Problem #3

A centrifuge is a circular apparatus with n equally spaced slots into which test tubes of equal weight may be inserted. In order to function properly, the center of gravity of the test tubes must be at the center of the circle. For example, as shown below, when n = 6 one can insert 2, 3, 4, or 6 test tubes in a balanced manner, but not 1 tube or 5 tubes.

Scientists use a set of extra tubes filled with water (which are the same weight as the sample tubes) to use as counterweights in order to balance the centrifuge if necessary. For example, when n = 6, only one extra tube is needed (to balance things in the case when there are 1 or 5 test tubes).

This month's puzzle has two parts:

• For each n = 1,2, ..., 30, what are the "allowable" numbers of test tubes (i.e. those that can be balanced in the centrifuge)? For example, when n = 6, the numbers are 2,3,4, and 6.

  What can you say about arbitrary n?

• For each n = 1,2, ..., 30, what is the minimum number of extra test tubes required? For example, when n = 6, only one extra test tube is required, but when n = 5, four are needed.

  What can you say about arbitrary n?

Source: Ivy Fitzgerald

The solution will be posted shortly.

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