A165738 Rank deficiency (= dimension of the null space) of the n X n "Lights Out" puzzle on a torus.

0, 0, 4, 0, 8, 8, 0, 0, 4, 16, 0, 16, 0, 0, 12, 0, 16, 8, 0, 32, 4, 0, 0, 32, 8, 0, 4, 0, 0, 24, 40, 0, 44, 32, 8, 16, 0, 0, 4, 64, 0, 8, 0, 0, 12, 0, 0, 64, 0, 16, 20, 0, 0, 8, 8, 0, 4, 0, 0, 48, 0, 80, 52, 0, 56, 88, 0, 64, 4, 16, 0, 32, 0, 0, 12, 0, 0, 8, 0, 128, 4, 0, 0, 16, 24, 0, 4, 0, 0, 24

Offset: 1

Author

Max Alekseyev, Sep 25 2009

Keywords

• nonn

Comments

The number of solutions to the puzzle is 2^a(n). If a(n)=0 then the puzzle has a unique solution.

References

• See A075462 for further references.

Links

• Max Alekseyev and Thomas Buchholz, Table of n, a(n) for n = 1..1000. (Terms n=91 through n=1000 from Thomas Buchholz, May 20 2014)
• M. Anderson and T. Feil, Turning Lights Out with Linear Algebra, Mathematics Magazine, 71 (1998), 300-303.
• Andries E. Brouwer, Lights Out and Button Madness Games. [Gives theory and a(n) for n = 1..1000, Jun 19 2008]
• Eric Weisstein's World of Mathematics, Lights Out Puzzle

Crossrefs

Cf. A159257.

Formula

a(n) <= 2n.

a(n) is a multiple of 4 and satisfies a(2n) = 2a(n). a(n+1) = 2 * A159257(n) + 4 if n = 2 (mod 3) and a(n+1) = 2 * A159257(n) otherwise. - Thomas Buchholz, May 22 2014

Extensions

More terms from Thomas Buchholz, May 20 2014