Alcuin’s River Crossing Puzzles and Common Sense
Test your common sense.
Posted June 14, 2010
As a puzzlist, I often emphasize that many puzzles are solved simply by using common sense or what the American pragmatist philosopher Charles S, Peirce (1839-1914) called "practical logic." When we understand how to do certain things practically, without having to be told or shown how to do them, we are employing this form of logic. Here's a classic puzzle that brings out the power of this type of instinctual thinking:
A traveler comes to a riverbank with a wolf, a goat, and a head of cabbage. To his delight he sees there a boat that he can use for crossing over to the other bank, but to his dismay, he notices that it can carry no more than two-the traveler himself, of course, and just one of the two animals or the cabbage. As the traveler knows, if left alone together, the goat will eat the cabbage and the wolf will eat the goat. The wolf does not eat cabbage. How does the traveler transport his animals and his cabbage to the other side intact in a minimum number of back-and-forth trips?
Try solving it before reading on. Incidentally, I have noticed over the years that people who have never come across this puzzle react to it typically in two ways when they first do: (1) they feel that they have somehow known it all their lives (indicating that its structure is archetypal?), and (2) they take pleasure in the fact that they are able to solve it with only "common sense" reasoning.
The traveler cannot start with the wolf, since that would leave the goat alone with the cabbage, and the goat would eat it. That is the key insight in solving the puzzle. So, practically speaking, the traveler can only start by taking the goat with him on the boat to the other side, leaving the wolf safely with the cabbage on the original side. After dropping off the goat on the other bank, he then rows back alone. Overall, this constitutes his first round trip. Back on the original side, he picks up the wolf and rows with it to the other side, leaving the cabbage by itself. Upon reaching the other bank he drops off the wolf, but rows back with the goat, so that wolf cannot eat the goat for lunch. Again, this decision is, clearly, part of common sense. This makes up the traveler's second round trip. Back on the original side, he leaves the goat there, taking the cabbage across with him on the boat. When he gets to the other bank, he drops off the cabbage, leaving the wolf and cabbage safely together there as he rows back alone. This is his third round trip. He then picks up the goat on the original side and rows across with it. When he gets to the other bank he will have his wolf, goat, and cabbage intact and, so, can continue on with his journey.
There is a second solution, which nonetheless starts off in the same way. The difference is that the traveler picks up the cabbage instead of the wolf at the start of the second round trip. The end result is the same-three round trips (or seven back-and-forth trips in total). As can be seen, this puzzle brings out the power of practical logic to minimize and even eliminate trial and error. That, in my view, is the cognitive backbone of what we call common sense.
The puzzle is one of a set of three called the "river crossing puzzles," posed originally by the famous English scholar and ecclesiastic Alcuin (735-804 CE), who became an adviser to the Holy Roman Emperor Charlemagne in 782. It is believed that Charlemagne became so obsessed over puzzles that he hired Alcuin primarily to create them for his enjoyment. The ingenious Alcuin put his puzzles together into an instructional manual for young students titled Propositiones ad acuendos juvenes ("Problems to Sharpen the Young"). Some editions of the text contain 53 puzzles, others 56. It was translated into English by John Hadley and annotated by David Singmaster. The translation was published in volume 76 (pp. 102-126) of The Mathematics Gazette in 1992.
The above puzzle is actually a paraphrase of number 18 in Alcuin's manual. Here is a different version of this puzzle for you to solve. Again, though more complicated, it can be solved simply by applying common sense to it.
The traveler reaches the same riverbank, with the same boat there. Along with him are his wolf, goat, head of cabbage, and this time a mythical monster called the Wolf-Eater. The Wolf-Eater eats only wolves. Moreover, when the Wolf-Eater is present on either side, he intimidates the goat, who will thus not eat the cabbage. How does the traveler get them all across safely?
Numbers 17 and 19 complete Alcuin's set of river crossing conundrums. A fourth one (number 20) also involves river crossing, but it has come down to us in incomplete form. Number 17 is about three men, each with an unmarried sister, who wish to cross the river using the two-seat boat, with each man "desirous of his friend's sister." There is an obvious, albeit unconscious, sexist subtext to the puzzle (given the historical era in which it was conceived). That notwithstanding, the puzzle again brings out what common sense is all about. Here is a paraphrase of the puzzle.
