## Monday 19 September 2022

### A Brief History of the River Crossing Problem

Recently reading through some new math blogging educators on Sue Van Hattum's web page, Math Mama Writes, and one used variations of the "River Crossing" problem to link algebra to their middle school student's arithmetic learning and reminded me that I wanted to gather together my notes on the history of this great old problem.

For those who actually have not heard of the river crossing problem, a common version is something like this example from a rather nice Wikipedia posting:
The jealous husbands problem, in which three married couples must cross a river using a boat which can hold at most two people, subject to the constraint that no woman can be in the presence of another man unless her husband is also present.

David Singmaster, one of the foremost (if not THE foremost) historian on the history of recreational mathematics credits the 9th Century Alcuin of York, an English scholar, ecclesiastic, poet and teacher from York, Northumbria. He was born around 735 and became friend and adviser to Charlemagne and the leading teacher at his court.
His Propositiones ad Acuendos Juvenes (Problems to sharpen the young contains the first written record of River Crossing Problems, apparently of three versions. Wikipedia lists the fox, goose and bag of beans puzzle and the Jealous Husbands problem above. (If you know the third version in his book, I would love to be informed).

The book also includes the first Explorer's Problem; first Division of Casks; first
Apple-sellers' Problem; first Collecting Stones; unusual solution of
Posthumous Twins Problem; first Three Odds Make an Even; and the first Strange
Families problem.

Professor Singmaster notes Pacioli's 1500 De Viribus has the next essential variation in the game, a river crossing that contains boats to hold more than two people. Then a few years later (1556), Tartaglia's General Trattato introduces the first River Crossing with four
couples. In his Science News blog, Ivars Petersen adds "With four or more couples, however, it's impossible to accomplish the crossings under the required conditions."
Several hundred years later, In volume 1 of Eduard Lucas' Recreations Mathematiques he gives De Fontenay's idea of couples crossing a river with an island. This would solve the four-couple Jealous Husbands problem. Lucas is also well remembered for his Towers of Hanoi Puzzle (briefly referenced in the picture at top with the tower of discs, and perhaps the turtle is a reference to Lo Shu and Magic Squares) and a Fibonacci like series that bears his name.

Today the forms of the puzzle have spread to reflect more modern additions to cultures. In the 19th Century Cannibals and Missionaries were a popular subject. A warfare version enters in this version from, I believe, a Russian version in the early 20th Century :
A detachment of soldiers must cross a river. The bridge is broken, and the river is deep. The officer in charge spots two boys playing in a rowboat by the shore. The boat is so tiny, however, that it can only hold two boys or one soldier. All the soldiers succeed in crossing the river in the boat. How?

And finally, this tongue-in-cheek quote from Ivars Petersen's Science News Article:
"One must be a little careful with some of these problems, as past cultures were often blatantly sexist or racist," Singmaster warns. "But such problems also show what the culture was like. . . . The river crossing problem of the jealous husbands is quite sexist and transforms into masters and servants, which is classist, then into missionaries and cannibals, which is racist. With such problems, you can offend everybody!"

Just to show a little of the popularity of these puzzles, I recently found a web page, Puzzle Museum, that had images of a Turnbridge Ware box of Puzzles sold by Rudolph Ackerman in the first quarter of the 19th Century. Inside one of the envelopes are cut-outs for the Wolf/goat/Seed and Jealous Husbands river crossing problems. Two other games were included, the Josephus (Turks and Christians count-out game) problem, and one of the dissected Cross.

Charles Dodgson (Lewis Carroll) often sent puzzles to his child-friends and included the river crossing problem with Fox, Goose, and Corn to young Jessie Sinclair. His interest in the problem seems to include the creation of an original variation on the problem, although there seems to be no direct evidence that he first created it.
His version is varioulsy called "The Captive Queen", or just the tower problem. Here is one version:
A captive queen and her son and daughter were shut up in the top room of a very high tower. Outside their window was a pulley with a rope around it, and a basket fastened to each end of the rope of equal weight. They managed to escape with the help of this and a weight they found in the room, quite safely. It would have been dangerous for any of them to come down if they weighed 15 lbs more than the content of the other basket, for they would do so too quick, and they also managed not to weigh less either.

The one basket coming down would naturally of course draw the other basket up.

The queen weighed 195 lbs, daughter 105, son 90, and the weight 75 lbs.

How did they all escape safely?

About three years after I wrote the above, Jim Wilder posted a version I had overlooked, involving a ball of twine.

This is from a wonderful collection of puzzles from a mathematician who has undoubtedly sold more mathematical puzzle books than anyone in the world. It is a Wonderful book to keep on the desk for a (sometimes) quick challenge.

River crossing problems seem to show across cultures. In Africa Counts: Number and Pattern in African Cultures, Claudia Zaslavsky mentions a version told by Kpelle children in Liberia in which a man must ferry a leopard, a goat, and a bunch of Cassava leaves. No period or origin of the puzzle is given, and the similarity to the fox, goat, and beans in Alcuin's problem makes one wonder if one influenced the other.

Bengali--> Bagh-pan-chhagol (tiger, leaves, goat) version too. Punjabis had a lovers version-I've forgotten!

Over the years I have come across several tongue-in-cheek solutions to river crossing type problems, so here are a couple of nice examples:
The first is from the wonderful XKCD site:
The Second is a business approach from Dilbert.Com:

If you  have other notes, information, or references to the history of these problems, I would love to hear from you.