A new upper bound for the football pool problem for nine matches

Abstract

Consider the set V 3 n of all n-tuples x = ( x 1, …, x n ) with x i ϵ {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 ϵ V 3 n there exists at least one element y ϵ W at a Hamming distance d H ( x, y)⩽ 1, σ n can also be considered as the minimum 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 the new upper bound σ 9 ⩽ 53 · 3 3. The bound has been obtained by an approximation algorithm which performs, alternatingly, phases of iterative improvement and randomized backsteps.