It's a Super Bowl tradition that goes back further than I can remember. Families gather together and risk their love for one another for the sake of winning a prize: the Super Bowl Squares game.
The game is fairly simple. A 10x10 checkerboard is signed by the initials of everyone who wants to play in the game- the wager is usually a couple dollars here or there- until the whole board is filled. Once all the names are signed, digits 0-9 are drawn randomly and filled out on the top and on the side of the board, with each set of numbers representing the score of one of the Super Bowl teams.
The goal is to have the game's score at the end of each quarter match up with your square. Imagine you have signed up and were randomly assigned the squares Ravens 4, 49ers 7. You hope is that the score at the end of each quarter is some variation of Ravens X4 - 49ers X7. That means you can hope for the score to be Ravens 4, 14, 24, 34, etc, and for the 49ers to put up 7, 17, 27, 37, etc. If you match up the score with your square, you win a prize; if you don't match, you lose and wait until the next quarter of play ends and hope for the best.
Got it?
Well, clearly not all squares are created equal. Some games will have automatic prizes for peoples assigned 2-2, 5-5, and 8-8 due to the lack of chance that the score will result in those matches at the end of each quarter of play. Is that fair? We'll see.
I decided to look into these Squares, drawing data from every football game from the past 6 years (I wanted to hit 3000 games worth of data points), in order to see the probably of winning in each quarter and, in the end, the relative value of each square. While not as perfect as it could be, I'm treating teams in a vacuum, where points scored are not weighted based upon the level of competition. This means that I will say that the Ravens and the 49ers have the same chance of putting up points throughout the game.
[Note: This means that the odds of the Ravens X4 - 49ers X7 will be the same as the Ravens X7 - 49ers X4. Probably not exactly correct, but it's what we've got with the numbers available]
What I found was interesting. Each quarter results with different odds of numbers winning, which makes sense, as the different variations of scoring throughout a 60 minute game is much greater than the expected outcomes in the first 15 minute quarter.
This fact, however, makes it quite clear: some squares are at a definitive advantage. Imagine the possible ways that teams can get to X1 in the first quarter. Technically, they could get a touchdown and two safeties. Maybe a touchdown with a 2PT conversion and a field goal. Point is, it's extraordinarily difficult to post an X1 in the first quarter without scoring three touchdowns. In all of this data, (over 3000 games), X1 happened 19 times after the first quarter- a 0.62% chance. In contrast, X1 has happened 293 times over the course of a full game- a 9.57% chance. As a result, any person with a square X1 has a much greater chance at winning the final pot, than they have at winning the first quarter prize.
On the other hand, a team scoring X0 happens 44.9% of the time after the first quarter- an astronomically high number when it comes to games of chance. Those odds drop to 16.3% at the end of the game. Still, any player with X0 gets to ride the whole game with fairly strong odds of winning a quarter at some point of the game. Tallying the numbers, X0 is expected to win 22.87% of the time- that's not to say that if you have an X0 square, you should win that often, but those are your odds if you own all of the X0 squares.
Now not everyone has the same payouts for the game, but a fairly common outcome is the 1:2:1:4 approach. This means that the prize for the halftime score is twice as much as the first and third quarters, while the final outcome wins the grand prize. The payout schedule changes the weights of each quarter and is important when determining the value of each square- but for this practice, we will look at the 1:2:1:4.
[For example, the odds of X7 winning is greater in Q3 and Q4, but are less in Q1 and Q2, which means that if the payout schedule is weighted heavier on second half results, the value of X7 will increase.]
Here's a table of the winning results in a 1:2:1:4 schedule:
As you can see, over the course of a full game, the most valuable squares will be the X0, X7, X3, and X4 squares. X2 and X5 are the two weakest. Surprisingly, to me at least, is the weakness of X9.
Using our expected results for each quarter, and weighting them by the 1:2:1:4 schedule, we can determine the fair value of each square- and if you're savvy, maybe you can get away with offloading some of your junk squares for a square with a higher chance at winning. As we're weighting both teams with equal chances of winning, you can treat these results as reversible (a percentage for X3/X7 can be taken as if either team scores the X3 or X7).
The single square with the greatest chance of winning is the X0-X0 square with a 6.09% chance of winning. Understand that each square, in a random world, would have a 1% chance of winning as there are 100 different combinations. The numbers point to X0-X0 being 6.09X as valuable as the expected return of a random square.
Here's a table of outcomes with greater than 2.5% chance of winning in a 1:2:1:4 schedule:
These squares all represent an expected return of 2.5X or greater. These outcomes make sense as they're all multiples of traditional scores. X0-X0 gets the jackpot as scoreless games through one quarter aren't uncommon, while scores of 10, 20, and 30 are very possible. X0-X7 is also a winner as teams are one touchdown away from reaching this multiple. If you get your hands on any of these squares, hold on for dear life.
