Problems
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.
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?
A very important tool in maths is to use symmetries to make problems easier. For today, define a symmetry of a shape as a movement that leaves the shape looking exactly the same as initial. For example, rotating a square by \(90^\circ\) (spinning it by a quarter turn) is a symmetry. Imagine you are playing lights out on a board that has no quiet plans. Explain why if a light pattern has a certain symmetry, then its corresponding plan will also have the same symmetry.
A \(3\times 3\) “Lights Out" board starts with all the lights off. Explain why \(5\) is the smallest number of presses you need to turn the whole board on.
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?
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 work out the \(4\) different 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 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.
