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E4. Very Proper Fractions

A proper fraction is one where the numerator is no larger than the denominator. We'll call a fraction with denominator 1 a VPF (very proper fraction). Imagine a civilization where the only way of representing fractions is as a VPF or a sum of (two or more) VPFs. To keep things interesting, they don't allow using the same denominator twice. So, for example, they can't write 2/5 as 1/5 + 1/5. However, 2/5 CAN be represented as a sum of distinct VPFs: 1/3 + 1/15. Since the denominator of these fractions are all 1, we can simplify the notation by simply writing the numerators. For example, we've shown that we can convert from our system to their system by 2/5 = (3,15).

Problem: Write 1143/1170 as it would be expressed in this civilization.

Extra challenge: Describe an algorithm for expressing any proper fraction as a sum of distinct VPFs, and prove that your algorithm works. (In particular, show that it terminates in a finite number of steps.) This challenge is an advanced problem.

E3. A Fraction Sequence

Let S be the sequence of all proper fractions with denominator ≤ 20, arranged in order from smallest to largest, so the first fraction is 0/1 = 0 and the last one is 1/1 = 1. Represent each fraction in lowest terms. Define the gap between two consecutive fractions in this sequence to be the difference of the smaller from the larger.

1. Determine the smallest gap in the sequence, and find two fractions that have this gap between them.

2. The gap between the first two fractions is 1/20 – 0/1 = 1/20, as is the gap between the last two fractions. Not counting these fractions, find two fractions that have a gap that is as large as possible.