A classic brainteaser concerns a hunter out to bag a bear. The hunter walks 1 mile south. He then turns left and walks 1 mile east, then turns left again and walks 1 mile north. He ends up back where he started and spots a bear. What color is the bear?
The obvious answer is “white.” If you’re at the North Pole and you go south along any meridian, turn left and go 1 mile east along a circle of latitude, then 1 mile north, you end up where you started. And you’re more likely to find a polar bear there than any other kind.
But, if you ignore the requirement of seeing a bear, there are infinitely many possible solutions—if you look for additional starting points near the South Pole. The South Pole itself isn’t a solution because the only direction in which you can go to start with is north.
In fact, there is “an infinity of infinitely many solutions,” Eli Maor of Loyola University Chicago points out in the September issue of Math Horizons. “And this is in addition to the one obvious solution, the North Pole.”
Suppose that you draw a circle centered at the South Pole with a circumference (not radius) of 1 mile. Start at any point along this circle and go 1 mile north. You reach a point that satisfies the requirements of the problem. Proceeding south from this point puts you on the circle, turning left and walking 1 mile along the circle takes you all the way around it, so walking 1 mile north then puts you back in the original spot.
There are infinitely many such points, all lying on a circle centered at the South Pole and having a radius of (1 + 1/2pi) miles.
But there are even more solutions. Suppose you draw a circle with a circumference of 0.5 mile. Any point 1 mile north of this circle is also a solution point. This time, you would end up going around the circle twice to end up at your starting point. Again, there are infinitely many solution points.
And you can do this for circles of infinitely many different circumferences.
So, altogether, you have an infinity of infinitely many solutions, plus one. In Georg Cantor’s arithmetic of infinity, that’s simply uncountably infinite.