Scoring Possibilities and the Squares

The same 26 possible configurations that have been described for a player to choose 1, 2, 3, or 4 squares are relevant for the scoring possibilities - how the winning squares themselves are distributed. We study these possibilities to gain insight as to why selecting squares in a column preferentially results in a larger likelihood of winning two or more squares. In fact, we find the scoring possibilities to even be more skewed if the column direction represents not the visiting team but the underdog and that these distribution also depends on how many total points are expected to be scored. As both the line and over/under are often known before a player picks squares, that information can be used as part of the selection strategy as will be described in a following post.

Recall all 26 possible configurations for a player to select up to four squares, where squares next to each other represent any two on a row, squares above and below represent and two on column, and squares diagonally separate have a unique row and column. These possible configurations also represent all possibilities of how a football game can have scoring in the first three quarters and the final score, Q1-Q3,FS. Configurations with less than four squares represent games that have single squares being winning squares multiple times on the grid.

The configuration labeled A represents a game where in each quarter, both teams have a unique final digit. The Home/Favorite team scoring 10, 7, 7, 7 in each quarter has Q1-Q3,FS final digits of (0, 7, 4, 1) and the Visiting/Underdog team scoring 0, 14, 7, 7 has Q1-Q3,FS final digits of (0, 4, 1, 8) is an example of such a game. The configuration labeled Z, a single square, represents a game where the final digits of the game never change. An example would be the Home team with quarter by quarter scores of 7, 10, 0, 10 with Q1-Q3,FS final digits of (7, 7, 7, 7) and the Visitors with scores 3, 0, 10, 0 and final digits of (3, 3, 3, 3). The other configurations represent all possibilities and we can assign each of the 4953 games from 1994-2012 and tabulate the configurations are the most likely to turn up. The order of the configurations displayed above is the order from most likely to least likely when the column represents, not the visiting team as usual, but the underdog ... the team least likely to win according to the line.

The order of the first row, configurations V-D, remains the same whether the column represents the underdog or the visiting team. For the column being the underdog, configurations V, F, S, R occur 12.4%, 11.5%, 8.4%, 6.7% of the 4953 games. Configurations J thru W occur from 5.7% to 3.0%, N thru L occur from 2.6% to 1.0%, and G and Z are both very rare at 0.28% and 0.24% respectively. Studying the order of the configurations, a simple pattern emerges. In all cases, when a pair of patterns look as a 45 degree mirror reflection, the one along the column direction is favored. S appears before R, J before I, E before D, C before B, P before O, U well before T, Y before X, M before L, and finally H before G.

Besides being the most likely configuration, pattern V has a unique distinction in that it looks symmetrical, but as the four winning squares Q1-Q3,FS are distributed among the three squares in the pattern, V can remain symmetrical with respect to rows and columns only if the "corner square" appears as the winning square twice. Otherwise, pattern V is skewed either along the column or row depending on which of the "outer" squares represents two winning squares.

Besides pattern V being symmetrical only if the corner square represents two winning quarters, the other other patterns that are symmetrical are F, Q, W, A, K, and Z. All other patterns are either skewed along the underdog column or the favorite row. We can tabulate the fractions of games that are symmetrical or skewed along columns or rows. We find the following:

The table has six columns. The first two represent the number of games (out of 4953) and the fraction for having a football game with scoring skewed along the column (42.01% if the column represents the underdog), 29.62% along a row. For 28.37% of the games, the scoring is symmetrical about the row and column. This is why choosing your picks along a column gives you a higher chance of winning two or more squares. The next pair of numbers are the games and fraction if the column represents the visiting team which is, more often than not - but not always, also the underdog. This suggests adopting a strategy that if you want to take advantage of the correlated scores (like winning two or more squares), then you should pick along the column being the underdog (that is, if the Squares sheet happens to be arranged with the underdog along the rows, turn the sheet by 90 degrees and always pick along the column corresponding to the underdog). The final column is like the first, the column represents the underdog, but in this case, we evaluate the 2037 games where the over/under is less than or equal to 40 points. Note a slight increase in the fraction of games (now 43.40%) skewed towards a configuration that represents a column.

