## Paint by Number and Equation

Artistic inspiration comes from anywhere even painting by numbers. However, in this case, the numbers are a table or matrix of numbers. How? First, take a look at this image that was part of the mathematical art exhibit at the

To understand this portrait, created by Dr. Bruce Torrence of Randolph-Macon College, you need to know the Lights Out game. We’ll play on a 5 by 5 grid of lights. When the game starts, a random number of these lights is switched on. In the game below, the upper left light is on.

Pressing any of the lights will toggle it and the adjacent lights. For example, pressing the lower right light, as seen below, toggles the lights above and to the right.

If we then press the light in the fourth row and fourth column, the puzzle becomes the configuration seen below. Why all the toggling? The goal of the puzzle is to switch all the lights off, preferably in as few button presses as possible.

This puzzle lies behind Torrence’s portrait of Tibor Gallai seen above. Gallai, a Hungarian mathematician, was the first to prove that for any sized grid in the Lights Out game, it is always possible to get from the all-on to the all-off state.

Where are the numbers in this art piece? The portrait is pieced together from 28 by 28 squares which come from reachable states of the Lights Out game on a 28 by 28 grid of lights. How do you know which configurations of lights can be acheived in a game? That’s where Torrence used math and equations. A 28 by 28 grid becomes a 28 by 28 grid of numbers where 1 and 0 —> Read More