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The wolf, the goat and the cabbage

Its strength lies in a minimum restriction: the boat can cross many times, but each shore must always be safe. Solving it is not about moving pieces, but about keeping an eye on which pairs are left alone.

A farmer must cross a river with a wolf, a goat and a cabbage. The boat can only carry the farmer and one of the other three.

If you leave the wolf alone with the goat, the wolf eats it. If you leave the goat alone with the cabbage, the goat will eat it.

How can everyone cross without either of those things happening?

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Hints

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The problem is not only what you carry, but what you leave behind on each shore.
The goat is the critical element: it cannot be left alone with either the wolf or the cabbage.
That's why the goat will have to cross more times than the other two.

Solution

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Answer: a minimum valid sequence is: 1. The farmer takes the goat to the other side.

The farmer returns alone.
The farmer takes the wolf to the other side.
The farmer returns with the goat.
The farmer carries the cabbage to the other side.
The farmer returns alone.
The farmer takes the goat to the other side again. It also works by exchanging the order of the wolf and the cabbage: take the cabbage first, bring back the goat, then take the wolf and finish by carrying the goat. What doesn't change is the structure: the goat must cross first, come back once, and cross again at the end. This way they are never alone, wolf-goat or goat-cabbage. In the end, wolf, goat and cabbage are on the other side.

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