%I M1040
%S 0,1,2,4,7,11,16,22,27,35,43,54
%N Solution to Berlekamp's switching game (or lightbulb game) on an n X n board.
%D Brown, Thomas A.; Spencer, Joel H. Minimization of +1  matrices under line shifts. Colloq. Math. 23 (1971), 165171, 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
%D Komlos, J., & Sulyok, M. (1970). On the sum of elements of +1 matrices, in Combinatorial Theory and Its Applications (P. ErdÅ‘s et al., eds.), NorthHolland, 721728. Apparently gives exact solution on m X n board for m and n large.  _N. J. A. Sloane_, Jun 29 2014
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H J. Carlson and D. Stolarski, <a href="http://dx.doi.org/10.1016/j.disc.2004.06.015">The correct solution to Berlekamp's switching game</a>, Discrete Math., Vol. 287 (2004).
%H P. C. Fishburn and N. J. A. Sloane, <a href="http://dx.doi.org/10.1016/0012365X(89)901416">The solution to Berlekamp's switching game</a>, Discrete Math., 74 (1989), 263290. (Apparently our value a(10) = 34 is incorrect!)
%H N. J. A. Sloane, <a href="/A005311/a005311a.jpg">Solutions for 3 X 3 through 10 X 10 boards.</a> Note that the socalled "solution" for n=10 is incorrect.
%H N. J. A. Sloane, <a href="/A005311/a005311b.jpg">The box built by Elwyn Berlekamp in the 1960's.</a>
%e According to Calson and Stolarski, the following position with 35 lights on cannot be reduced:
%e xxx00xx000
%e xx0xx000x0
%e 0xxx00000x
%e x0x0x00x00
%e x00x0x0x00
%e 0x00xx0000
%e 000xx0x000
%e 0x0000xx00
%e 000x0000x0
%e x00000000x
%K hard,nonn,more,nice
%O 1,3
%A _N. J. A. Sloane_
%E Corrected and extended Dec 09 2003; last line of example corrected Dec 30 2004
