A005311 Solution to Berlekamp's switching game (or lightbulb game) on an n X n board. (Formerly M1040)

0, 1, 2, 4, 7, 11, 16, 22, 27, 35, 43, 54

Offset

1,3

References

Brown, Thomas A.; Spencer, Joel H. Minimization of +-1 - matrices under line shifts. Colloq. Math. 23 (1971), 165--171, 177 (errata insert). MR0307944 (46 #7059). Gives asymptotic bounds, some exact values, and generalizations to an m X n board. - N. J. A. Sloane, Jun 29 2014

Komlos, J., & Sulyok, M. (1970). On the sum of elements of +-1 matrices, in Combinatorial Theory and Its Applications (P. Erdős et al., eds.), North-Holland, 721-728. Apparently gives exact solution on m X n board for m and n large. - N. J. A. Sloane, Jun 29 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..12.

J. Carlson and D. Stolarski, The correct solution to Berlekamp's switching game, Discrete Math., Vol. 287 (2004).

P. C. Fishburn and N. J. A. Sloane, The solution to Berlekamp's switching game, Discrete Math., 74 (1989), 263-290. (Apparently our value a(10) = 34 is incorrect!)

N. J. A. Sloane, Solutions for 3 X 3 through 10 X 10 boards. Note that the so-called "solution" for n=10 is incorrect.

N. J. A. Sloane, The box built by Elwyn Berlekamp in the 1960's.

Example

According to Calson and Stolarski, the following position with 35 lights on cannot be reduced:
xxx00xx000
xx0xx000x0
0xxx00000x
x0x0x00x00
x00x0x0x00
0x00xx0000
000xx0x000
0x0000xx00
000x0000x0
x00000000x

Crossrefs

Sequence in context: A002789 A083204 A061784 * A296202 A126613 A024224

Adjacent sequences: A005308 A005309 A005310 * A005312 A005313 A005314

Keyword

hard,nonn,more,nice

Author

N. J. A. Sloane

Extensions

Corrected and extended Dec 09 2003; last line of example corrected Dec 30 2004

Status

approved