Super Bowl math puzzle

In honor of the upcoming Super Bowl, here’s a math puzzle:

Some scores cannot be achieved in American football without a safety or a two-point conversion — i.e., just using 3-point field goals and 7-point touchdowns (and 6-point touchdowns, though they don’t matter for the puzzle, since any score that can be gotten with 6-point touchdowns can be gotten with double that number of 3-point field goals).

You can’t, for instance, have a side getting only 1 point (unless it’s a forfeit), or 2 points (that would require a safety, which the puzzle doesn’t allow). You can’t get 4 points, either, if all you can score are field goals and ordinary touchdowns.

What is the highest such score that mathematically cannot be gotten using just 3-point field goals and 7-point touchdowns? Past that point, all possible scores can be gotten using those two ways of scoring. (Of course, assume that the teams have infinite scoring capacity — I’m asking a math question, not a question about human stamina or speed.)

Relatedly, what is the highest score that cannot be gotten using just 6-point touchdowns and 7-point touchdowns (i.e., if neither safeties nor field goals nor two-point conversions are scored)?

I’m talking here just of one team’s score; the other team’s score doesn’t enter into the puzzle. (Thus, if you think the highest score is 4, you should just say 4, not 4-4.)

For the answer — and a cool formula that yields the answer, together with a follow-up problem — see here. But if you come up with an answer yourself (no peeking!), please feel free to post your answer in the comments to this post.

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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