David Singmaster

The Jealous Husbands and "The Missionaries and Cannibals"

The classical river crossing problem of the jealous husbands involves three couples who have to cross a river using a boat that holds just two people. The jealousy of the husbands requires that no wife can be in the presence of another man without her husband being present. This can be accomplished in 11 crossings (i.e. one-way trips). Tartaglia gave a sketchy solution for four couples but Bachet pointed out that this was erroneous and that four couples could not get across the river. In 1879, De Fontenay pointed out that four or more couples could cross the river if there was an island in the river and gave a solution for n couples in 8 n – 8 crossings. Dudeney improved the solution for n = 4 and Ball noted that this gives 6 n – 7 crossings for n couples.

Integral solutions of ass and mule problems

In [1], Tomislav Doslic studies a special case of the classic Ass and Mule Problem and determines when integral data give integral solutions. The question of when integral data gives integral solutions in the general case intrigued me some years ago; I even published the assertion that I could not see any simpler result than computing the answers and seeing if the results were integral [2]. However, I later found a satisfactory result for this and some variants of the problem, which are given in [3, 4]. Here I give this result, which describes an infinite set, and show it leads to a simpler derivation of Doslics interesting result that his variation has only 16 positive integral solutions. A different method permits finding the nonpositive integral solutions. I will then state a similar but unsolved problem.