Revisiting Super Bowl squares strategy

Scoring in (American) football offers excitement for the analytically inclined because of the uniqueness of having only a few building blocks with which to build up particular scores. They usually come in bunches of three and seven, sometimes six, sometimes eight, and more rarely, two. That gives us an interesting lens to look through for America's favorite party game, Super Bowl squares.

Back in 2018 I wrote about Super Bowl squares strategy if your rendition of the game picked the numbers before participants bought in. While this is usually not the case it's still helpful to understand your chances of winning given the squares you've been saddled with.

If you’re unfamiliar with how the game works here’s a primer:

You start with a 10-by-10 grid with the Rams and Bengals -- this year's Super Bowl contestants -- team labels on the bottom or on one side (or vice versa). From there people buy individual squares. Once every square has been purchased, the numbers zero through nine are randomly chosen and placed across the bottom and side of the grid. The y-axis of the grid represents the last digit of the Ram’s score, while the x-axis represents the last digit of the Bengals’s score. (Note: there are variations on how the numbers/squares are chosen).

At the end of the game, or perhaps at the end of each quarter, the winner is the player who owns the square where the last digit of the Rams' score intersects with the last digit of the Bengals’ score. For example, if the game finishes Bengals 27-21, the player who had picked the square sitting at the intersection of the column with seven and the row with one would win.

Since we have easy access to NFL game data all the way back to the 1999 season thanks to nflfastR nowadays, we can calculate the likelihood of winning Super Bowl squares based on historical results and which of the 100 squares you've chosen. The aforementioned home team X7 - away team X1 score1 has happened one percent of the time since 2015.

Using game data back to the 2015 season2, we see that sevens and zeroes are the most desirable numbers to own for end-of-game payouts. A Bengals win with their score ending in a seven and the Rams with a zero is the most common scenario to expect next Sunday, one that has happened in 3.3 percent of games since 2016. The vice versa scenario -- Rams ending in a seven and the Bengals a zero -- has also happened 3.3 percent of time (technically it's 3.33 percent vs 3.27 percent, respectively).

Let's put some dollar amounts on this. On a $100 bet per square you'd be expected to get paid out the following amounts. This is called Expected Value.

Earlier I said seven and zero are the most desirable number because they're the ones that occur most often. To show that here's the frequency of each digit3 occurring in all NFL games dating back to 2015 Week 1. We clearly can see that seven and zero are the most popular numbers, followed by three and four. Digits six, one, eight, and nine are not that common, while two and five are the dreaded values you hope to avoid.

As pointed out by David Robinson in this blog post, even though seven and zero are the most common digits, 7-7 and 0-0 are not the most common pairs of digits. This makes sense because games usually don't end in a tie.

The most common overall score in the NFL is 20, which has occurred 251 times since 2015. The common way you get there is by scoring two touchdowns (and the corresponding extra points) and two field goals; pretty standard scoring sequences in football. Other common scores include 17 (237 times), 24 (232) and 27 (223). Overall, the most common game score is 20-17, but while those two numbers occur frequently, 17-17 or 20-20 are relatively rare compared to other combinations of common numbers because of how few games go into overtime.

Your office or Super Bowl party likely also pays out on end-of-quarter scores, not just the game's final score4. So let's take a look at what the probabilities look like on a quarterly basis. For this analysis I used data going all the way back to 1999.

What stands out is how good of a chance you have at winning money after the first quarter if you've got the magic zero or seven values. Since 1999 18.3 percent of first quarters have ended 0-0, the most common pair of any digits of any quarter. Overtime is also interesting because we can see how relatively few combinations have actually occurred since 1999. That makes sense given few NFL games go into the extra period, and the ones that do don't end in ties all that often.

Let's look at the histograms of most common digits by quarter now:

Again, zeroes and sevens show up in large numbers, although don't represent as high a share of the overall pie in later quarters. Having at least one zero in particular is a gold mine, but having 0-0 for per-quarter payout games is even better. In fact, if you had that combination you have approximately a 33 percent of winning money in that scenario; more than half that value comes from the first quarter alone. So while scores where the two teams' final digit is zero isn't as common as many other scenarios it will pay out the most money in a game where winners are chosen at the end of each quarter.

If the world of NFL scoring combinations excites you, I highly encourage you to watch what is possibly my favorite YouTube video ever: an introduction to Scorigami by Jon Bois:

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1. Despite the game being played at the Rams' SoFi Stadium, Cincinnati will be considered the home team. This is a predetermined designation and rotates between the AFC and NFC each year. Funny enough the Rams will be allowed to use their usual locker room.

2. In 2015 the NFL moved the distance for the point after kick to the 15-yard line.

3. Code to recreate the above plots can be found on my GitHub.

4. The most common payout is one winner for each of the first 3 quarters and a 4th winner for the final score. The payouts can be equal or they can increase each quarter. For example, if each square is sold for $10, then each winner would receive $250 in the equal payout scenario. Alternatively the payout structure could look like: 1st Quarter $100; 2nd Quarter $175; 3rd Quarter $275; Final Score $450.