The

ancient egyptian notation for

fractions. All fractions were expressed as a

sum of

reciprocals. For instance,

`2/3 = 1/2+1/6`, and

`7/13=1/2 + 1/26 = 1/3 + 1/5 + 1/195`. It is

interesting that there is always such a representation for a fraction. In fact, always taking the largest reciprocal still smaller than the remainder is guaranteed to

terminate! (This is the

greedy algorithm for the problem)

This naive algorithm is, however, very awkward in practice, as it requires huge denominators. For instance, it gives `47/60 = 1/2 + 1/4 + 1/30`, whereas `47/60 = 1/3 + 1/4 + 1/5` would be much better.

There are many hard problems related to this representation. For instance, if we restrict ourselves to using reciprocals of odd numbers, we'll always get a fraction with an odd denominator. But is it true that any fraction with odd denominator can be expressed as an egyptian fraction using only odd numbers? It turns out that this is true (a previous version of the writeup claimed this is an unsolved problem, but it isn't one).