Football puzzle answer, plus a follow-up

The highest football score that cannot be gotten using just 3-point field goals and 7-point touchdowns is 11.

The highest football score that cannot be gotten using just 6-point touchdowns and 7-point touchdowns is 29.

Note that 11 = 3 x 7 – (3 + 7).

Likewise, 29 = 6 x 7 – (6 + 7).

This is no accident! For extra credit, prove that whenever there are two ways of scoring, one worth p points and one worth q points — and p and q have no common divisors (above 1) — the highest score that cannot be gotten using these two ways of scoring is p x q – (p + q).

This turns out to be a version of the Frobenius coin problem,

Eugene Volokh teaches free speech law, religious freedom law, church-state relations law, a First Amendment Amicus Brief Clinic, and tort law, at UCLA School of Law, where he has also often taught copyright law, criminal law, and a seminar on firearms regulation policy.
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Eugene Volokh · January 22, 2014