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 with all the light being turned off. How can you turn on the top-left and bottom-right squares at the same time?

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

Let’s 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 top-middle light, the top-left corner, or the centre. 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.

After some playing with the \(3\times 3\) board, Sam guessed that there were \(900\) different light patterns that could be obtained by playing on this board. Was he right?