# A more equitable statement of the jealous husbands puzzle

Every time I use the jealous husbands river crossing problem, I prefix it with a waffly apology about its formulation. You’ll see what I mean; here’s a standard statement of the puzzle:

Three married couples want to cross a river in a boat that is capable of holding only two people at a time, with the constraint that no woman can be in the presence of another man unless her (jealous) husband is also present. How should they cross the river with the least amount of rowing?

I’m planning to use this again next week. It’s a nice puzzle, good for exercises in problem-solving, particularly for Pólya’s “introduce suitable notation”. I wondered if there could be a better way to formulate the puzzle – one that isn’t so poorly stated in terms of gender equality and sexuality.

There’s a related, but not identical, problem – but this doesn’t help as it has its own, different issues. Here’s the version of the missionaries and cannibals problem given by Wikipedia:

Three missionaries and three cannibals must cross a river using a boat which can carry at most two people, under the constraint that, for both banks, if there are missionaries present on the bank, they cannot be outnumbered by cannibals (if they were, the cannibals would eat the missionaries). The boat cannot cross the river by itself with no people on board.

Wikipedia says the jealous husbands problem is older, dating back in some forms in Europe to the 800s, with the ‘husbands and wives’ formation coming between the 13th and 15th centuries.

Anyway, absent of a clever revelation I asked Twitter. There are minor spoilers below, so you might want to have a go at the puzzle first if you haven’t seen it before.

First, Christian Lawson-Perfect suggested simply to replace each wife with a heavy, inanimate object that belongs to one person and is coveted by the others. The object must be heavy, or at least bulky, in order that the boat can only hold one person and one object on each journey. I pointed out that whatever those coveting the object want to do with it must be done during a boat ride. In the classic formulation, I suppose each husband fears his wife would be charmed during time alone with another man. Christian suggested unlocked suitcases and Colin Beveridge suggested that these could contain top-secret information. Matthew Arbo pointed out what I had missed: at some point in the solution, we’d require one of these suitcases to row the boat.

Christian suggested replacing the wives with people who know TV spoilers. It’s a nice thought, but I think this would be very complicated to state because of the pairing of characters in the puzzle. We’d need each person who knew spoilers to know different spoilers and be paired with one of those who don’t know spoilers known by the others.

Ian Preston suggested a formulation that I wrote up like this:

Three children, each accompanied by one of their parents, each want to cross a river in a boat that is capable of holding only two people at a time. Children behave very well with each other and with their parent, but misbehave in the presence of other adults when their parent is not present. Everyone must therefore cross the river with the constraint that no child can be in the presence of an adult who is not their parent unless their parent is also present. How should they cross the river with the least amount of rowing?

This is longer than the classic statement and more convoluted. The requirement that children behave together is necessary so that we don’t think they need to stay with the parent at all times, but it’s a big hint that at some point some children are going to be left alone. Even so, there is a further problem. James Grime was confused about whether the children could row the boat, suggesting I replace children and their parents with dogs and their owners. Since at some point we require children to row the boat, perhaps I should say they can do this in the statement – yet another hint.

James Grime also suggested prison wardens and prisoners on a boat to Alcatraz. This is a creative idea, but at some point in the solution we have all the guards at Alcatraz and the prisoners, with the boat, on the shore at San Francisco. Plus, I think this is closer to the missionaries and cannibals than the jealous husbands because of the lack of pairing.

Alison Kiddle suggested a formulation in which we have three mods and three rockers, with each mod having a rocker sibling. People tolerate their own clique or their own sibling, and in a mixed group they won’t kick off if their sibling is present. I think this is a good statement of the problem and I like it quite a lot, though the cultural reference might need updating and its a bit more complicated to explain what will happen if the two groups are allowed to mix.

out of the norm said he’d heard it with Harry, Ron and Hermione with three ogres, or three nuns and three ogres, since overpowering is equivalent to jealousy. Karen Hancock suggested the allergies puzzle at the bottom of this list of interesting river-crossing problems. Nice statements, but I don’t think either is equivalent to the jealous husbands.

Then we came to the suggestion I think I am happiest with. James Sumner made a suggestion that I’ve written up as the following:

Three actors and their three agents want to cross a river in a boat that is capable of holding only two people at a time, with the constraint that no actor can be in the presence of another agent unless their own agent is also present, because each agent is worried their rivals will poach their client. How should they cross the river with the least amount of rowing?

This maintains the jealousy, so is hopefully easy to understand and should minimise the need for additional explanation. As James pointed out, we might wonder why on earth they’re crossing a river in a boat made for two, but I think that’s a minor quibble.

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### 11 Responses to “A more equitable statement of the jealous husbands puzzle”

1. Ian Preston

It’s not so much that the children have to behave in the absence of adults as that it doesn’t matter for what you are trying to prevent. You could think that the aim is to avoid annoyance to adults or that the problem is that parents are mortified by having their children show them up while in the care of other parents. It’s just like the original jealous husbands being unconcerned by the possibility of any impropriety the wives might get up to among themselves.

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• Peter Rowlett

Thanks. I see that, but I think if you are setting up that children are likely to misbehave, the reader might think that leaving the children on their would lead to non-completion of the goal. I think the husbands extreme jealousy wrt other men but complete lack of concern about the wives among themselves is one the niggles I have with the puzzle as originally stated. I like the actors/agents version because I feel it avoids some of these extra explanation issues.

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• Ian Preston

Maybe this is obvious but if it were just the sexism that were problematic then you could just relabel the original version as being about jealous spouses, couldn’t you? Have both husbands and wives worried about leaving their spouse with anyone else of the other sex and the gender equality issues would disappear, albeit that there’d still be an implicit stipulation of exclusive heterosexuality. The key features of the problem (besides all participants being able to row a boat) seem to be the existence of pairings, the danger of mispairing and the division into types who can’t interact inappropriately among themselves. The nice thing about the performers and agents interpretation is that those are all conveyed naturally.

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• Peter Rowlett

The nice thing about the performers and agents interpretation is that those are all conveyed naturally.

Nicely put, I agree. On Twitter, someone had suggested making everyone jealous, but that isn’t ideal.

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2. Zeno Rogue

What about: Josephine’s problem, Hall’s marriage theorem, the marriage problem from probability? There is a form of gender inequality in all the statements.

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• Peter Rowlett

Sure, there are other examples. On the sheet I’m using next week I have a puzzle with a woman who bakes cakes and another with a cunning trickster who is a man, whose genders I arbitrarily swapped for my version, and a population control problem where parents prefer male to female children that we decided to leave as it is. None of this is ideal, but something about the jealous husbands particularly stood out (from the collection I plan to use — which doesn’t include the ones you named).

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3. Andy

I use Zombies and soldiers with my students. They are crossing a river to get to a hospital that can cure zombification.

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4. Jesse McKeown

How ’bout solicitors and witnesses? (I would also point out that the “jealous” husbands need not be suspcious of their wives’ fidelity, but only of other husbands’ propensity to bore)

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