Football Math

Introduction: Who doesn’t love football? Every autumn, every week, there are dozens of games, good games, on national television. Often, you have no allegiance to either team and, mercifully, for once, neither team is depending on your armchair cheering to tilt the outcome of the game in its favor.

At last, an opportunity to watch football in peace! No responsibilities. Just beer, pretzels and picking out the next Tom Brady. Incredible feats of strength and speed, paradigms of grit and determination, and coaching stratagems worthy of a chess master. What’s not to love!

And then there’s the game itself, the ebb and flow, the score. Early score differentials (‘spreads’) are sometimes amplified as the game progresses, but just as often they are dampened, and in some cases, they are actually reversed. Football is a 21st century cultural phenom. In this age of social fragmentation, it’s just about the only thing we still have in common. Almost everyone speaks football. If baseball is our national pastime, football has become our national language and every fall weekend we celebrate our secular liturgy in the vernacular.

A football game is all about the unfolding of patterns (every ‘play’ is really a pattern), and after watching dozens of games this season, it occurred to me that while there are numerous patterns inside each game, the games themselves might also form patterns.

For example, can the scores of multiple games between various opponents tell us anything? Or do they just vary randomly? Do scores evolve arithmetically over the course of a game; or is there something else, something non-linear, at work?

Football Math: To explore this, I looked at score differentials at the end of the first half and compared them with score differentials at the end of the same games. Easy-peasy, right? Well, no! In fact, solving the problem requires us to develop (or deploy) a whole new mathematics.

A football game is an example of a discontinuous process. We deal with whole numbers only (no fractions). Plus, unlike most other sports (soccer, baseball, e.g.), those numbers do not increase iteratively. Say a team scores 5 times in the course of a game (5 runs, 5 goals, etc.). In most sports, that would result in a score of “5” for that team. Not in football.

In ‘football math’ there are only 3 digits: 2, 3, and X where the value of X can be 6, 7, or 8. Crazy? Yes, but if you are a regular reader of Thoughts While Shaving (TWS) and Aletheia Today Magazine (ATM), you already know about civilizations that ‘play the game of life’ with number systems very different from the one you learned in grade school. In fact, we’ve studied one culture that has no numbers whatsoever. Other cultures have limited inventories of numbers (e.g., 1, 2, X where X refers to any collection of 3 or more items). Hot Link Compared to these societies, ‘football math’ gives us a lot to work with.

Granted, we only have 3 digits (2, 3, X), but one of those digits (X) can represent any one of three different values (6, 7, or 8). So, in essence we have 5 digits (2, 3, 6, 7, 8), Plus, we can ‘add’ those digits together to generate higher, ‘secondary’ numbers.

So far so good, but please, don’t get too comfortable! It turns out that the score of a football game is much like the result of a road race, according to Zeno. Zeno-math applies in universes, like football games, where quantity is not infinitely divisible.

In our search for patterns, we need to look at an event (e.g., the game) from 3 perspectives: pre-game, game, and post-game. Pre-game began at Big Bang and won’t end before kick-off. (If you’re tailgating, you might want to take an Uber…and invest in a port-o-potty. 15 billion years equates to a lot of Budweiser.)

Pre-game, the so-called ‘score’ is always 0 – 0, of course. But ‘0 – 0’ is just a short way of saying, ‘The game’s not afoot yet, my dear Watson’. 0 – 0 is not a score; it looks like a score but in fact it denotes the absence of a score. Rather, it’s a state of Being, i.e., pre-being. A whistle blows: the kick-off –finally, a play that could result in points. Seemingly, we’ve moved from pre-game to game…but in fact, we’ve merely transitioned from pre-game to potential-game, ‘being-in-waiting’, which is still a flavor of pre-being.

Remember, for the purposes of this exercise, we are not concerned with a 60 yard rope, a one-handed snag, a blocked punt, or a pick-six. We are only tracking score and so far, we have no score. We say that the score is still 0 – 0 but again, that is just a convention.

As we saw above, a score of 0 – 0 corresponds to the state of being we call ‘pre- game’. The ‘game’ begins when pre-game ends and pre-game ends when someone scores points. As Yogi Berra might have said, “We have no score until we have a score.”

