Problems

Imagine a \(2\times 2\) “Lights Out" board. If every light is off at the start, how can we turn on just one of the squares? Can you notice something about the order in which we press squares?

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?

A \(2\times 2\) “Lights Out" board starts being all off. How can you make the top-left and bottom-right squares be on at the same time?

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

Imagine a “Lights Out” board that starts with all the lights off. A plan that is not empty (it has at least one button in it) is called quiet if, after pressing all the buttons in the plan, the board ends up all off again (remember that plans can’t have buttons being pressed twice). Now take a \(3\times 2\) “Lights Out” board. Can you find two different ways to turn every light on? How can this help you to discover a quiet plan?

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.