M: Math is Beautiful

Imagine a row of 50 numbered light bulbs, each of which is operated by a toggle button. If you push the button, the light bulb turns on; if you push the button again, it turns off. At first, all the light bulbs are off. Now send a sequence of students along the row. Student 1 pushes every light bulb’s button: now all the bulbs are on. Student 2 pushes every button next to a multiple of 2, turning off bulbs 2, 4, 6, etc., up to 50. Student 3 pushes every button next to a multiple of 3, and so on, until finally student 50 pushes the button next to 50. When all the students have gone through the sequence, which light bulbs are shining?

Take a break before reading further, and try to solve the problem. What do the shining light bulbs have in common? Do you find this result surprising?

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The shining light bulbs are 1, 4, 9, 16, 25, 36, and 49: all the perfect squares (numbers that are the result of a whole number being multiplied by itself) less than 50. Why should this be? Take another break and see if you can come up with an explanation.

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When I looked at the problem from the students’ perspective, I was amazed, but when I looked at it from the light bulbs’ perspective, it started to make sense. Take bulb number 12, for example. Its switch was pushed by students 1, 2, 3, 4, 6, and 12. An even number of times. It was turned off as many times as it was turned on. Or a prime number – bulb number 17, for example – which was only pushed by students 1 and 17, also an even number of times. But bulb number 9’s switch was pushed by students 1, 3, and 9 – an odd number of times – meaning the last time its switch was pushed, it was turned on. Only the perfect squares have an odd number of factors (numbers that divide into them without leaving a remainder). 

Why do perfect squares have an odd number of factors, and other numbers don’t? See if you can answer this one yourself.

Factoring numbers is an exercise often given to upper elementary or middle school math students. They complete worksheets and move on with their lives. But this problem approaches the concept from a unique angle – first, notice a fascinating pattern, and then try to figure out what is responsible for the pattern. After spending some time with this problem, which I hope you did, you have a much deeper understanding of the concept of a factor than you did before.

This is what I love about math. There are patterns all around us, which we often ignore or take for granted, but when we look at them more closely, we discover that they point to other patterns (like the pattern of students turning on light bulbs pointing to the pattern of how many factors any number has).

A key is to be playful. When my oldest children were small, I got them (to be honest, I got myself and let them play with) a set of pattern blocks. As my daughter used the blocks to build pictures while her brother napped, I grabbed all the triangles and started some further explorations into the concept of perfect squares – could I arrange any perfect square number of triangles into a larger triangle? It turns out I could.

1 layer: 1 triangle, 2 layers: 4 triangles, 3 layers: 9 triangles, 4 layers: 16 triangles, 5 layers: 25 triangles

I noticed that some of the triangles were pointing down and some were pointing up. I thought it would be aesthetically pleasing to remove the downward pointing triangles. And while I was at it, I might as well just move them to the side the same distance to make another triangle.

That was interesting! Each of my new triangles was a sequence of 1+2+3+…; also known as a triangle number. Was every perfect square the sum of two consecutive triangle numbers? It seemed that it had to be, just from the geometry of the pattern blocks. Why should that be? Could I prove it with numbers instead of pattern blocks? I grabbed a pencil and piece of paper, let my little girl have the green triangles back so she could make leaves for her pattern block tree, and went off to explore the patterns behind the patterns.

What makes math addictive is that it takes something beautiful and orderly and unveils an even deeper and more orderly pattern that explains or gives rise to the first beautiful thing you saw. The deeper you go, the more clearly you see patterns everywhere. Math is beautiful.