GeekDad Puzzle of the Week Solution: Football Squares

Geek Culture

Congratulations to the region that hosts the sports team that won the recent televised sporting event! Here is last week’s puzzle:

Watching this evening’s televised sporting event (which will remain nameless, so as to save me visits from intellectual property lawyers), I remembered that I took part in the curious office practice of wagering on the score of the game at the end of each quarter. Specifically, the bet is called “Football Squares,” and it all starts with a 10×10 grid. Gamblers purchase each of the 100 squares in the grid, and after all the squares are sold, the numbers from 0-9 are randomly assigned to positions in each column and row. As the football game progresses, if the last digits of each teams’ scores intersect on that square at the end of a quarter, you win the amount allocated for that quarter. A sample grid is shown below:


In this grid, the owner of the green square wins the pot for any quarter that the red team’s score ends in 4 (4, 14, 24, etc.) and the blue team’s score ends in 7 (7, 17, 27, etc.)
It is important to note that the numbers for each row/column are randomly selected *after* all of the squares are purchased — there is no such thing as a “good square” or a “bad square” when they are selected. After the numbers are assigned, however, people invariably gloat or complain about their chances of winning money at the end of a quarter (e.g., “5-5? Worst. Square. EVER!”).

But is there such a thing as a “good” or “bad” square? And if so, what are they? Answering that question will get you a chance at this week’s prize, and perhaps increase your winnings in next year’s Football Squares event.

A few assumptions/explanation of football scoring:

  • There are four quarters in a football game, and the last digit/ones digit of each team’s score at the end of each quarter will determine which square wins.
  • For the sake of this puzzle, each team has the same odds/performance, so assigning one team to a column or row shouldn’t matter.
  • For the sake of this puzzle, each team has four “possessions” during each quarter of the game.
  • During any given possession, there are a variety of different types of things that can happen.
  • There is a 40% chance that nothing interesting will happen (no points scored) during your team’s possession.
  • There is a 33% chance that your team will score a touchdown for 6 pts. Of these, some 85% are successful in getting an extra point (+1 pt), and 3% make a two point conversion (+2 pts.)
  • There is a 25% chance that your team will score a field goal for 3 pts.
  • There is a 2% chance that your team will give up a “safety,” giving the other team 2 pts. If this happens, this counts only as your possession and not theirs, and they get the ball next.

Each team has the same performance/odds for each of their 4 possessions, and they maintain this performance for each of the four quarters of the game.

There were several approaches that people took to get answers: some people took more of a stochastic approach, calculating probabilities and combinations, while other people used a Monte Carlo approach and simulated the game 100,000 to 500,000 times. One thing on which they all agreed? That while not a good square by any means, (5-5) is not the worst square possible. That “honor” goes to (8-8), and (5-8) and (8-5) are even worse than (5-5.)

The best square was universally determined to be (0-0), with (3-0), (0-3) and (3-3) close behind. Congratulations if you draw those squares in your office pool!

Also, many thanks and congratulations to Andy Arizpe, who presented one of many correct solutions. Andy will soon be the proud owner of a $50 ThinkGeek gift certificate. Everyone else that submitted a solution, or has simply scrolled down this far in the post, can save $10 off a $50 order at ThinkGeek by using the coupon code GEEKDAD21FK.

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