Configuration strategy

While it has been shown that Football Squares is random such that, over the long term, there is no way to adopt a strategy to improve your odds for how many total squares you win, a player can invoke a strategy that will impact the likelihood of various possibilities for any one football game.

For any one football game with four winning squares after Q1-Q3, and after the final score (FS), either you win zero squares, 1 square, 2 squares, 3 squares, or 4 squares. By making choices on the configuration of which squares you choose, how many squares you choose, and even whether you play certain games or not, you can adjust how likely each of those categories turn up. In essence, you can trade a small increase in the chance that you win zero squares for a greater chance that you win 2, 3, or 4 squares (2 or more).

If your strategy is to play the game and minimize the chance that you win no squares, then you should pick your squares on unique rows and columns. With this approach, you'll also have the highest probability of winning exactly one square. The downside, is that you'll have the lowest probability of winning 2, 3, or 4 squares. There is merit to this strategy.

Another strategy that I will focus upon is one where you maximize your chances of winning big by taking 2, 3, or all 4 squares that day even if means increasing the chance that you walk away with no winning squares. You have the best chance to claim total victory over all the other players ... I like this strategy.

The first simulations show a clear winning way to play your squares for this strategy ... pick them all in a column corresponding to the visiting team. Picking them all in a row corresponding to the home team isn't too bad either, but especially to maximize the chances for the rare day of winning all four squares, pick them all in a column.

Now, it's all about the simulations in order to quantify how much of a difference can you actually make (it turns out to be pretty small, so if you just want to fill in a sheet and not worry about any of this, go ahead). Let us consider playing Football Squares where you get choose four squares. Reading left to right and top to bottom, here are the 16 possible configurations of choosing four squares in order of being most likely to win 2 or more squares (remember, four vertical squares represents any four squares in the same column and squares next to one another just have to be anywhere in the same row):


Graphically, and using numbers from multiple simulations, one billion trials, here are the results for picking four squares:

The chart has a horizontal axis corresponding to the order of the players as shown for the different configurations based upon the likelihood of winning two or more squares. The all vertical choice (Player H) listed first and the choice of picking squares in unique rows and columns (Player A) listed last. The blue line represents the probability of winning at least one square with that choice of configuration. This is smallest for Player H at 12.46% and rises to 13.46% for Player A. The darker blue line, representing the strategy to try to win two or more squares is shown to be maximum for Player H at 2.92% while Player A's chances for this are 2.24%. Even more dramatic is the green line, that utilizes the zoomed in right axis, showing the (small) chance for claiming total victory and winning all four squares ... Player H at 0.062% and Player A at 0.014%. So Player H trades a little bit of the blue line (winning at least one square) for a higher probability of winning two or more squares.

Comparing the strategy to be dominant and have the best chance to win at least two squares, Player H to Player A, Player H reduces the chance of winning anything at all by only about 7.4% but then increases the chances of winning two or more squares by 30% along with a 337% increase (more than 3x likely) for the rare chance that all four squares are to be won!

First Simulation Results

What do these numbers mean?  100,000,000 random games of Football Squares were simulated with squares filled out with players A-Z as shown above. The astute reader will recognize that Players A-P correspond to the 16 possible ways one can uniquely choose four squares (see the Post on "Player Selections"), Q-V are the 6 ways one can choose 3 squares, W-Y, 2 squares, and Player Z has only one square.

The first column with numbers around 16,000,000 are the total number of squares that a player wins in all the simulations. The second column with numbers between 1.9 and 2.3 million represent the number of simulations where the player wins two squares in a single simulation. The third column (varies between 272 and 500 thousand) has the number for each player winning three squares, and the fourth column (varies between 14 and 66 thousand) represents the player winning all four squares in the simulation.

So with a single square (Player Z), you have a probability of 2432/100,000,000  or 0.0024% that you pick the right square and the game is such that you win all four squares. Based on this number and the fact that there were 4953 simulated games, I can predict that 12 out of those 4953 games are such that the same ending digit appears in the scores after all Q1-Q3 quarters and the final score. Indeed, I verified that there were 12 such games where in Q2-Q4, both teams scored either 0, 10, or 20 points such that the ending digit never changed.

