Football has kind of weird scores. Even though it is common to most readers of this blog, a sport where scoring 3 or 7 is common while 2 is rare is kind of weird in the scheme of sports (most other sports are strictly one point at a time other than basketball where 1, 2, 3 are each fairly common).

I was always interested as a kid in figuring out what possible scores can happen in a football game. Certainly multiples of seven are common: 7, 14, 21, 28, etc. along with one or two field goals thrown in for good measure. I wanted to know exactly which scores are possible and which are absolutely forbidden.

Each team’s score is independent of the other: how many points I can score doesn’t depend on how many the other team scores, so we need only look at a single team’s possible scores. For numbers less than seven, zero is clearly possible. Next, a safety gives two and a field goal three. Four, five, and six are made up of combinations of safeties and field goals. Then anything seven or greater can be scored by following simples rules (along with many other possible combinations): keep subtracting seven point touchdowns as long as possible. If the remaining score is zero, you are done. If it is one, switch one of the extra points to a two point conversion. Otherwise, if it is two through six, add safeties and field goals as necessary. For example, if a team had its heart set on scoring 43 points in a game, we would see that six touchdowns takes us to 42, one point short, so five regular touchdowns plus one with a two point conversion gets us there. (For the adventurous reader, this sort of math is known as modular arithmetic.)

This leaves us with any score accessible except for one. This was always a bit disappointing because that is infinitely many possible inaccessible scores. But so it goes.

Except, not. Actually this is not true at all. Continue reading →