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?

Let’s now play with the \(2\times 3\) “Lights Out’’ board. You can take it as a fact that there are exactly \(3\) quiet plans for this board (you can read example \(3\) again to remind yourself of quiet plans). Can you find them all? Using this, or otherwise, can you work \(4\) plans that produce the following pattern?

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.

George is playing on the \(5\times 5\) “Lights Out" board and says that since each light can be on or off, and there are \(25\) lights on the board, the number possible light patterns that can be achieved by playing the game is \(2^{25}\). It turns out that the number is much smaller, it is \(2^{23}\). Can you explain why? You may take it as a fact that these three are the only quiet plans of the \(5\times 5\) board: