Superbowl square strategy

For Superbowl 56, I’ll be rooting for the mod 10 operator. The mathematical function outputs the remainder of a number after dividing by 10. For example, mod 10 of 21 equals 1, or mod 10 of 14 equals 4. Modular arithmetic - when using a larger base than 10, say a long prime number - forms the basis for many encryption algorithms. In fact, most of your important data is encrypted and protected with this nifty mathematical function. But, obviously more important than secure financial records or confidential health claims, the modular operator is used for the popular bar game: Superbowl squares. 

Here’s how the game works: The Cincinnati Bengals and the Los Angeles Rams will play football for one quarter. During this time, you will indifferently observe the game while you eat your Superbowl snacks and chat with friends. At the end of the quarter, while you reconvene in on the tv for the real action, the mod 10 operator is applied to each team’s score. So, if the score was 13 to 6, we’d have an output of (3 ,6). For a score of 0 to 10, we’d output (0, 0). You then look to see if your Superbowl squares cover that bucket. If yes, you win a cash prize. If no, you wait until the end of the next quarter, cross your fingers, and hope the modularized score has moved into one of the squares that you do own.

You have a potential to win four times, one for each quarter. As compared to watching for solely football related reasons, rooting for football derivatives is much better because you can win multiple times. It’s much more satisfying to win two quarters of Superbowl squares than it is to have your chosen team win the Superbowl, because 2 > 1. Math.

The possible modular scores result in a grid of 100 squares. The Bengal’s score could end a {1, 2, 3, 4, 5, 6, 7, 8, 9, 0} and the Ram’s score could end in a {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}, and 10 * 10 = 100. Usually you’re allowed to select multiple squares. At my company, for the office version of the game, we can choose up to five squares. So, given all the aforementioned rules, a player would naturally ask, “Which squares should I select?”

***

Assuming you know which squares you’re selecting (in some formats of the game you don’t - more on this later) these are the top 15 most likely squares:

The frequency represents the probability that the square will be hit during the game. I pulled the data from the last 55 superbowls for each of the 4 quarters, so 55 * 4 = 120 data points. As a methodology note, the buckets were made by taking an average of both the NFC and AFC’s modular score, which is to say, the bucket of (0, 7) represents an average of (0, 7) and (7, 0). 

You probably won’t be able to get both the (0, 7) and (7, 0) buckets, so just grab whichever one you’d prefer if the option presents itself. As guiding advice, the Bengals have a good kicker, so you want squares that end in three for them. E.G., for the (0, 3) option, I’d recommend Rams 0, Bengals 3 before continuing with the option of Rams 3, Bengals 0.

***

As useful as it may be, the above analysis is rather basic, easily answered with a quick Google search. The real question with a more complicated analysis is, “If the numbers are randomized, which squares should I choose?” In this circumstance, you would know where on the grid your squares sat, but not the score combination they represented. The blow grid shows more clearly what I mean.

You may think to yourself, “If all the squares are random, it wouldn’t matter which squares you selected since you don’t know what they represent anyway. Wouldn’t you have an equal probability of winning regardless of your choices?”

No.

While you can’t control your square selections, you can control the pattern you choose. For example, you could choose one of the possible patterns:

While maybe not intuitive, some of the above patterns have better probabilities of winning than others. The reason for this is that the superbowl squares are more likely to transition in certain directions, less likely to transition along diagonals and more likely to transition along files. Follow me into the statistical weeds, and I’ll show you the winning pattern.

***

The first part of my analysis, I calculated the transitional probabilities between quarters. I.E., the probability of being in one square and transitioning into one of the 99 other squares. For example, if the modular score at the end of Q1 is (7, 3), what is the probability the score will be (7, 6) at the end of Q2? Or (4, 3)? Or (1, 3)? Or (0, 3)? Or (0, 0)? Or any transition. I calculated all 17,424 of these probabilities. 

Historically the Superbowl has only fallen into 66 buckets, so 66 of the possible 100 squares. Even though it’s an unreasonable assumption, I assumed history is representative of the future, and squares that haven’t happened historically won’t happen this year. It made my calculations simpler and saved room in my excel workbook. After generating all the transitional probabilities, I was able to calculate the probability of any mod 10 score at the end of any quarter. 

The second part of my analysis, I hooked up excel’s rand() function to a grid which allowed me to try out different selection patterns. I would input a pattern, then each time excel calculates, a random set of five squares outputs. The five squares would flow through the transitional probabilities to output an average payout. (Payout = Cash Winnings - Cost to Buy Squares). If you snagged some of the glory squares - think (0, 7), (0, 0), (3, 0), (7, 3) - you’d have an expected payout greater than zero; whereas if you got stuck with some of the nasty squares - think (5, 2), (2, 2), (8, 2), (5, 5) - you’d have expected payout less than zero. 

However, since the numbers are randomized, in some situations, with the same pattern strategy, you’d have a positive payout, but in others, you’d have a negative payout. For example, using the thumbs up pattern, the expected payout of five simulations equaled -$4.18, -$3.45, -$1.27, +$1.64, and +$3.82. The average of these simulations is -$0.69, so maybe you’d think it’s a poor strategy. If you’re a statistician, you’d want a larger sample size, and use VBA code to generate 20,000 simulations for each strategy. 

The cool part of statistics is being able to make claims based on small amounts of data. Some real life applications include accurately predicting the next president based on a sample of only 5% of the population, accurately predicting the 94th Oscar winners based on the previous award ceremonies, and accurately predicting winning superbowl square strategies based on the Central Limit Theorem. In the statistics religion, the Central Limit Theorem is the first pillar of the faith. It can be used to model the distribution of just about anything. Surprisingly, a sample of samples, with genuine random creation, will follow a normal distribution. The Central Limit Theorem proves this, and can be used to reject different hypotheses based on the variation of your samples. In my case, the null hypothesis was that strategy X (random, line, plus …) had an average payout greater than zero.

Running my 20,000 simulations on each strategy and recording important aggregate statistics, I was able to use the Central Limit Theorem to prove if a strategy was good or bad. In many cases, the variation wasn’t large enough to make a statistically significant claim, so I couldn’t say if anything. But, in the end, with an alpha of 0.05, I was able to statistically prove that the plus strategy has an average payout greater than zero. In the long run, you’ll make money with this selection. Go with the plus.