Definitions

# Fox, goose and bag of beans puzzle

The fox, goose and bag of beans puzzle 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., p. 26

## The story

Once upon a time a farmer went to market and purchased a fox, a goose, and a bag of beans. On his way home, the farmer came to the bank of a river and hired a boat. But in crossing the river by boat, the farmer could carry only himself and a single one of his purchases - the fox, the goose, or the bag of the beans.

If left alone, the fox would eat the goose, and the goose would eat the beans.

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 must be to bring the goose across the river, as any other will result in the goose or the beans being eaten. When the farmer returns to the original side, he has the choice of bringing either the fox or the beans across. If he brings the fox across, he must then return to bring the beans over, resulting in the fox eating the goose. If he brings the beans across, he will need to return to get the fox, resulting in the beans being eaten. Here he has a dilemma, solved by bringing the fox (or the beans) over and bringing the goose back. Now he can bring the beans (or the fox) over, leaving the goose, and finally return to fetch the goose.

His actions in the solution are summarised in the following steps:

1. Bring goose over
2. Return
3. Bring fox or beans over
4. Bring goose back
5. Bring beans or fox over
6. Return
7. Bring goose 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. Other cosmetic variations of the puzzle also exist, such as wolf, sheep and cabbage;, p. 26; fox, chicken and grain; 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, Russia, Romania, Scotland, the Sudan, Uganda, Zambia, and Zimbabwe., pp. 26–27;

The puzzle was a favorite of Lewis Carroll, and has been reprinted in various collections of recreational mathematics., p. 26.

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.

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