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,[1] and has entered the folklore of a number of ethnic groups.[2][3]

The story

Once upon a time a farmer went to a 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 rented 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 beans.

If left unattended together, the fox would eat the goose, or 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

Visualisation of the fox, goose and bag of beans puzzle. Uppercase letters denote objects at the destination and lowercase ones denote them at the origin. Movement of each object is represented by a coordinate axis. All the 8 valid and invalid placements are shown as vertices of a cube, and all 12 movements as its edges. Invalid moves are crossed out, leaving the 2 solutions shown in blue and purple.

The first step must be to take 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 taking either the fox or the beans across next. If he takes the fox across, he would have to return to get the beans, resulting in the fox eating the goose. If he takes the beans across second, he will need to return to get the fox, resulting in the beans being eaten by the goose. The dilemma is solved by taking the fox (or the beans) over and bringing the goose back. Now he can take the beans (or the fox) over, and finally return to fetch the goose.

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

  1. Take the Goose over
  2. Return
  3. Take the beans over
  4. Return with the goose
  5. Take the fox over
  6. Return
  7. Take 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;[4][2], p. 26 fox, chicken, and grain;[5] fox, goose and corn;[6] and panther, pig, and porridge.[7] 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.[2], pp. 26–27;[8] 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.[9]

The puzzle was a favorite of Lewis Carroll,[10] and has been reprinted in various collections of recreational mathematics.[2], 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 Nintendo DS puzzle game Professor Layton and the Curious Village, and in the 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.

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.[2], p. 27.

See also

References

  1. ↑ Pressman, Ian; David Singmaster (June 1989). ""The Jealous Husbands" and "The Missionaries and Cannibals"". The Mathematical Gazette. The Mathematical Association. 73 (464): 73–81. JSTOR 3619658. doi:10.2307/3619658.
  2. 1 2 3 4 5 Ascher, Marcia (February 1990). "A River-Crossing Problem in Cross-Cultural Perspective". Mathematics Magazine. Mathematical Association of America. 63 (1): 26–29. JSTOR 2691506. doi:10.2307/2691506.
  3. ↑ Gurdjieff, G. I. (1963). Meetings with Remarkable Men (1st English ed.). London: Routledge & Kegan Paul. pp. 4–5.
  4. ↑ Alcuin's Transportation Problems and Integer Programming, Ralf Borndörfer, Martin Grötschel, and Andreas Löbel, preprint SC-95-27 (November 1995), Konrad-Zuse-Zentrum für Informationstechnik Berlin.
  5. ↑ The Classic River Crossing Puzzle
  6. ↑ Mary Jane Sterling, Math Word Problems for Dummies, P.313
  7. ↑ Stewart, Ian (1998). The Magical Maze. Phoenix. ISBN 0-7538-0514-6.
  8. ↑ Evans-Pritchard, E. E. (1962). "235. Three Zande Texts". Man. 62: 149–152. JSTOR 2796709. doi:10.2307/2796709.
  9. ↑ "Carrying a Wolf, a Goat, and a Cabbage across the Stream. Metamorphoses of ATU 1579", Piret Voolaid, Folklore: Electronic Journal of Folklore 35 (2007), pp. 111–130. Tartu: Eesti Kirjandusmuuseum.
  10. ↑ p. 17, Rediscovered Lewis Carroll Puzzles, Lewis Carroll, compiled by Edward Wakeling, Courier Dover Publications, 1996, ISBN 0-486-28861-7.
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