New upper bounds for the football pool problem for 6, 7, and 8 matches

Abstract

Consider the set V 3 n of all n-tuples x = ( x 1, …, x n ) with x i ∈ {0, 1, 2}. We are interested in σ n , the minimal size of a subset W of V 3 n , such that for any element x ∈ V 3 n there exists at least one element y ∈ W at a Hamming distance d H (x, y) ⩽ 1. σ n can also be considered as the minimal number of forecasts in a football pool of n matches, such that at least one forecast has at least n − 1 correct results. In this note we present new upper bounds on σ 6, σ 7, and σ 8: 73, 186, and 486, respectively. The bounds have been obtained by an approximation algorithm based on simulated annealing. A closer analysis of the result for the 8-matches problem has led to a simple way to construct a large number of subsets W of V 3 8, each consisting of 486 8-tuples and each having the aforementioned property.