Here's a table of outcomes with less than 0.15% chance of winning in a 1:2:1:4 schedule:
As you can see, X2 and X5 are the weakest squares and it makes sense for them to have automatic payouts. It's quite difficult for a team to hit X2 or X5 without an irregular score (safety, two point conversion, a ton of field goals), so the outlook is quite dim if you get these squares. Of course, both the Ravens and the 49ers are known for their smash mouth defenses, so there could be plenty of field goals. Just note that the odds aren't smiling in your favor.
Squares are a great way to find yourself invested in a game if your team isn't represented. The results should have been fairly unsurprising- multiples of touchdowns and field goals should be expected, while scores that need more unique circumstances happen less frequently. Still, the expected value of each square can help when determining your odds of winning when the game starts- and you can adjust your enthusiasm accordingly.
Below is a table that represents the fair value of a 1:2:1:4 schedule with a $10 square price. You can see the winners and losers and the expected value of each square, based upon the historical results for each quarter.
Best of luck and enjoy the Super Bowl!
[I sure ain't responsible for people taking this advise. Do this at your own risk.]
The game is fairly simple. A 10x10 checkerboard is signed by the initials of everyone who wants to play in the game- the wager is usually a couple dollars here or there- until the whole board is filled. Once all the names are signed, digits 0-9 are drawn randomly and filled out on the top and on the side of the board, with each set of numbers representing the score of one of the Super Bowl teams.
The goal is to have the game's score at the end of each quarter match up with your square. Imagine you have signed up and were randomly assigned the squares Ravens 4, 49ers 7. You hope is that the score at the end of each quarter is some variation of Ravens X4 - 49ers X7. That means you can hope for the score to be Ravens 4, 14, 24, 34, etc, and for the 49ers to put up 7, 17, 27, 37, etc. If you match up the score with your square, you win a prize; if you don't match, you lose and wait until the next quarter of play ends and hope for the best.
Got it?
Well, clearly not all squares are created equal. Some games will have automatic prizes for peoples assigned 2-2, 5-5, and 8-8 due to the lack of chance that the score will result in those matches at the end of each quarter of play. Is that fair? We'll see.
I decided to look into these Squares, drawing data from every football game from the past 6 years (I wanted to hit 3000 games worth of data points), in order to see the probably of winning in each quarter and, in the end, the relative value of each square. While not as perfect as it could be, I'm treating teams in a vacuum, where points scored are not weighted based upon the level of competition. This means that I will say that the Ravens and the 49ers have the same chance of putting up points throughout the game.
[Note: This means that the odds of the Ravens X4 - 49ers X7 will be the same as the Ravens X7 - 49ers X4. Probably not exactly correct, but it's what we've got with the numbers available]
What I found was interesting. Each quarter results with different odds of numbers winning, which makes sense, as the different variations of scoring throughout a 60 minute game is much greater than the expected outcomes in the first 15 minute quarter.
This fact, however, makes it quite clear: some squares are at a definitive advantage. Imagine the possible ways that teams can get to X1 in the first quarter. Technically, they could get a touchdown and two safeties. Maybe a touchdown with a 2PT conversion and a field goal. Point is, it's extraordinarily difficult to post an X1 in the first quarter without scoring three touchdowns. In all of this data, (over 3000 games), X1 happened 19 times after the first quarter- a 0.62% chance. In contrast, X1 has happened 293 times over the course of a full game- a 9.57% chance. As a result, any person with a square X1 has a much greater chance at winning the final pot, than they have at winning the first quarter prize.
On the other hand, a team scoring X0 happens 44.9% of the time after the first quarter- an astronomically high number when it comes to games of chance. Those odds drop to 16.3% at the end of the game. Still, any player with X0 gets to ride the whole game with fairly strong odds of winning a quarter at some point of the game. Tallying the numbers, X0 is expected to win 22.87% of the time- that's not to say that if you have an X0 square, you should win that often, but those are your odds if you own all of the X0 squares.
Now not everyone has the same payouts for the game, but a fairly common outcome is the 1:2:1:4 approach. This means that the prize for the halftime score is twice as much as the first and third quarters, while the final outcome wins the grand prize. The payout schedule changes the weights of each quarter and is important when determining the value of each square- but for this practice, we will look at the 1:2:1:4.
[For example, the odds of X7 winning is greater in Q3 and Q4, but are less in Q1 and Q2, which means that if the payout schedule is weighted heavier on second half results, the value of X7 will increase.]
Here's a table of the winning results in a 1:2:1:4 schedule:
| Score | Odds |
| X0 | 22.87% |
| X1 | 7.40% |
| X2 | 2.09% |
| X3 | 15.86% |
| X4 | 13.22% |
| X5 | 2.57% |
| X6 | 7.35% |
| X7 | 20.73% |
| X8 | 4.23% |
| X9 | 3.68% |
As you can see, over the course of a full game, the most valuable squares will be the X0, X7, X3, and X4 squares. X2 and X5 are the two weakest. Surprisingly, to me at least, is the weakness of X9.
