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.