Egyptian fractions

The ancient Egyptians and Babylonians mainly used fractional units, the fractions with 1 as the numerator. Other fractions were written as sums of fractional units, now called 'Egyptian fractions'. For example, 19/24 = 1/2 + 1/6 + 1/8, because 12/24 + 4/24 + 3/24 = 19/24.

The book's puzzle: it is easy to make 1 from fractional units that are all the same, like 1/2 + 1/2 or 1/3 + 1/3 + 1/3. But can fractional units that are all DIFFERENT add up to 1? With two units it is impossible. 1/2 is the largest fractional unit, and 1/2 + 1/2 is already exactly 1. To make the two units different, at least one 1/2 must be replaced by a smaller unit, and then the sum is less than 1.

What about three different units? Start from 1/3 + 1/3 + 1/3 = 1. To make them different, one 1/3 must get bigger, and the only fractional unit bigger than 1/3 is 1/2. So 1/2 is one of them. Now 1/2 + 1/4 + 1/4 = 1, and one of the 1/4s must get bigger: the only unit bigger than 1/4, other than 1/2, is 1/3. So the first two are 1/2 and 1/3.

1/2 + 1/3 = 3/6 + 2/6 = 5/6, so the third unit must be 1/6. That gives 1/2 + 1/3 + 1/6 = 1, and this is the only way to write 1 as a sum of three different fractional units (in any order).