Using our expected results for each quarter, and weighting them by the 1:2:1:4 schedule, we can determine the fair value of each square- and if you're savvy, maybe you can get away with offloading some of your junk squares for a square with a higher chance at winning. As we're weighting both teams with equal chances of winning, you can treat these results as reversible (a percentage for X3/X7 can be taken as if either team scores the X3 or X7).
The single square with the greatest chance of winning is the X0-X0 square with a 6.09% chance of winning. Understand that each square, in a random world, would have a 1% chance of winning as there are 100 different combinations. The numbers point to X0-X0 being 6.09X as valuable as the expected return of a random square.
Here's a table of outcomes with greater than 2.5% chance of winning in a 1:2:1:4 schedule:
| Outcome | Chance |
| X0-X0 | 6.09% |
| X0-X7 | 5.05% |
| X7-X7 | 4.44% |
| X0-X3 | 3.77% |
| X3-X7 | 3.36% |
| X0-X4 | 2.74% |
| X4-X7 | 2.64% |
| X3-X3 | 2.56% |
These squares all represent an expected return of 2.5X or greater. These outcomes make sense as they're all multiples of traditional scores. X0-X0 gets the jackpot as scoreless games through one quarter aren't uncommon, while scores of 10, 20, and 30 are very possible. X0-X7 is also a winner as teams are one touchdown away from reaching this multiple. If you get your hands on any of these squares, hold on for dear life.
Here's a table of outcomes with less than 0.15% chance of winning in a 1:2:1:4 schedule:
| Outcome | Chance |
| X2-X2 | 0.06% |
| X2-X5 | 0.07% |
| X5-X5 | 0.10% |
| X2-X9 | 0.10% |
| X2-X8 | 0.12% |
| X5-X9 | 0.12% |
As you can see, X2 and X5 are the weakest squares and it makes sense for them to have automatic payouts. It's quite difficult for a team to hit X2 or X5 without an irregular score (safety, two point conversion, a ton of field goals), so the outlook is quite dim if you get these squares. Of course, both the Ravens and the 49ers are known for their smash mouth defenses, so there could be plenty of field goals. Just note that the odds aren't smiling in your favor.
Squares are a great way to find yourself invested in a game if your team isn't represented. The results should have been fairly unsurprising- multiples of touchdowns and field goals should be expected, while scores that need more unique circumstances happen less frequently. Still, the expected value of each square can help when determining your odds of winning when the game starts- and you can adjust your enthusiasm accordingly.
Below is a table that represents the fair value of a 1:2:1:4 schedule with a $10 square price. You can see the winners and losers and the expected value of each square, based upon the historical results for each quarter.
Best of luck and enjoy the Super Bowl!
| 49ers | |||||||||||
| X0 | X1 | X2 | X3 | X4 | X5 | X6 | X7 | X8 | X9 | ||
| Ravens | X0 | $ 60.88 | $ 14.12 | $ 3.81 | $ 37.67 | $ 27.43 | $ 4.55 | $ 15.29 | $ 50.51 | $ 7.49 | $ 6.91 |
| X1 | $ 14.12 | $ 6.42 | $ 1.88 | $ 11.26 | $ 10.69 | $ 2.34 | $ 5.91 | $ 14.28 | $ 3.89 | $ 3.22 | |
| X2 | $ 3.81 | $ 1.88 | $ 0.58 | $ 3.09 | $ 3.08 | $ 0.74 | $ 1.68 | $ 3.89 | $ 1.22 | $ 0.97 | |
| X3 | $ 37.67 | $ 11.26 | $ 3.09 | $ 25.57 | $ 20.52 | $ 3.73 | $ 11.49 | $ 33.62 | $ 6.16 | $ 5.51 | |
| X4 | $ 27.43 | $ 10.69 | $ 3.08 | $ 20.52 | $ 18.39 | $ 3.82 | $ 10.22 | $ 26.41 | $ 6.29 | $ 5.35 | |
| X5 | $ 4.55 | $ 2.34 | $ 0.74 | $ 3.73 | $ 3.82 | $ 0.95 | $ 2.07 | $ 4.68 | $ 1.56 | $ 1.23 | |
| X6 | $ 15.29 | $ 5.91 | $ 1.68 | $ 11.49 | $ 10.22 | $ 2.07 | $ 5.72 | $ 14.80 | $ 3.40 | $ 2.94 | |
| X7 | $ 50.51 | $ 14.28 | $ 3.89 | $ 33.62 | $ 26.41 | $ 4.68 | $ 14.80 | $ 44.40 | $ 7.71 | $ 6.98 | |
| X8 | $ 7.49 | $ 3.89 | $ 1.22 | $ 6.16 | $ 6.29 | $ 1.56 | $ 3.40 | $ 7.71 | $ 2.57 | $ 2.03 | |
| X9 | $ 6.91 | $ 3.22 | $ 0.97 | $ 5.51 | $ 5.35 | $ 1.23 | $ 2.94 | $ 6.98 | $ 2.03 | $ 1.66 | |
[I sure ain't responsible for people taking this advise. Do this at your own risk.]
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