A very important tool in maths is to use symmetries to make problems easier. A symmetry of a shape is a movement that leaves it looking exactly the same. 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 being all off. Explain why \(5\) is the smallest number of presses you need to turn the whole board on.
Alice and Jamie each have an identical “Lights Out” board (same size, same rules). Both boards start with all lights off, and on this board size there are no quiet plans. Alice presses a plan \(A\); Jamie presses a different plan \(B\) (not the same set of buttons). Could they end up with exactly the same final pattern of lights?