm seems to be a clearinghouse for stuff about variations of Lights Out. $\endgroup$ Jul 23, 2015 at 13:34
  • $\begingroup$ Yes, there are discussions and articles using linear algebra. The question is whether there is some way to characterise solvability in terms of tiling using dominoes and monominoes as in quoted Theorem 7 for game over GF(2) $\endgroup$ Jul 23, 2015 at 13:34
  • $\begingroup$ I don't know. Have you looked at all the links I have posted? If so, you should list in the body of your question all the sites you've looked at, so people don't waste their time telling you things you already know. And if you haven't looked at all these links, well, what are you waiting for? Here's another one: units.miamioh.edu/sumsri/sumj/2001/GraphTheory2001.pdf $\endgroup$ Jul 23, 2015 at 13:37

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It might be of interest to you:

Hunziker, Machiavelo and Park prove that there are infinitely many square grids which are not completely solvable for $p = 3$ and formulate a conjecture which would prove the result true for all primes. Here is the link: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.394.7187&rep=rep1&type=pdf

Maithreya

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