How to Get Rich off Finnish Football Fans

by Andrew Pimlott '97

The following uses material from the previous article on error correcting codes

How can a perfect, ternary code provide a good way of winning a football pool? Suppose 11 games are being played and you participate in a pool in which you must bet on the results of all 11 matches (each can be a win by either team, or a tie). You wish to place the minimun number of bets such that, no matter what the results of the matches, one of your bets picks thecorrect outcomes in at least 9 of them. This is exactly what the (11, 3^6, 5) ternary Golay code provides! Since it corrects two errors and is perfect, every vector (here, interpreted as a possible outcome of the 11 games) is at a Hamming distance of no more than 2 from some codeword. Therefore, if you bet the various combinations of wins, losses and ties indicated by the 3^6 = 729 codewords, one of your bets will have only 2 (or fewer) matches wrong.

Here's the code itself. The three possible outcomes will be represented by the symbols (-1, 0, 1). The 729 bets you should place are represented by the following matrix ('-' represents -1):

To obtain the codewords, take all sums of the rows, each multiplied by either -1, 0, or 1. Since each of the six rows gets one of the three operators, there are 3^6 = 729 possible combinations, as expected. The addition should be done coordinate-wise (do not carry from one column to the next), with 1 + 1 = -1 and -1 + -1 = 1. Now, interpret the numbers as bets, call up your cousin Olaf in Finland, and get ready for the markka to come rolling in!

Of course, this strategy does not circumvent the laws of chance. If the outcomes of the matches are completely random, the chance of correctly picking 9 or more matches with one bet is

Therefore, anyone who places 729 random bets on several occasions will, on average, get 9 matches correct once on each occasion. The beauty of the Golay system is that it guarentees this happening exactly once each time.

When is this advantageous? Generally, in a pool covering 11 games, it is not to your benefit. As was just shown above, if the outcomes are determined by chance, you will come out even every time actually, you will lose the percentage taken by the organizers of the pool and the long distance charges to Finland). In the more realistic situation in which many matches appear likely to go one way or the other, you will lose money by placing bets on improbable combinations of upsets (unless you are very lucky and the upsets occur). However, the system can be used to your advantage if you are betting on more than 11 games, and are fairly confident about all but 11 of them. Then, simply place 729 bets, all of them containing the sure things with the remaining matches filled in by the Golay code. You thus assure that you will not lose much on the 11 questionable games, allowing you to profit in the matches you are sure of.

There is much room for further exploration and optimization of the Golay code as a betting strategy. For example, if you are sure of the outcome of one of the 11 games, should you leave out the codewords which have different outcomes for this game? Can you fine tune the codewords by carefully picking what each symbol (-1, 0, 1) stands for in each match, so that they tend to be closer to the outcomes that you think most likely? Have fun with it, and perhaps you may realize the hope of the man who publis hed it: "To enrich the possibilities of the players, and perhaps the players themselves."[1]

Please direct inquiries to pimlott@math.harvard.edu. (dead link)

Appeared in volume 1 in 1994 Tangents: Harvard-Radcliff Math Bulletin.

posted 5-13-2004 wdj