Impossible scores identity?
Short version: I found an unexpected identity,
Floor(a/b)+Floor(2a/b)+...+Floor((b-1)a/b)
= Floor(b/a)+Floor(2b/a)+...+Floor((a-1)b/a)
where a, b>0 are integers. Is there a "nicer" way to write this sum? I'm particularly interested in the case when a and b are relatively prime. For example, restricting to such pairs and letting a=2^k, these sums are (a-1)(b-1)/2.
For context, with the Super Bowl today, I was wondering what scores were theoretically possible. You can score 2 or 3 points (among others), which is sufficient to score any non-negative integer except 1. If a, b>0 are relatively prime, the sums above represent the number of exceptions, i.e. the number of scores that are not theoretically possible in a game where you can score a or b points at a time. (If they are not relatively prime, a similar interpretation works, but it's complicated by the fact that infinitely many scores are unobtainable.)
...sigh. Y!A's horrible typesetting strikes again. The formulas are:
Floor(a/b) + Floor(2a/b) + ... + Floor((b-1)a/b)
= Floor(b/a) + Floor(2b/a) + ... + Floor((a-1)b/a)
Ah cool, thanks Pauley Morph. It follows roughly from your picture that my sum in general is (a-1)(b-1)/2 + (gcd(a,b)-1)/2, which is (a-1)(b-1)/2 when they are relatively prime. Much nicer than I expected.