Welcome back from the summer holidays! We hope you’re rested and ready for an exciting year of problems. Today we’re going to dive right into our first topic with a fun game called “Lights Out."
Imagine a grid of square buttons. Each square has a light that can be ON or OFF. When you press a square, that light flips, and so do the lights directly above, below, left, and right. In today’s pictures, dark blue means ON and light blue means OFF. If you want to try it at home, search “Lights Out (game)” on Wikipedia for an interactive version. It may be useful for some of the harder problems.
Before we start, let’s quickly define two key words:
A light pattern is just which lights are ON or OFF on the board.
A plan is the set of buttons you choose to press to make a pattern. In Examples 1 and 2 we’ll see why a plan only needs to say which buttons are pressed: the order doesn’t matter, and each button is pressed either once or not at all.