# Math puzzles

My brother sent me a picture of a book that one of my nephews was planning to bring with him on a long plane flight:

I was thrilled to see this, because Martin Gardner’s books are a huge reason that I am the way I am today. I suspect that lots of math and science nerds would say the same. I love the idea of yet another generation being exposed to them.

In honor of Martin Gardner, here’s a math puzzle I just saw. It was posted on the wall of the UR math department. One of the faculty there posts weekly puzzles and gives a cookie to anyone who solves them. I don’t know if this one is still up there, but if it is, and you hurry, a cookie could be yours.

A mole is on a small island at the exact center of a circular lake. A fox is at the edge of the lake. The fox wants to catch and eat the mole; the mole wants to avoid this. The mole can swim at a steady speed. The fox can’t swim but can run around the edge of the lake, four times as fast as the mole can swim. If the mole makes it to the edge of the lake, she can very quickly burrow into the ground. Can the mole escape? (That is, can she reach a point on the edge of the lake before the fox gets to that same point?)

You can assume that the fox and mole can see each other at all times, and that they can change their speed and direction instantaneously.

This is a great Gardner-esque puzzle, because you don’t need any advanced mathematics to solve it. All you need is distance = rate x time and the rule for the circumference of a circle.

If you want something harder, you can try to figure out the minimum speed that the fox must have in order to be able to catch the mole. (That is, what would you have to change the number four to, in order to change the answer?) I think I’ve worked that one out, although I could have made a mistake.

### Ted Bunn

I am chair of the physics department at the University of Richmond. In addition to teaching a variety of undergraduate physics courses, I work on a variety of research projects in cosmology, the study of the origin, structure, and evolution of the Universe. University of Richmond undergraduates are involved in all aspects of this research. If you want to know more about my research, ask me!

## 4 thoughts on “Math puzzles”

1. Presumably the mole has the best chance by moving directly away from the fox. Since 4 > pi, however, the fox can still reach the other side before the mole. But this is presumably too easy, and you indicated that they can change their speed and direction instantaneously. That makes it more complicated. Does the mole even have to reach the shore? Why not stay on the island, or in the lake?

I recommend Gardner’s annotated versions of Lewis Carroll’s works.

2. Once the fox starts to run around the lake, either clockwise or counterclockwise, the mole’s optimal strategy is to turn and swim at some angle that will force the fox to run more than pi radians.

3. test says:

You need to change 4 to pi + 1

4. I’m disabling comments, because for some reason I’m getting barraged with spam comments on this post. In the unlikely event you’re a real person who wants to comment on this post, let me know.