Problems

Suppose we have a \(2\times 2\) board where all the lights start being turned off, how can we turn on the top two lights?

Now let’s imagine a \(3\times 3\) board, how can you turn on just the middle light?

Below you see three light patterns and their matching plans. Each plan turns all lights on except one square: the centre, the top-left corner, or the top-middle light. Explain why, using these plans, and one fourth plan that you need to find, any light pattern on the \(3\times3\) board can be made by a suitable plan.