Coin problems

Here’s a problem that a friend posed to me a while ago. Apparently this is called the Coin Problem, but I’d never heard of it before.

McNuggets used to come in boxes of 6, 9, or 20. Given that scheme, what is the largest number of nuggets that cannot be ordered exactly?

Here’s a similar problem:

In American football, teams can score points in increments of 3 (field goal) and 7 (touchdown plus extra point). What is the largest number that can’t be a valid football score? (I’ve ignored other possible ways of scoring — 2-point safeties, 6-point touchdowns without the extra point, 8-point touchdowns with a two-point conversion — because the problem is utterly trivial with these extra options.)

I’m not going to give the answers (if you want to cheat, see the above link), but I suggest questions like these as a way of engaging elementary-school students (who have mastered addition and multiplication) with a non-traditional math question.

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