Four puzzles about getting everyone to the far bank. Three of them are the oldest we have: problems 17, 18 and 19 of the Propositiones ad Acuendos Juvenes — Problems to Sharpen the Young — attributed to Alcuin of York, who died in 804. The fourth, missionaries and cannibals, is the modern descendant of the same idea.
Getting everyone across is not the hard part. Doing it in the fewest crossings is, and knowing when you have stopped doing it is harder still.
- 7 Wolf, goat and cabbage — crossings, and exactly two solutions
- 11 Missionaries and cannibals — crossings, 8,100 shortest routes
- 11 Brothers and sisters — same length, only 486 routes
- 9 Man, woman, two children — crossings, eight ways to do it
How do you solve the wolf, goat and cabbage puzzle?
Seven crossings, and there are exactly two ways to do it. A farmer with a wolf, a goat and a cabbage, a boat that holds him and one other thing, and two rules: the wolf eats the goat, the goat eats the cabbage, and neither happens while the farmer is watching.
You will also see it called the wolf, sheep and cabbage puzzle. Same puzzle, same seven crossings — Alcuin's Latin says capra, a goat, and the sheep came in with later English retellings.
- Take the goat across — it is the only one that cannot be left with either of the others.
- Come back empty — the wolf and the cabbage are perfectly safe together.
- Take the wolf across — or the cabbage; this is the only real choice in the whole puzzle.
- Bring the goat back — the step everybody refuses, and the one the puzzle is built on.
- Take the cabbage across — leaving the goat alone on the near bank.
- Come back empty — the wolf and the cabbage are together again, which is fine.
- Fetch the goat — and all four are across in seven crossings.
Why do you have to take the goat back?
The goat is the only one of the three that conflicts with both of the others. So it goes over first, alone with nothing to eat. Whatever you carry second has to be left with something on the far bank — and it cannot be left with the goat.
So the goat comes back with you. That single reversal is what people are missing when they decide the puzzle is impossible.
Why are there only two solutions?
Because there is almost nothing to decide. The picture above is every position the puzzle can reach, and it gives the answer directly: no choice at all for the first two crossings, and none for the last two.
The only decision in the entire puzzle is at step three, where the route splits — and the two branches are mirror images, so it does not even matter which you take.
How do you solve missionaries and cannibals?
Eleven crossings, and unlike the wolf and goat there are 8,100 ways to do it. You do not need to find the one right route; you need to avoid a fairly small number of wrong ones.
Three missionaries, three cannibals, a boat for two, and one rule: on neither bank may the cannibals outnumber the missionaries — unless there are no missionaries there at all, in which case nobody is at risk.
What is the jealous husbands problem?
It is Alcuin's problem 17, and it is usually retold wrongly.
The familiar version is three couples, with no wife left in the company of another man unless her husband is there. Alcuin wrote brothers and sisters: three men each travelling with his sister, and no woman may be with a man unless her brother is present. Same mathematics, a different century's anxieties.
Which is harder, the missionaries or the siblings?
Alcuin's, by a distance — even though both need eleven crossings with the same boat and the same six people.
His rule cares about which brother is present. The missionary rule only counts heads. So 44 of the 128 arrangements are legal in his version against 68 in the modern one, and there are 486 shortest routes against 8,100.
Is the man, woman and two children puzzle really a puzzle?
Problem 19, and the odd one out. Nothing here eats anything and nobody is jealous. A man and a woman each weigh as much as a full cart; their two children weigh half that each; and the boat carries one cartload.
So the two children can cross together, or one adult can cross alone. That is the entire constraint, and it takes nine crossings.
It is worth playing precisely because it feels so unlike the other three. There is nothing to be careful about — only something to be efficient about, which makes it the most modern-sounding of them by about twelve hundred years.
How many crossings does each puzzle need?
| Boat | Fewest crossings | Legal positions | Shortest routes | |
|---|---|---|---|---|
| Wolf, goat, cabbage | 1 passenger | 7 | 20 | 2 |
| Brothers and sisters | 2 people | 11 | 44 | 486 |
| Missionaries and cannibals | 2 people | 11 | 68 | 8,100 |
| Man, woman, two children | 1 cartload | 9 | 32 | 8 |
Every one of those minima is proven rather than observed. The distance from each reachable position to the goal was computed in advance, for all four puzzles — which is what makes the light on the board trustworthy.
How do you play these on this page?
- Tap the figures — they step into the boat. Somebody who can row has to be among them.
- Send the boat across — the button stays disabled if the load will not float.
- Watch the light — green means every crossing so far is on a shortest route; amber means you have taken a detour, and says how many crossings you now need.
Each sibling pair shares a colour, because that pairing is the rule of Alcuin's puzzle and no icon can carry it. Best move is genuinely the best move rather than a plausible one, for the same reason the light is honest: every reachable position's distance to the goal is known before you start.
Who invented river crossing puzzles?
Nobody knows, and Alcuin is the earliest name attached to them. The Propositiones is a collection of fifty-three problems and the earliest surviving book of what we would now call recreational mathematics. Some of them are arithmetic, some are jokes, and three are about a river.
Whether Alcuin wrote them or gathered them from older sources is not settled — he was Charlemagne's schoolmaster, and a collector as much as an author. What is certain is that people have been refusing to take the goat back for at least twelve hundred years.
If you like this one
Towers of Hanoi is the other puzzle here where the minimum number of moves is a fact rather than a target, and Nim hides a rule that decides the game before it starts. The rest are in math and logic.
