The Bone Man Problem is an intriguing and aggravating and wonderful and awful problem posed by Clifford Pickover in his book, The Mathematics of Oz, Mental Gymnastics from Beyond the Edge, Cambridge University Press, 2002. You can find more information about his book here.

Dorothy and Dr. Oz peer into a deep hole in the ground. The bone man comes closer and opens and closes his mouth spasmodically. "In the pit," he says, "are 10,000 leg bones. I have cracked each bone at random into two pieces by throwing them against a rock. What do you think is the average ratio of the length of the long piece to the length of the short piece for each time I crack a bone? You can reason from a purely theoretical standpoint. If you cannot find the solution within two days, I will add your leg bone to the pit."

This question is not an easy one to solve, and many intelligent people approach it from many different angles. Debates rage over its interpretation and its answer. The main thing to keep in mind is that you are not trying to determine the average ratio of the long bone to the short bone. That is obviously unknowable without measuring the bones yourself. The Bone Man is essentially asking "what is the most likely average ratio of this pile of 10,000 bones." In other words, what average ratio should you tell him such that you have the best chance of not losing your leg?

After some work, however, I believe that I have solved the problem, assuming that "average" indicates "mean" and assuming that the Bone Man will tell you to what accuracy he can measure the bones.

Unfortunately, the analysis entails many equations and graphs that would be a hassle to put on my web page here, so I wrote up a Microsoft Word document that happens to be over 20 pages long. The gist of it is that the answer depends on the accuracy at which the Bone Man can measure the lengths of the bones. Here is a brief report of the answer, but if you want an explanation, click on one of the two links below.

Bone Man's Precision

Your Best Guess to save your leg

1/10th

4.208

1/100th

8.141

1/1000th

12.609

1/10,000th

17.192

1/15,002th

18.002

Anything More Precise

18.002

Even Infinite Precision

18.002

If you have any questions or find any problems with my analysis, please let me know! mark@nandor.org