How to Pick the Perfect Super Bowl Squares

Super Bowl Squares are a popular past time where sports fans guess (bet on) the outcome of one of game. But what numbers should people chose?
Historical numerical frequencies for NFL games.
The way this bet works, a user must chose what the last digit of the final score will be for both teams. For example, when the Seahawks beat the Broncos with a score of 17-24 (this is my actual prediction), then the person with the square at the intersection of 7 to 4 would win. I'm optimistic since I know the Vegas spread favors the Broncos by roughly 3 points. So if the final score were 24-17 Broncos, then the person with the [4,7] square would win. The graph above shows the most likely outcomes from all 14,000+ historic games. Going up and down are the odds for the scores from the winning team. Going left to right visualizes the scores from the losing team. Dark blue indicates an outcome three times as likely as the average outcome. Teal indicates average likelihood. Yellow Indicates below normal odds. These colors are not colorblind safe, but they do go well with Seakhawks colors.


Notice several things: there are patterns in the numbers

Historical numerical ranks for NFL games.

The purpose of the chart above is to help make the first chart clearer. In order to make decision making more meaningful, this chart forces some distinct between similar values through ranking each from most likely to least likely. This should make it even clearer to the reader which positions are more or less likely to win. Here dark blue represents the number 1 position and bright white represents the worst position. Again, similar patterns are present as in the first chart but there is now increased variation.


So what? Perhaps others have already analyzed the Super Bowl squares problem. What frustrates me is that none of these predictions seem to account for a common and important principle. Mathematicians have known about this secret for over a hundred years. Simply put, the smaller the number is, the more likely it is to occur. So, when it comes to the perfect squares problem this is important. The number 0 will occur more often by chance. Then 1 will be the next most common and so on. Actually the charts below summarize this effect.
Natural distribution of the trailing digit of a two-digit number in a table.Natural distribution of the trailing digit of a two-digit number in histogram chart.

Armed with this new knowledge, recreate the previous chart to account for these likelihoods.
Natural distribution of the trailing digit of a two-digit number.

Notice that many of the same trends still occur. However, the region in the bottom-right is now more important than the top-right. Specifically the numbers [7,6],[7,7], and [8,7] are now more common than [0,0],[1,0], and [1,1]. This represents a quantifiable advantage useful for decision-making. Actually, the image below shows the top 10 squares to consider choosing both before and after accounting for the Benford frequencies.
Top 10 picks before and after Benford normalization.

Notice the how the number one spot changes from [0,7] to [4,7]. So, if the most likely outcome is supposed to be [4,7], what will the full score be? Will it be 14-17, 17-24, or 7-14, etc? And which team will win? It turns out that mono-frequency football analysis favors the winning team having a score starting with a 2 and the losing team having a score staring with a 1. This suggests a score such as 24-17. (I can post this if there is interest). In terms of which team will win... Lynch and Sherman are two of the reasons why I choose to make this prediction in favor of the Seahawks.


Data Sources: Football Scores, Frequency of Natural Numbers (Benford's Law).

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