Counting is Mathematics Too

2 September 2017

Have I mentioned that I love mathematics? I love mathematics so much that I really believe most people love math too, even if they don’t know it yet. Maybe people who “don’t like math” just haven’t seen a problem that is interesting to them.

The advantage of believing that everyone could like mathematics as much as I do is that I have an ongoing desire to make any mathematics I’m doing interesting to friends who “don’t do math.” (I’m not endorsing that phrase in quotation marks, for the record.)

All of this preamble is to be transparent about the fact that I am trying to convince you that you, too, can like math. Take it or leave it: I won’t be offended if you do not agree.

So now, the math stuff: in my Abstract Algebra class, I wanted to make my final paper contain at least some problems that were understandable to my friends. Thus, my math-loving friend and classmate (shoutout to Maya Banks) worked with me on the final paper with that goal in mind. Our paper, “Counting in the Presence of Symmetry: The Ballad of Cecilia” contained theorems and formulas, and nothing rings “mathematics” quite like theorems and formulas (lies). But it also included some fun problems, presented below, which I invite you to think about if you’re so inclined.

  • Imagine you are babysitting your three-year-old cousin, Cecilia, and she is playing with blocks. You decide it would be fun to paint each face of every block either turquoise or magenta. There are 6 faces and 2 colors, so you figure there are 2 possible colors for each of the 6 faces and thus 2^6 = 64 total colorings. So off you and the dedicated Cecilia go, painting 64 blocks, each one slightly different from the others.

    But once the 64 blocks dry, Cecilia begins to play with them, rolling the blocks this way and that. Suddenly, certain blocks begin to look the same! The block you painted with turquoise on only the top and bottom faces looks the same as the block you painted with turquoise on only the right and left faces. Of course, these blocks do not only look the same; they are the same! Inconceivable.

    So now, you and Cecilia begin another activity: given a certain block, can you find the other blocks that are the same? How many distinct sets of blocks will there be at the end? In other words, how many nonequivalent colorings are there of the set of faces of a cube?

  • Fast forward a few years to Cecilia’s sixth birthday. She is playing in the grass, and we are all making flower crowns out of dandelions, clovers, and violets. We decide to make each crown using five flowers. How many different designs are possible, such that no two designs are equivalent? (For people who didn’t make flower crowns as kids, first of all, I’m sorry. Please look them up. But I do need to share an important detail: unless you’re a true flower-crown-making professional, the flower crowns you will probably make can be flipped over for the same effect.)

  • Cecilia has developed a passion for baking. We have made a square pan of brownies, and we cut the brownies into 3x3 squares for 9 brownies in all. Her favorite colors are turquoise and magenta (from her block- painting days long ago), so we wish to keep the brownies in the pan and frost each with either turquoise or magenta. How many different (or “nonequivalent”) ways could we frost them? (Note: Two ways of frosting are the “same” if you can start with one frosting, rotate the pan, and get to the other frosting. But there’s no flipping the pan! Can you flip a pan of brownies upside down and still see the frosting on them? No, you can’t, mister.)

  • Cecilia is now 11 years old, and a budding glass artist. We are working with her to make stained glass windows for her mother’s house. The stained glass windows will be made 3 squares wide and 4 squares long. One option is to use only turquoise and magenta squares. Another option is to use turquoise, magenta, and goldenrod squares. How many distinct ways are there to make the stained glass windows using only 2 colors? With 3 colors? (Note: In this one, we are dealing with glass that looks the same on both sides. As such, you can flip the glass around!)

So you see? These problems are fundamentally about counting things. And yes, counting things is mathematics. Counting things is also accessible. I hope it’s fun too. What are you counting?

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