Three men, each one accompanied by his unmarried sister, come to a riverbank. The small boat that will take them across can hold only two people. To avoid any compromising situations, the crossings are to be so arranged that no sister shall be left alone with a man-on the boat or on either side-unless her brother is present. How many crossings are required, if any man or woman can be the rower?
A famous later version of this puzzle is known as the missionaries and cannibals puzzle. Can you solve the following paraphrase?
Three missionaries and three cannibals must get across a river. At no time on either bank can the cannibals outnumber the missionaries, since this uneven number would lead to one of the missionaries being eaten up. How do they get across with a boat that can hold only two, if either a missionary or a cannibal can operate the boat?
Number 19 in Alcuin's anthology is slightly different in make-up, but it too requires the same kind of common sense to solve. The following is, again, a paraphrase of the original puzzle.
A man and a woman who weigh the same, together with two children, each one half the weight of an adult, come to the same riverbank and the same boat. The boat can carry two people, but it can only hold, as a maximum, the weight of one adult, otherwise it would sink. How do they get across?
More complicated versions of river crossing puzzles, involving different combinations of people, animals, and victuals have come down to us through the ages from across the world, indicating a universal fascination with this form of archetypal logical thinking (as it can be called). It is not clear if any of these predate Alcuin's puzzles. For this reason, the latter are still considered to be the first ones of their kind. Incidentally, not all kinds of river crossing puzzles turn out to be solvable. For instance, the puzzlists Sam Loyd (1841-1911) and Henry E. Dudeney (1847-1930) discovered that it is impossible to arrive at a solution involving four brothers and their unmarried sisters (or equivalently four jealous husbands and their wives). A solution is possible only if there is an island in mid-stream for use as a transit stop.
Actually, river crossing puzzles have turned to be much more than mere exercises or exemplifications in common sense thinking. Many mathematical historians trace the conceptual roots of combinatorics to Alcuin's river crossing puzzle. And it is easy to recognize the roots of modern-day systems analysis, which is based on critical decision-making logic, in these simple, yet intriguing paradigmatic puzzles.
There are several ways to solve the Wolf-Eater puzzle, all consisting of four round trips (nine individual back-and-forth trips in total). Here's one.
1.The traveler must start by taking the wolf with him to the other side, leaving the Wolf-Eater with the goat and cabbage on the original side. The Wolf-eater's presence ensures that the goat will not eat the cabbage.
2.Upon reaching the other bank, the traveler drops off the wolf there and rows back alone. This is his first round trip.
3.Back on the original side, he picks up the cabbage, leaving the Wolf-Eater and goat safely alone there, and rows with it to the other side.
4.Once there, he leaves the cabbage safely with the wolf and then rows back alone. This is his second round trip.
5.On the original side, he picks up the Wolf-Eater, leaving the goat alone there, rowing with the monster to the other bank.
6.Once he reaches it, he drops off the Wolf-Eater, but picks up the wolf for his trip back (so that the Wolf-Eater will not eat the wolf), leaving the Wolf-Eater safely alone with the cabbage. This is his third round trip.
7.Upon reaching the original side, the traveler drops off the wolf there, picking up the goat for the trip over.
8.Once he reaches the other side, he leaves the goat safely with the Wolf-Eater and the cabbage, who are already there. The Wolf-eater's presence ensures that the goat will not eat the cabbage. He rows back alone. This is his fourth round trip.
9.Back on the original side, the traveler picks up the wolf, rows with it to the other side. He gets off the boat with the wolf, and continues on his journey with all four.
Four round trips (nine individual trips) are also required to solve Alcuin's Number 17. Slight variations to the solution below are possible.
1.One brother and sister pair row across first, leaving the other two brother and sister pairs safely on the original side.
2.The brother is dropped off on the other bank and his sister goes back alone on the boat. This is the first round trip.
