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.
Answers
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.
