I love simple puzzles with elegant solutions that are perhaps unexpected. I particularly enjoy it when these puzzles can be solved without invoking a whole bunch of machinery to support the solution. Mathematical machinery is nice, but a great simple puzzle can be a great segway into that machinery. As such, invoking it right off the bat kind of misses the point.

Here’s an interesting question: what is a dimension? Dimensions can often seem like life. We know it when we see it, but defining it can be kind of tricky. As such, a good heuristic can be useful in thinking about dimensions, and, in particular, the number of dimensions a particular mathematical object has.

As a tutor at the secondary level, I get an inside look at the various issues that students face within mathematics. While the specific issues are different from person to person, I’ve come to see a pattern emerging. Anecdotally, I think that most of the issues students face could be summarized with one sentence.