A Combinatorial Proof for the Football Pool Problem for Six Matches

Consider the setF63of all 6-tuplesx0x1x2x3x4x5withxi∈{0, 1, 2}. It is known that there is a subsetCofF63with 73 elements such that, for anyx∈F63, there is a word inCthat differs fromxin at most one coordinate. We show that there exists such a setCwith a clear structure; this structure is used to give a combinatorial proof of the covering property of the set.