How would you play in this pool? As σ(3,3) = 5, for a $5 investment you can be guaranteed a 2 nd place prize!Ħ Coding Model We can consider the set of all forecasts as V(n,k), and then we see that σ(n,k) is the size of the smallest code having covering radius 1 (every forecast is within Hamming distance 1 of some codeword.) Since a unit ball contains n(k-1) + 1 elements, the sphere packing bound gives us: n, k k n n k 1 1.ħ Other Lower Bounds In 1970, Rodemich, by carefully analyzing the overlaps of the unit balls, was able to improve the sphere packing bound when n k: n, k kn 1 n 1 when n k. There are 3 3 = 27 possible forecasts on any given Saturday. The remainder of the cash is saved for an end of the season party. The top prize is 50% of the money generated (about $20) and 2 nd prize is 25%. Some of the parents have started a fusball pool. The teams are evenly balanced, any outcome in a game is possible.
Every Saturday morning 3 matches are played. (We see the European influence here, since football in Europe means soccer and ties are much more frequent in soccer games than in American football.) We may increase the generality of the problem and consider k possible outcomes in a game (perhaps thinking of a horse race as a game.) The minimum number of forecasts for n matches with k outcomes per match, so that the actual outcome differs by no more than one match from some forecast of this set will be denoted by: σ(n,k)ĥ Example The Louisville senior boys soccer league has 6 teams. Generally, there are no prizes after the fourth.ģ Football Pools As the only way to guarantee a 1 st place is to bet on all possible forecasts, a more interesting question is: What is the minimum number of bets needed to guarantee a 2 nd place prize?Ĥ Football Pools In the traditional literature, emphasis has been placed on games where there are 3 outcomes home team wins, away team wins or a tie. Third and fourth place prizes also exist. The second place prize goes to those who get only one game wrong (again, split amongst multiple winners). The first place prize goes to anyone whose forecast matches the actual outcomes of the games (and is split amongst multiple winners).
In a football pool, each player wagers on a forecast.
1 Math Meets the Bookies: or How to win Football PoolsĢ Football Pools Given n games (matches) to be played by 2n teams, a forecast is an n-tuple consisting of a prediction for each of the n matches.