Towards a More Uniform Football Squares System

Football Squares is a football betting system in which the participants buy squares on a 10 x 10 grid labeled with the digits 0 through 9 on each axis. The last digit of each team’s score at the end of the game (or after each quarter) correspond to the row and column of the winning square. Regardless of which teams are playing, some squares are much more likely to win than others. Is there a way to organize football squares to get a more uniform distribution of outcomes?

Historically, bettors have preferred 0s, 3s, 4s, and 7s, and with good reason: both winning and losing scores are more likely to end in these digits. Below is the distribution of winning and losing football scores for all 15,474 games in the history of the NFL.


Now let’s look at the distribution of the last digit of the score. The value of squares that have 0s, 3s, 4s, and 7s are even more apparent in this visualization:


One possible way to get a more uniform distribution of outcomes would be to convert the scores to a different base before taking the last digit. For example, converting the scores to base 12 would result in a 12 x 12 grid with the digits 0 – B on each axis. Below are the historical probabilities (with teams randomly assigned to each axis) of each outcome in base 12. Darker squares are more likely to occur.


As you can see, square (0, 0) is much more likely to occur than any other square, so base 12 isn’t the best choice if we are aiming for uniformity. Let’s look at base 10, since that’s what everyone uses:


Bettor’s already know that base 10 isn’t uniform, but this is still a useful way to justify the desire for 0s, 3s, 4s, and 7s.

In my disappointing quest to find a base in which squares were roughly equally likely, base 4 ended up being pretty okay. I wanted to find a metric to quantify how uniform the score distribution was for each base but didn’t have much luck. The sample variance would be useful to see which of two different sports had a more uniform score distribution on a board like this (as long as the base is the same on each board), but it is not useful here. Another thing I tried was to compute Pearson’s chi-squared goodness-of-fit test statistic, but that resulted in extreme values for all bases I tried (up to 35) which basically says that none of them are uniformly distributed. In the end, I just looked at these 2D histograms and picked the one that looked the most uniform (other than base 2, because that would be really boring). Anyway, here is what base 4 looks like:


Maybe football squares just aren’t meant to be uniform. Even if there were a base that made football scores uniformly distributed, good luck getting your friends to agree to doing this with you: “Okay guys, I made this cool 16 by 16 grid for the Super Bowl. You just convert the Patriots score to hexadecimal and put it on this axis, and same for the Rockets and put it on the other axis. That way, the scores…”

If you want to have a uniform gambling party with your friends, just flip coins or something.


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