Many Ways To Arrive At A Simple Answer

A couple of weeks ago, I appended to one of the Monty Hall paradox articles a classic teaser that seems as though it cannot be solved. In fact, the very simplicity of the puzzle is the secret of its lasting charm. This teaser puzzle is extremely simple, and it can be solved.

Assume a hole is bored through the center of a sphere, and that the hole is one inch long. How much material remains in the sphere after the hole is drilled?

So simple, and yet so cunning! I can’t remember the first time I ran across this little gem, but it was at least forty years ago when I was in high school and studying geometry and algebra. Since those were the tools I had, and being an impetuous youth, I jumped into it and used the various formulae for spheres, cylinders, and end caps to get the right answer after much work. It was straightforward to set up the equations with the diameter of the hole as an unknown.

After simplifying the terms and re-arranging to get the answer, I had one of those classic “gotcha” moments when you realize that you have just gone down the wrong road. Knowing the answer, the correct way to solve it was obvious - and a lot less work than I had just been through.

Later, when I was in college, I saw the same puzzle in a somewhat different form in Martin Gardener’s Mathematical Recreations column in Scientific American. But by now I was heavily into calculus, and having these powerful new tools, I decided to use them and solve the problem in a new way. Sure enough, using relatively formulation, the same answer popped out.

So now we have at least three ways of solving a puzzle that at first appearance looks to be insoluble. There is no reason to explore the two difficult ways unless you want to for the fun of it. The key to the solution is to assume the presenter of the problem (me, Martin Gardener, whoever) is honest. I assured you the problem could be solved. It has a unique, well-formulated solution. So you can reason that if it has a solution, and if I did not specify the diameter of either the sphere or the hole, then you can specify the diameter of the hole to meet your needs. An obvious choice is to make the hole have zero diameter. Then the length of the hole is just the diameter of the sphere. And since we all remember the formula for the volume of a sphere is (4/3)πr3 , the answer is obvious. Just put in r = ½ and that’s it.

Again, as long as you believe the puzzle has an answer, this is the correct one. You will get the same answer for any diameter greater than 1. Of course, if you assume a larger diameter of the sphere, the diameter of the hole will depend on that choice in a simple way determined by the Pythagorean theorem. It’s pretty sobering to realize that the answer will be the same for a sphere the diameter of the Earth as it would be for something the size of a large bead.

If you have any doubts about this, please feel free to work it out the long way, and you will see that the diameter of the hole falls out of the final expression.

Since the problems presented here are oriented around decision theory and how to set up an environmental description such that a computer can make judgment calls, why is the hole in the sphere problem here? It has nothing to do with statistics or decision theory.

What the sphere puzzle does tell us is that the computational load can change greatly (whether by high school student, college student, or computer) depending on how closely we listen to the given conditions. In school, one might have to go the long way around to get an A. In real life, anything you can use to lighten the computational load is fair game. Assuming a solution exists in small potatoes compared to some assumptions that have to be made to do such common tasks as driving an automobile.

So when setting up a program or trying to teach a machine to make decisions, be very careful about not building in your own prejudices. Why force a machine to use solid geometry or calculus when there is an easier way?

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here .