![]() Table[First[Dimensions[NullSpace[AdjacencyMatrix[GridGraph[ \\ Max Alekseyev, CROSSREFSĬf. LIGHTS OUT PUZZLE FULLZhao Hui Du, edited by Max Alekseyev, EXAMPLEįor n=2, matrix is which is of full rank. So their GCD is x(1 x) and the degree is 2, that is a(5)=2. Then a(n) equals the degree of gcd(f(n,x), f(n,1 x)). Let f(k,x) = U(k,x/2), where U(k,x) is the k-th Chebyshev polynomial of the second kind over the field GF(2). Martin Kreh, "Lights Out" and Variants, The American Mathematical Monthly 124:10 (2017), 937-950.Įric Weisstein's World of Mathematics, Lights Out Puzzle FORMULA Brouwer, Lights Out and Button Madness Games William Boyles, Table of n, a(n) for n = 1.25000 (first 1000 terms by Max Alekseyev and Thomas Buchholz).Īndries E. Max Alekseyev, Sep 17 2009Ī(n) is even and satisfies a(n) = a(n - 1). Rank deficiency is 2 for that game, but only initial configurations that admit a solution are given.Ī(n) is nonzero iff n is in A117870 a(n) is zero iff n is in A076436. The size 5 game can be played at the link given below. The sequence is the rank deficiency (corank) of the matrix, such that the zero terms correspond to the sizes for which the general case admits a solution. It requires inverting a n^2 by n^2 matrix in Z/2Z. ![]() The general problem is to turn all pixels ON given any initial configuration. Touching a pixel switches its state and the state of the adjacent pixels. (Greetings from The On-Line Encyclopedia of Integer Sequences!) ![]()
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