Several years ago, I got a 5x5 lights out game for Christmas. It is a puzzle game in which the goal is to turn off all the lights which are on at the start (from 1 to all 25). Lights are turned on and off by pressing them. However, pressing any one light will change both that light and all the orthogonally neighboring lights.

But now, two of the lights are broken. The two lights still light up correctly, but pressing either of them will not have any effect. However, if a puzzle was solvable with all 25 lights working, it was still solvable with the two broken lights.

How many different pairs of lights can be broken like that?

Hint:

The lights out game I got had 50 preprogrammed puzzles.  The first one was (X=on):

. . . . .
. . . . .
X . X . X
. . . . .
. . . . .

The manual even gave the solution for this one, showing how the puzzle changed by pressing 6 different buttons on the bottom half.

Posted by Brian Smith on 2008-04-09 21:37:12