Key Finding 1: There is no advantage to your overall expectation value (how many squares one wins divided by how many simulations) no matter what you do. Indeed, for a single square, Player Z wins about 4 million times, all the two squares players win 8 million times, 12 million for the 3 square players, and 16 million times for picking four squares. The small differences are consistent within statistical uncertainties (sometimes when you flip a coin, you'll get four heads in a row). Chance of winning is 4% per square as there are four winners on a 100 square grid.

So, it does not matter whether you pick squares all in a row or all in a unique row and column in terms of your overall long term probability of winning squares. There is no strategy that can make you more money over the long term.

Here's a simple example that convinced me that this is true. Imagine playing a smaller squares game with a 2x2 grid and imagine a game played with a first half and a second half where you knew 100% that the visiting team would have the same digit and the home team would have 2 unique digits (the two winning squares would be one on top of the other (denoted by "X"). Let the player pick 2 squares in the three ways that they can uniquely do this. This is taking the non-random factors present in correlated scores between quarters to the extreme. All the possibilities are shown below:

Notice that if the Player picks two side-by-side squares or picks squares on a distinct row and column, they will always win one square no matter what. If the Player picks two squares on top of each other, the player will win two squares 50% of the time and win no squares 50% of the time. In either case, the Player is on average, over a long time, expected to win one square per game.

Key Finding 2: In the simple example above, one can see that the strategy of picking squares will result in a large variation of what's possible and likely to happen in any given game (either you always win one, or you can have a strategy that gives you some chance of winning both squares - though you now have a chance of  not winning any). In the first simulation results, you can observe significant variations in the 2nd, 3rd, and 4th column of numbers representing the number of times of winning two, three, or all four squares. It's this Key Finding that the following posts will study in great detail to develop strategies that can be employed for Football Squares.

Player Selections

A player of Football Squares decides how many squares to obtain and where to position the squares assuming there are plenty of squares in the 10 x 10 grid still available. If the player plays only 1 square, the player should feel free to place it anywhere. Since the rows and column numbers are chosen after the squares, it doesn't matter where one would place a single square.

If the player plays two squares, there are three ways the player can decide to play. Each square on an independent row and column, or both on the same row, or both on the same column.

If the player plays three squares, there are six ways to arrange the squares knowing that interchanging rows and columns would not matter as each have its numbers chosen at random. The six ways include placing a single square randomly and then placing the other two squares as above (3 ways). The other other three ways have all the squares connected by row or column either all vertical, all horizontal, or such that there is a corner square and then one above and one next. The figure below shows the 1, 3, 6 possible ways to choose 1, 2, or 3 squares. Note: squares sitting side-by-side represent any choice where those two squares sit on the same row and squares above-and-below similarly represent any choice where two squares are on the same column. A square that is diagonally offset represents any choice where it has a unique row and column relative to the other squares. The letter above a configuration is just a unique name to be used to refer to that specific configuration.


If the player plays four squares, it turns out there are 16 possible ways to uniquely arrange four squares such that interchanging rows and columns do not matter. Those patterns are shown below.
 

If the player selects five squares, there are 34 possible ways. Those include 16 ways where an additional square with a unique row and column is chosen along with one of the patterns above for four squares. Then there are these 18 additional  patterns.
 

It is an interesting mathematical problem to work out how many different ways could squares be taken where interchanging rows and columns doesn't matter. The series has been worked out. For six squares, there are 90, seven 211, and eight squares can be arranged 558 unique ways. The numbers grow quite fast beyond that (see http://oeis.org/A049311).

The Non-random Factors

Football squares is not entirely random for several reasons. These factors will be studied and will result in a description of a strategy for you to choose your squares.

1) The teams corresponding to the rows and columns are known. In my example, the home team corresponds to the rows and the visiting team corresponds to the columns. It turns out, the home team on average scores about 2.5 more points per game than the visiting team. This implies different numbers are are more or less likely on rows versus columns. Similarly, you might only play differently depending on which team was favored or you might play differently if you knew the expected total number of points that might be scored.

2) Scores in Q1-Q3,FS are correlated. It will be studied whether picking squares that are similarly correlated might have advantages. It's not obvious whether picking 4 squares in a row might actually be preferred to picking 4 squares with a unique row and column. The simulation can address this as it uses the historical data that encompasses these correlations.

3) The player often has the choice of how many squares to acquire. If it's a single square, the game is certainly random as the numbers for the rows and columns are chosen randomly. If a player can choose multiple squares, how one makes the picks given factors 1) and 2) gives rise to a strategy.

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.