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?

(In reply to Hint by Brian Smith)

It sounds like an important part in solving is to figure out how to turn on (or off) lights without pressing those two lights themselves.

For example, in the hint, we can turn those three lights off without pressing them by pressing the lights below them instead.

I think it's also important to note that if we can reason about which pairs of lights such a solution can be constructed for, there is no need to explicitly find such a solution for the pairs.

In the comments for http://perplexus.info/show.php?pid=1344 there was some discussion about how to solve the general case, if anyone was interested in looking at other cases.

Posted by Gamer on 2008-04-10 16:05:09