Now suppose the game ends in a 0 - 0 tie (after overtime): then for our purposes, there was no game. Disagree? Check out the standings. A team with a tie in its record is the same as a team that has played one less game.

Eventually, sometimes mercifully, the final whistle blows, the stadium clock reads 00:00 and there are no flags on the field. The game is over. No further points can be scored…this week. Only now can we talk about a winner and a loser. The final score is not part of the game itself; it is part of the post-game.

Let’s check the time. Pre-game began at Big Bang and post-game doesn’t end until Big Crunch (or Heat Death), so I recommend you head home as soon as the game ends.

The Game: Blue scores: the game has begun. And Blue leads, right? Wrong! Blue does not lead. As long as there is still time left on the game clock, the game is still ‘statistically tied’. How come?

After Blue scores, the scoreboard reads 2 – 0 or 3 – 0 or 6 – 0 or 7 – 0 or 8 – 0. But either team can score 8 points on any one play, and there is at least one play left. So, a lead of 8 points or fewer is actually no lead at all because it can be erased at any second so long as the ball is still in play (i.e., the game clock reads something other than 00:00).

British philosopher, Alfred North Whitehead, ‘the process philosopher’, describes every event in the real world, the way we just described a football game. What we call pre-game, he calls ‘the actual world’; what we call post-game, he calls ‘objective immortality’.

Every event (game) arises out of an actual world (pre-) and dissolves into an objective world (post-). Every football game begins at the end of pre-game and ends at the beginning of post-game. Assume there is an 8 point differential heading into the final play of the game. The final score will reflect a differential somewhere between 14 points and 0. (A Touchdown scored on the last play of a game can only be worth 6 points to the leading team.)

In the language of statistics, e.g., political polling, we would say that the so-called ‘score’ at any point in the game has a margin of error of up to 8 points. Therefore, when the ball is still in play and Blue leads Red by 8 points, the game is statistically tied.

The Search for Patterns: Ground rules in place (and hopefully agreed), we can now get back to our search for patterns; I examined the box scores of the 13 FBS games played during week #7 of the season in which at least one of the teams playing was ranked in the Top 25 (quality control).

Of those 13 games, 5 ended with a differential of 8 points or less (one score), 6 ended with a differential of 16 points or less (two scores) and 2 games ended with a differential of more than 16 points (three scores or more).

Now let’s look at the scores of those same games at the end of the first half.

Hypothesis: On average, the differential in points at the end of the first half should be half of what it is at the end of the game. If so, 11 games (out of 13) should have been ‘statistically tied’ (point differential of 8 or less) at the half. Observation: The total number of points scored was roughly the same in both halves, as expected; but only 9 games were statistically tied at the half (vs. the 11 anticipated).

This means that there is a centripetal force at work in a football game that offsets, at least in part, that ubiquitous centrifugal force we know as ‘time’ (or duration). In English, please? Ok, scores tighten, not absolutely but relative to time played.

Confirmation: Unwittingly, Miami Dolphins head coach, Mike McDaniel, recently gave Football Math a big boost. In a 2022 game against the Buffalo Bills, Buffalo scored a touchdown at the end of the first half, making the halftime score 21 – 13.

The announcer asked McDaniel for his reaction, which I paraphrase: “It doesn’t matter; it’s still a one score game.” In other words, the game is still within the 8 point margin of error, so it remains statistically tied. Hear what McDaniel had to say in his own words: https://youtu.be/7bTAjZ728eI

Application: Can we learn something from this analysis that we can apply beyond the universe of football?

A football game is an example of a single event with conflicting objectives. Like any system in a state of quantum coherence, it often manages that inherent conflict by delaying its ‘winner reveal party’ until after the last play of the game. While ‘there can only be one winner’, the game itself is shaped by both sides. Objectively speaking, it doesn’t matter which team wins; it’s a zero-sum game. Subjectively speaking, of course, it makes all the difference in the world; it’s an all or nothing proposition!

Process is self-modifying. Things diverge less than expected, based on traditional arithmetic. Interaction favors convergence, not divergence. W. B. Yeats notwithstanding, things do not fall apart as rapidly as expected. Interactivity inserts another variable into the cosmic equation. Hope, even in the face of inexorable entropy - that’s the hidden meaning of football.