Tuesday, October 15, 2013

The Numbers

This study is meant to address questions of strategy and choices that I have not found elsewhere. Many studies of the mathematics of football squares concern themselves with determining what are the best numbers to have. As part of the game, this information is a little fun because your hopes for a positive outcome can rest upon which numbers you end up with. However, as the rows and columns are chosen at random, there is nothing you can really do with the information. One exception is a variation of this game where the rows and columns are chosen before the game and players bid on what numbers to take.

I do tabulate, over my data sample, the frequency of winning squares in Q1-Q4,FS. This information is provided but is not the subject of actual strategy that one can employ. The numbers that I find agree with other such tabulations that can be found on sites where Football Squares is analyzed.

The tables below, show grids with the frequency at which squares were winning squares in the quarter shown. For example, about 20% of the time, the score after Q1 ended in 0-0 as final digits. This includes scores of 0-0, 10-0, 0-10, 10-10 etc. Scores ending in 7 and 3 are also likely as expected. In the later quarters, as more scoring takes place, there is a larger variation as to which numbers are likely to come up.


The table below shows a grid and the frequency of winning in any of the Q1-Q3,FS. The numbers in the table could sum to 400% as the variation of the game under consideration has four winning squares after Q1-Q3, and the final score; however, I normalize this table to 100% for easier relative comparisons. It is also noted that this tabulation does not take into account correlations between the scores in the different quarters. This correlation is taken into account by the simulation.

It is not uniquely defined how one would value individual numbers as you hold separate squares corresponding to a number for a row and for a column. My tabulation of the relative value of the numbers is shown below showing 0 and 7 are clearly the best numbers, 3 and 4 are good, 6 and 1 are OK, and 8, 9, 2, and 5 are not so good relatively speaking (unless that day, the final score is 29-25). This tabulation treats 0-0 (and other double-digit possibilities) twice just as it treats 7-0 on equal footing as 0-7. In fact, the mistake of not doing this would lead some to believe 7 is a better number than 0. With the table below, you can see how valuable are your numbers for each of the quarters, the final score, and overall if winning squares are awarded Q1-Q3,FS.

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