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,