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. It *is* possible to score one point in a game, although it is exceptionally rare – a fact that I became aware of thanks to this post. In 2004, Texas was awarded one point on an extra point attempt against Texas A&M in which the ball was not kicked through the uprights, and in the 2013 Fiesta Bowl, Oregon was awarded one point in similar play against Kansas State. (Personally I think that this play between Rice and Alabama in the 1954 Cotton Bowl is actually the rarest play.) After searching our extensive football databases here at ALDLAND, in addition to Google and Wikipedia, I believe that these are the only two instances of this rare play. The simplest description of what happens comes from this one line score summary from the 2013 instance: “Alejandro Maldonado blocked PAT recovered by defense outside endzone and tackled in endzone for one-point conversion safety.” That is, the team attempting to kick the extra point still got one point, so for the sake of discussion of possible scores this changes nothing: in each case a team still gained seven points after the touchdown and point-after attempt.

Except there’s another caveat: that has never happened, as far as I know. It is possible for the defense to block a PAT kick and run it back. This is a two point play. If, however, the defense fumbles the ball just before making it all the way and the offense/kicking team recovers the ball outside their own endzone and is then tackled in their own endzone, the result would be a one point safety with the score awarded to the defense. This opens up the possibility of a 6-1 score (note that the other team must have scored in order for it to apply). From 6-1, other possible scores that are now allowed are 8-1 (add a safety), 9-1 (add a field goal) and so on for any number 8 or higher. The result of this new (to me), bizarre rule is that the only truly disallowed scores are 1-0, 1-1, 1-2, 1-3, 1-4, 1-5, and 1-7. Every other score can be reached.

Before continuing, let’s check the rule book just to make sure we’re not completely off the rails. The NCAA football rulebook can be found here. The relevant part is rule 8 on page FR-80, which allows for a safety on a “try down” for one point. An additional rule is mentioned, a forfeit results in a score of 1-0 (unless the forfeit happens after a game suspension and the forfeiting team was already losing in which case the score stands). This allows for one more disallowed score to be ticked off. Only six scores are disallowed now. For the NFL, rule book here, a one-point safety is also allowed (see rule 11 on page 57), although interestingly in the case of a forfeit, the official score is 2-0 (which does not count towards tie-breakers and such). So a score 1-0 is possible in NCAAF but not in the NFL.

There is one more case that seems worth discussing, although necessarily from a more casual point of view, and that is the maximal allowed score. Certainly no team can score a million points, but 100 is likely possible. The most points scored by one team was not accomplished by Michigan, despite a history of 100+ point games in the beginning of the last century, but was a feat accomplished by Georgia Tech over Cumberland in 1916 with a score of 222-0. (There is a rich, fun history of 100+ point games in college football, but my favorite is Rice’s. Rice went down early 3-0 to SMU in 1916 but then came back to clinch the game 146-3.) From a more practical standpoint, the only game in the last 50 years between major football programs with a triple-digit score was in 1968 between Houston and Tulsa (Houston trounced those Golden Hurricanes 100-6). The NFL records are much tamer. Only twice has a team breached 70, in 1950 and 1966. As such, for the modern, era, it feels safe to put an extreme upper limit of 100 (keeping in mind that many scores are still impractical: 100-99 and such).

Practical limits are for engineers and probably lawyers and other people. physguy is only satisfied with the *actual* limit. Suppose each team scores on each kickoff. Since they kick from the 35 (penalties can help here) and the defense must start behind the 45, if the fastest (say a 4.4 40 yard dash) player lines up at the 45 and sprints into the ball, catches a line drive of a kick, and runs for the endzone, we are looking at between 4 and 5 seconds to score assuming that he runs in a straight line. If we conservatively take 5 seconds and assume that each team goes for two each time, we would have scores of roughly 2880-2880. Other than adding penalties on the kickoffs (e.g., celebration), I can’t see how a team could score faster than this without moving to even more ridiculous scenarios (note that the NCAA defines an effective minimal length per play of two seconds on page FR-48 of their rule book in a discussion of spiking).

Apparently there still are infinitely many inaccessible scores. Alas.