3.Back on the original side, the sister picks up a second sister and rows back with her to the other side. The remaining sister on the original side is safe, of course, because her brother is still there with her.
4.Once on the other side, the first sister lets herself off to stay with her brother, who is already there. The second sister rows back alone. This completes the second round trip.
5.When the second sister gets to the original side, she picks up her brother and rows with him over to the other side.
6.On that side she drops off her brother and rows back alone. Since the first brother and sister pair are already there, no problems ensue from the presence of the second brother. This constitutes the third round trip.
7.When the second sister gets back to the original side, she picks up the third sister and rows with her across to the other bank, leaving her brother alone on the original side.
8.Once they get to the other bank, the second sister drops herself off to stay with her own brother, who is already there. There are now two brother and sister pairs on the other side. The third sister rows back alone to the original side. This is the fourth round trip.
9.Back on the original side, the sister picks up her brother and rows over with him to join the others.
A solution to the missionaries and cannibals puzzle also produces the same result-four round trips (nine individual back-and-forth trips). Again there are other slight variations to the pattern. In this version, one of the cannibals is the rower for all back-and-forth trips.
1.Two cannibals start it off by rowing together over to the other side.
2.One is dropped off on the other side and the other one goes back alone. This is the first round trip.
3.On the original side, the rower cannibal picks up a missionary and rows with him across to the other side. No danger results from this because the missionary is together with the rower cannibal on the boat, while on the original side there are two missionaries and one cannibal. Thus, the cannibals do not outnumber the missionaries anywhere in this scenario.
4.Once they reach the other side, the cannibal drops off the missionary and then rows back alone. Again, no danger results from this, since on the other bank there is one missionary and only one cannibal. This completes the second round trip.
5.Back on the original side, the cannibal picks up the second missionary and rows with him over to the other side. In this scenario there is a missionary-cannibal pair on both sides and on the boat. So, again, no danger results from it.
6.On the other bank, the cannibal drops off the second missionary and rows back alone. There are now two missionaries on the other bank with one cannibal, while back on the original side there is a missionary and a cannibal waiting. This completes the third round trip.
7.When the rower cannibal gets back to the original side, he picks up the last missionary and rows across with him to the other bank. No danger results from this, of course.
8.Once there, he drops off the missionary. There are now three missionaries and one cannibal on the other side. So, the cannibal rows back to get him. This completes the fourth round trip.
9.Back on the original side the rower cannibal picks up the last cannibal and rows over with him to the other bank to join the others.
Solving the adults and children puzzle also produces the same pattern, with variations. Here is one specific solution.
1.The two children start it off by rowing together over to the other side. The boat can hold both their weights, of course, because they add up the weight of one adult.
2.One child stays on the other side, while the other one goes back alone. This completes the first round trip.
3.On the original side, the rower child gets off, and one of the adults gets on the boat and rows over to the other side alone. The boat can hold at most an adult weight.
4.Once on the other side, the adult gets off and the child who was already there gets on the boat and rows back alone. This completes the second round trip.
5.Back on the original side, the child picks up the other child waiting there and together they row across to the other side.
6.Once on that side, one of the children gets off and the other one rows back alone. There is now an adult and a child on the other side, while back on the original side there is one adult waiting. This completes the third round trip.
7.When the rower child gets to the original side, the child drops himself off, and the second adult can now get on the boat and row alone safely to the other bank.
8.Once there, the adult drops himself off to join the other adult already there. The child who is also there gets on the boat and rows over to get the child who is waiting on the original side. This completes the fourth round trip.
9.Once the rower child reaches the original side, the other child gets on the boat with the rower to go across and join the adults.
For technical discussions of river crossing puzzles, the interested reader should consult: Benjamin L. Schwartz, "An Analytic Method for the Difficult Crossing Problems," Mathematics Magazine 34 (1961), pp. 187-193; Ian Pressman and David Singmaster, "The Jealous Husbands and the Missionaries and Cannibals," The Mathematical Gazette 73 (1989), pp. 73-81; and Ivars Peterson, "Tricky Crossings," Science News, 164 (2003).