Football Squares
It’s September and that means the NFL (American Football) is back! Every year my mother-in-law hosts a game of football squares — also known as Super Bowl Squares, since it’s frequently played at Super Bowl parties. In football squares, people can buy a square in a 10 x 10 grid. Each square represents a digit from 0 – 9 for the home team and a digit from 0 – 9 for the away team. These numbers reflect the last digit of all possible score outcomes for the game. For example, if the home team has a score of 13 and the away team a score of 28, then the person who purchased the square corresponding to 3 for the home team and 8 for the away team would win the moolah.
Common NFL Scores
Before examining the data, we should first understand how points are accumulated in the NFL. The most common ways to score are field goals (FGs) (3 points) or touchdowns (TDs) followed by successful conversion of the extra point (6 + 1 = 7 points). Less common ways to score are safeties (2 points) or TDs with successful 2-point conversions (6 + 2 = 8 points), which are generally not attempted early in a game. Finally, a TD with a failed conversion will only yield 6 points (a multiple of 3). From this, we can see that scores such as 1 are essentially impossible (there is an obscure rule which makes this theoretically possible, although it has never happened), 2 is very rare (a safety occurs about once every 14 games), 4 is extremely rare (requiring two safeties), 8 is unlikely (requiring a TD and successful 2-point conversion or 2 FGs and a safety), 11 is unlikely (multiple possible ways to get there), etc. Other scores are intuitively more probable and include multiples of three and/or seven: 0, 3, 6, 7, 9, 10, 12, etc. I would predict that good digits to have in football squares would be 0, 3, 7, and maybe 4, while less valuable digits would be 1, 2, 5, 6, 8, and 9. If we look at all final scores from 1988* – 2020, we see this is the case. Only “1” and “4” have never been part of a final score (although apparently the Chicago Cardinals finished with 4 points in 1923).
The Best Numbers
My mother-in-law (a Chicago Bears fan) purchased a square for my daughter. I wanted to know what the probability was that we’d win. In my mother-in-law’s version of the game, the score at the end of each quarter represents an opportunity to win if you hold the corresponding square. The game is played for the entire regular season (17 games), but your numbers/squares are different for each game. There are 68 opportunities to win (17 games x 4 quarters), therefore different probabilities exist from week to week. So, what numbers are best? To answer this I looked at quarter-by-quarter scores for every game from 1988 – 2020, pulling the data from pro-football-reference. I then plotted how frequently each score comes up at the end of each quarter.
Wow! At the end of the first quarter, almost 20% of all scores end in 0 – 0. This is not surprising since many teams fail to score any points, or commonly reach 10 points. There are only around 2 to 3 drives per quarter per team and so combinations of 0/3/7 are also frequent. Note that although the 0/7 pair shows up more frequently than 0/0, the 0/7 pair is across two squares. Thus, you’re still best off holding the 0/0 square.
By the end of the second quarter, the number 4 starts to show up more frequently in the score (two TDs or 3 TDs and a FG).
At the end of the 3rd quarter, either of the 0/7 squares is better to hold than 0/0, and the 0/3 squares are not far behind.
By the end of the game, 0-0 isn’t looking like the optimal choice, although it’s not bad. The distribution is starting to look a little more uniform, but still favors 0/3/6/7. It looks like 7/0 is your best shot if you’re playing the variation where only final score matters.
Unfortunately, most variations of the game don’t let you choose your numbers (or I suppose fortunately, if you arrive late to the Super Bowl party and your pick of squares). But what about those only playing squares for the Super Bowl? Does the World Series of Football play out any differently? Usually, there isn’t a home team since the site is decided before the season starts (the 2020 Tampa Bay Buccaneers are the only team to play [and win] a Super Bowl in their home stadium). Here are the results for all Super Bowls from 1988 – 2020 (Pro Football Reference still lists a home and away team). Stay away from 2/5/6/8, I guess. 1’s look pretty good.
What is the probability I will win?
OK, back to my daughter’s numbers and the regular season. I used the numbers above to generate probabilities for every score combination in each quarter. Then I generated a table to show the squares she had and calculated the probability that the square would win in any quarter. If you have a chance to win every quarter, the best square to hold is 0-0 with a probability of winning of 0.301; 0-7 (0.256) and 7-0 (0.224) are not far behind. The worst square to have is 2-2 (probability = 0.00082). This season, my daughter’s best chance of winning is week 15, where she has the 0-4 square. Her overall chance of winning at least once during the entire season works out to 32.2%.
| Game | Site | Opponent | Bears | Opp. | Prob. |
| 1 | Away | Rams | 7 | 8 | 0.016 |
| 2 | Home | Bengals | 7 | 1 | 0.052 |
| 3 | Away | Browns | 9 | 3 | 0.022 |
| 4 | Home | Lions | 2 | 1 | 0.006 |
| 5 | Away | Raiders | 1 | 3 | 0.037 |
| 6 | Home | Packers | 1 | 0 | 0.034 |
| 7 | Away | Bucs | 9 | 3 | 0.022 |
| 8 | Home | 49ers | 1 | 5 | 0.008 |
| 9 | Away | Steelers | 4 | 5 | 0.008 |
| 10 | Home | Ravens | 5 | 7 | 0.011 |
| 11 | Away | Lions | 4 | 6 | 0.035 |
| 12 | Home | Cardinals | 2 | 1 | 0.006 |
| 13 | Away | Packers | 8 | 8 | 0.005 |
| 14 | Home | Vikings | 8 | 8 | 0.005 |
| 15 | Away | Seahawks | 0 | 4 | 0.097 |
| 16 | Home | Giants | 1 | 9 | 0.009 |
| 17 | Away | Vikings | 5 | 6 | 0.005 |
Outcome
My daughter didn’t win this season. In week 11 against the Lions the game ended 16-14 (Bears-Lions) but my daughter had 4 for the Bears and 6 for the Lions. Week 15 was her highest chance of winning, but none of the quarters ended with both numbers at the same time. Oh well, her college fund didn’t get a head start.
Thanks for reading!
Code and Other Details
This is my first post. As I move forward, I’d like to include code and other relevant details of my projects, but this project was started ages ago (updated analysis for this post) and the code is messy.
*1988 is an arbitrary starting year. The 1970’s saw a lot of rule changes (moving the goalposts, hash marks, etc.) that affected scoring, success rates, and play calling. Even now, small rule changes can have consequences as teams rely more on the pass or the run. Because I’m trying to analyze current probabilities, I don’t believe it’s important to include some of the older data.