The other observation from examining the order of the 26 rankings is that configurations with three winning squares happen quite frequently. That is, there are single squares that become two winning squares by the end of the game. Patterns Y and W, both being realized more than 3% of the games, with two squares, represent games where either the two squares that are winners have each winning twice or one square winning in the 3 times throughout Q1-Q3,FS and the other square once. Note that pattern H - a game where the underdog's final digit doesn't change (either 0's or 10's typically scored in Q2-Q4) and the favorite's score has four unique ending digits - is a rare pattern for a game to have such scoring. Yet, if you are picking four squares, pattern H is the pattern you want to maximize your chance to win two or more squares and, as was shown, is the best pattern to select to maximize the rare chance you win all four squares. It's unlikely that the game is like pattern H and your pattern H square selections exactly match all four winning squares. It's more likely that you win four squares using pattern H from a game with pattern Y (two in a vertical column) and by having four squares in a column, you are sometimes lucky and cover those two coveted sqaures. 

This post was rather lengthy but it goes into detail as to how the ending digits of professional football games end up in a Football Squares game as winning squares and shows how you can adopt a strategy in selecting your squares if you choose to benefit from the correlations that exist.

Number of Picks Strategy

How many squares should be selected? In many games, there are fewer than 100 people involved and so players are allowed to select multiple squares. Most games are meant to emphasize the fun and so, out of courtesy and etiquette, it is typical for a player to not pick more than four or five squares.

However, if the strategic goal is to maximize the chances for winning two or more squares, it has been determined that your squares should all be in the same column corresponding to the visitor's team. I have studied this particular strategy and have found that you should pick as many squares as possible in that column (all 10 if you can). If you can pick even more squares, start filling them up in another column too. Not surprisingly, if you want a higher chance of winning, pick more squares. What's not obvious is that the strategy of picking squares in the same column holds for these additional squares.

Now, let's consider picking 1-10 squares in a column. In tabulating the results, let us normalize to picking 10 squares. That is, if you pick one square, multiply the results by 10. For two squares, multiply by five. Multiply by fractions for most of the other numbers so that we can compare the strategic differences on the same footing. Here's a figure:

The figure shows that taking the optimal strategy to win 2 or more squares by selecting squares in a column, you still have to choose how many squares to select. Over the long term, there is no difference in what you do, for every 10 squares selected, you have a 10% chance of winning in any one Q1-Q3,final score. However, for any one game, you can trade off a have a lower probability for winning at least one square (this blue line goes up on the graph as you pick fewer and fewer squares) for a larger probability of winning two or more squares (red line that is highest for picking all 10 squares in a column). I also show the probability (axis on the right of the graph) of winning all four squares (very rare) that shows a very large advantage for picking your squares 10 in a row. In terms of numbers and comparing relative to picking 10 single square games, picking 10 squares in a single column will see your likelihood of winning anything at all drop by 26.4%, but your likelihood for winning two or more squares will increase by 59.3% with your chance of winning all four squares increasing by 402% - over 4x more likely!

I was a little surprised to notice that the lines in the figure are smooth and, in particular, there is no jump at selecting four squares. After all, in any one game, there are four periods to determine winning squares. Of the 4953 games under study, 12 have four winning squares in a single square, 155 in a vertical column of two squares, 154 in a vertical column of 3 squares, and only 40 in a vertical column of four squares. Most of the time that you win all four squares comes not from four in a column and four distinct rows, but having enough selected squares in a column such when a game has winners in two or three in a column, you happen to have the rows for those squares covered.

One interesting configuration is what to do if you were going to pick 13 squares. I consider four cases.

1. 10 in one column and 3 in another column
2. two columns of 9 and 4
3. two columns of 7 and 6
4. an 'L' shaped configuration with 10 in one column and the other 3 in a row 

The last choice is such that one row has a total of 4 squares including the extra square from the 10 column. Here are the results for simulating 100 million trials of these three configurations showing that you have the best chance for winning two or more squares by 10 in one column and 3 in another column. Case 1 wins two or more squares 13.30% of the time, case 2 wins two or more 13.11%, case 3 wins two or more 12.95%, and case 4 wins two or more squares 13.29% (just slightly lower than case 1). This study suggests that you do better in filling up one column and then, if allowed, start filling up a second column. I do not study all possibilities up the trivial case where a single player selects all 100 squares and thus wins everything. However, the general study is made of the strategy of selecting along a single column in order to maximize the chance that you win two or more squares.

Finally, if you choose a strategy that wants to minimize the chance of not winning anything, then the strategy is to pick squares in unique rows and columns. What's not obvious is whether you'd be better off picking four such squares in a game OR a single square in four separate games. This has been studied and it is found that you are better picking the four squares in a single game. The difference is small: 13.4% to 13.1% (2.3% difference) although, as expected, the choice of picking a single square in four separate games has a 8.3% larger chance of winning more than two squares. In fact, by entering more four games with a single square, you even have a very very small chance of about 1 chance in 20,000 of winning 5 or more squares between your entries.