I’m not responsible if you break your picture frames

January 04 2017

Here’s a puzzle: How can you hang a picture from two nails (nailed into a wall in a normal fashion) so that removing either one of the nails causes the picture to fall?

It may not look like it at first, but this is a math puzzle; in fact, it’s connected to some really cool (but advanced) mathematical ideas. A child could understand what this puzzle is asking, and a high school student in the process of solving this puzzle would almost certainly discover the gist of some of these ideas, whether they realise it or not.

Humans fundamentally like patterns, and our role as mathematicians is to seek out patterns that occur in the world around us, play with them to find out why they’re there, and ask questions about our findings to continue our explorations. When I mention I study math, I almost always hear some form of “I’m not a math person!” – but the picture-hanging puzzle likely piqued your interest; everyone’s fundamentally wired to be interested in finding patterns and solving problems!

Mathematics is presented to schoolchildren as a barrage of algorithms to be memorised and applied rotely and quickly. Because they get scared away by their teachers or fall off track at some point in their mathematical education, people are generally not exposed to mathematics for what it truly is. It is an art that so happens to find scientific methods handy, but the way that we teach it in schools now really is analogous to teaching art using Paint-by-Numbers kits! (Please take a read through Paul Lockhart’s “A Mathematician’s Lament” for a better exposition of this; it’s a fantastic article and says more than I can in 600 words.)

As technology continues to propagate in our society, there is an increasing need for people able to think in an adept manner mathematically, yet the education systems in place worldwide continue to fail to equip students with mathematical skills outside the scope of what they need to pass their exams. On the other hand, many of my fellow mathematicians feel that understanding how to play with mathematical objects has allowed us to develop an ability to problem-solve that has been useful in many aspects of our lives.

I contend that there is a way to fix this problem, but enacting a solution may not happen during our time. In my mind, it will require teachers to be trained at a much higher mathematical level, a radical change in the way policy-makers perceive mathematics, and a willingness to open up the mathematics classroom to teach in a completely different way, one that gives students the opportunity of participating in the art of mathematics through allowing students the freedom to play with and explore whatever patterns they think are interesting. On a microcosmic classroom level, I’ve seen this work, but it will take a lot more work to see this succeed on a wider scale. In the meantime, through my teaching work, I’m going to try my best to show others the beauty of the art within mathematics.

Just for fun, here are two more puzzles which might not look mathematical but actually are:
-If you remove two opposite corners of a chessboard, can you tile the remaining part of the board with dominoes?
-Google “Bridges of Königsberg” and look at the first image. Can you start at some point in the city and take a walk which crosses each of the seven bridges precisely once each?

Shoot me an email if you’re interested in talking more about math and math education. I’d be glad to chat.

Boston, MA

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