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Math problem

Bonzo

Scholar
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Joined
Feb 8, 2004
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Here is an interesting math problem.

You have a perfect, solid sphere. You drill a cylindrical hole completely through the center of the sphere, from one side to the other. What is left looks like a bead, with the end caps removed.

The hole is 6 inches long from end to end.

What is the volume remaining in the sphere? (Hint: there is only one answer, and the answer is not a function).
 
Oldie.

Answer (select to reveal)
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36 π

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Okay, I guess I'll be the idiot who doesn't get it.

How can you calculate the volume without specifying the diameter of the hole?
 
The volume turns out to be the same, regardless of the diameter of the hole. A larger diameter hole would seem to leave less of the sphere remaining, but the catch is that a sphere with a large diameter hole must itself be bigger in order for the hole to be the same 6 inches long. It evens out in the end.

If you're allowed to assume that it evens out, which is certainly not obvious (at least not to me), there's an easy way to find the answer: since the hole's diameter doesn't matter, we may as well say it's zero. If its length is 6 inches, the diameter of the sphere is 6 inches too. So the answer is just the volume of a 6-inch diameter sphere.

I once worked out the answer without making any assumptions. The algebra was a bit of a mess, as I recall, but the answer did turn out to be independent of the hole's diameter.
 
Thanks for the answers and link- I hadn't seen that one before. Where I was going wrong is that I was thinking that the length of the hole was the same as the diameter of the sphere, which is obviously not the case. I went through the algebra and I got the right answer in the end.

I guess I'll just put that one down to old age/tiredness/general stupidity.
 
Dont forget the units!

The units are cubic inches.

There sticklers out there who will say (with some justification) that any answer that should have units but omits the units is a wrong answer!

Trivia: Mr. Randi posed this question as a puzzle in 1999 or 2000 (he borrowed it from his friend Martin Gardner). I was one of the first to respond with the correct answer, and I received an e-mail from Mr. Randi congratulating me on my solution. What a thrill. If I remember right, when Mr. Randi published the solution a week later, he omitted the units from his answer.
 
Ceptimus is right that this is an old problem. I think I first saw it in one of Martin Gardner's books a long time ago. I have always liked it.

Does anyone know if Martin Garnder is still alive? I think he would be close to 90. I used to really like his column in Scientific American. After he left I quit subscribing.
 
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