Consider a 3x3 array of lighted buttons:

[O][O][O]
[O][O][O]
[O][O][O]

If any one of these buttons is pressed, it and all of its directly adjacent neighbors will toggle their status on/off. (Diagonals do not change.)

The goal is to get all of the lights to be off. Which buttons must be pressed to accomplish this?

Which buttons would accomplish this goal in the minimum number of presses for 4x4, 5x5, 6x6, 7x7, 8x8, and 9x9 arrays?

A nice site with the game from 3 x 3 up to 8 x 8 is here.

...is not very fruitful.

There are definitely some patterns of 1s that repeat.  Notably every 5th pattern.

The patterns for 4, 9 and 14 each is very sparse with individual 1s surrounded by 0s and blocks of 4. 

10 and 15 are both very regular with grids of 4x4 blocks and singles at their corners.

6 and 8 both have a solid octagon of 24 1s.

11 has 3 different patterns two of which have no symmetry.  All seem very irregular.  17 has a very irregular single pattern with diagonal symetry.

5, 7, 15, 16 and one of the 11s all feature extended blocks of 7 or 10 1s.

I was hoping o see how to extend these findings to solve an arbitrarily large nxn array.

Posted by Jer on 2009-06-04 15:08:52