The following notes arose from a talk given by Professor Halberstam to the South West Wales branch of the Association. In the marriage bureau problem we imagine that n boys and n girls are to be paired off in marriage, that we are able to establish preference lists for each boy of all the girls and vice versa. These preference lists may be displayed in a table such as that given below, which shows the rankings for four boys α, β, γ and δ, and for four girls A, B, C, and D. The boys’ rankings are the first members of the number pairs, in rows, and the girls’ are the second members, in columns.