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Solution to the knot problem in Mathematical Problems in the Biological Sciences: Proceedings of Symposia in Applied Mathematics, Vol. 14, 1962, pages 55—68.

If links are not required to be at right angles to their adjacent links, knots of 24 links are possible.

2. Figure 14 gives Fitch Cheney's answers to the problem of expressing the integers 1 through 20 by using pi, as few times as possible, and the symbols specified. He was able to express all integers from 1 through 100 without using more than four pi's in each expression.

Figure 14

How the first 20 integers can be "pied"

Hundreds of readers improved on Cheney's answers. Here are some typical ways of shortening six of the expressions:

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