Wolf, goat and cabbage problem


The wolf, goat and cabbage problem is a river crossing puzzle. It dates back to at least the 9th century, and has entered the folklore of a number of ethnic groups.

The story

Once upon a time a farmer went to a market and purchased a wolf, a goat, and a cabbage. On his way home, the farmer came to the bank of a river and rented a boat. But crossing the river by boat, the farmer could carry only himself and a single one of his purchases: the wolf, the goat, or the cabbage.
If left unattended together, the wolf would eat the goat, or the goat would eat the cabbage.
The farmer's challenge was to carry himself and his purchases to the far bank of the river, leaving each purchase intact. How did he do it?

Solution

The first step that must be taken is to let the goat go across the river, as any other actions will result in the goat or the cabbage being eaten. When the farmer returns to the original side, he has the choice of taking either the wolf or the cabbage across next. If he takes the wolf across, he would have to return to get the cabbage, resulting in the wolf eating the goat. If he takes the cabbage across second, he will need to return to get the wolf, resulting in the cabbage being eaten by the goat. The dilemma is solved by taking the wolf over and bringing the goat back. Now he can take the cabbage over, and finally return to fetch the goat.
His actions in the solution are summarized in the following steps:
  1. Take the goat over
  2. Return
  3. Take the wolf or cabbage over
  4. Return with the goat
  5. Take the cabbage or wolf over
  6. Return
  7. Take goat over
Thus there are seven crossings, four forward and three back.

Occurrence and variations

The puzzle is one of a number of river crossing puzzles, where the object is to move a set of items across a river subject to various restrictions.
In the earliest known occurrence of this problem, in the medieval manuscript Propositiones ad Acuendos Juvenes, the three objects are a wolf, a goat, and a cabbage, but other cosmetic variations of the puzzle also exist, such as: wolf, sheep, and cabbage;, p. 26 fox, chicken, and grain; fox, goose and corn; and panther, pig, and porridge. The logic of the puzzle, in which there are three objects, A, B, and C, such that neither A and B nor B and C can be left together, remains the same.
The puzzle has been found in the folklore of African-Americans, Cameroon, the Cape Verde Islands, Denmark, Ethiopia, Ghana, Italy, Romania, Russia, Scotland, the Sudan, Uganda, Zambia, and Zimbabwe., pp. 26-27; It has been given the index number H506.3 in Stith Thompson's motif index of folk literature, and is ATU 1579 in the Aarne–Thompson classification system.
The puzzle was a favorite of Lewis Carroll, and has been reprinted in various collections of recreational mathematics., p. 26.
In his 'Arabian Nights' memoir, Meetings with Remarkable Men, the metaphysical Magus, G. I. Gurdjieff cites this riddle as "The Wolf, the goat and the cabbage". He notes, "This popular riddle clearly shows that...not solely by means of the ingenuity which every normal man should have, but that in addition he must not be lazy nor spare his strength, but must cross the river extra times for the attainment of his aim."
Variations of the puzzle also appear in the adventure game , the Nintendo DS puzzle game Professor Layton and the Curious Village, and in The Simpsons episode "Gone Maggie Gone", where Homer has to get across a river with Maggie, Santa's Little Helper, and a jar of rat poison that looks like candy. In the Class of 3000 episode "Westley Side Story", Sunny and his students perform a similar exercise involving a chicken, a coyote and a sack of corn. The Between the Lions episode "Farmer Ken's Puzzle" portrays it being made into a computer game with a cat, a hen, and a sack of seeds.
In the Bull episode "Justice for Cable", Benny begins a riddle with "a man has a fox, a duck, and a bag of beans". Bull inexplicably declares "There is no answer", and everyone believes him.
In some parts of Africa, variations on the puzzle have been found in which the boat can carry two objects instead of only one. When the puzzle is weakened in this way it is possible to introduce the extra constraint that no two items, including A and C, can be left together., p. 27.