Problems to Sharpen the Young

Of all the Popes who have headed the Catholic church over the centuries, only one has been a mathematician. Gerbert of Aurillac (c. 950–1003) was Europe’s leading mathematician before taking over as Pope Sylvester II, starting in 997.

In an article in the November College Mathematics Journal, Leigh Atkinson of the University of North Carolina at Asheville notes that Gerbert received practically no instruction in mathematics when he went to school at a monastery close to Aurillac, a village in south-central France.

Among the very few mathematical works that might have been available to Gerbert was an oft-copied, widely circulated manuscript called Propositiones ad acuendos juvenes. The Latin title can be translated as “Problems to sharpen the young.” It’s usually attributed to Alcuin of York (735–804).

The text contains 53 (or so) word problems (with solutions), in no particular pedagogical order. Among the most famous of these problems are four that involve river crossings, including the problem of three jealous husbands (each of whom can’t let another man be alone with his wife), the problem of the wolf, goat, and cabbage, and the problem of “the two adults and two children where the children weigh half as much as the adults.” (See “Tricky Crossings” at Tricky Crossings.)

In Atkinson’s opinion, the most interesting problem in the collection is the following (translated from the Latin):

There is a ladder that has 100 steps. One dove sat on the first step, two doves on the second, three on the third, four on the fourth, five on the fifth, and so on up to the hundredth step. How many doves were there in all?

Alciun’s solution is to note that there are 100 doves on the first and 99th steps, 100 more on the second and 98th, and so on for all the pairs of steps, except the 50th and 100th.

Problem 43 was “composed for rebuking” troublesome students, and there’s no given solution.

A certain man has 300 pigs. He ordered all of them slaughtered in 3 days, but with an uneven number killed each day. What number were to be killed each day?

“One wonders how long the students needed to realize that three odd numbers can’t add up to 300,” Atkinson comments.

Problem 26 offers a bit of a calculus flavor.

There is a field that is 150 feet long. At one end stood a dog; at the other, a hare. The dog chased the hare. Whereas the dog went 9 feet per stride, the hare went only 7. How many feet and how many leaps did the dog take in pursuing the fleeing hare until it was caught?

Problem 12 is the forerunner of what are now known as barrel-sharing brainteasers.

A certain father died and left as an inheritance to his three sons 30 glass flasks, of which 10 were full of oil, another 10 were half full, while another 10 were empty. Divide the oil and flasks so that an equal share of the commodities should equally come down to the three sons, both of oil and glass.

Problem 5 from the Alcuin collection represents the first known appearance in Europe of a type often called the “hundred fowls” problem, from a 5th-century version that features 100 birds (cocks, hens, and chicks). By Alcuin’s time, the problem and several variants were already available in Indian and Arabic texts.

A merchant wanted to buy 100 pigs for 100 pence. For a boar, he would pay 10 pence; for a sow, 5 pence; while he would pay 1 penny for a couple of piglets. How many boars, sows, and piglets must there have been for him to have paid exactly 100 pence for the 100 animals?

The collection contains six other versions of this problem.

Problem 52 is another one that has survived in various forms to this day.

A certain head of household ordered that 90 modia of grain be taken from one of his houses to another 30 leagues away. Given that this load of grain can be carried by a camel in three trips and that the camel eats one modium per league, how many modia were left over at the end of the journey?

Modern versions are sometimes described as “jeep” problems because they involve a jeep in the desert with n cans of fuel and a distant destination.

Browsing the problems (and solutions) in Propositiones ad acuendos juvenes provides fascinating glimpses of various aspects of life in medieval times. And it testifies to the enduring power of puzzles in mathematical education.

“Recreational problems have immense pedagogic utility,” David Singmaster once commented. “Some of these problems have fascinated students for nearly 4,000 years and should continue to do so indefinitely.”

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