## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## June 27, 2007

### Problems to Sharpen the Young

One of the oldest collections of mathematical problems we know of is Problems to Sharpen the Young. No one knows who wrote the book, but some scholars say that the author might have been someone named Alcuin, who lived from about 732 to the year 804 (three-digit years!). Alcuin was born near the city of York in England and was a student, then a teacher, and then head of the Cathedral School at York.

Here's a problem from the book you might find interesting:

A staircase has 100 steps. On the first step stands a pigeon; on the second, two pigeons; on the third, three; on the fourth, four; on the fifth, five; and so on, on every step up to the hundredth. How many pigeons are there altogether?

Can you find a quick way to solve the problem that doesn't require you to add all the numbers from 1 to 100? You'll find the answer at the end of this article.

Problem 5 in the collection is one of six variations in the book on what was known as the "hundred fowls" problem, so-called after a 5th-century version that features 100 birds (cocks, hens, and chicks):

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 100 animals?

Problem 52 in the collection 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 (not necessarily all the way) and that the camel eats one modium per league but only eats when he is carrying a load, how many modia were left over at the end of the journey?

Forget about the modia. The units are irrelevant. Modern versions are sometimes called jeep problems because they describe a jeep in the desert with n cans of fuel and a distant destination.

Here's a really sneaky one:

A man has 300 pigs and orders that they are to be killed in three days, an odd number each day. What odd number of pigs must be killed each day?

There's no answer. According to the book, this problem was composed to show up smart-alecky or misbehaving schoolchildren. Wonder how long it took students to figure out that this was a trick question and that three odd numbers can never add up to an even number.

It seems that some things about school haven't changed much—even over hundreds of years.

ANSWER: Alcuin's clever solution to the staircase problem is to count the pigeons in pairs, starting with the one on the first step and the 99 on the 99th step. That makes 100. The two on the second step and the 98 on the 98th step also make 100. Remember the 50th step and the 100th step have no pairs. The grand total? 5050 pigeons, which is a lot of feathers.

If you want to try other problems in the collection, go to logica.ugent.be/albrecht/alcuin.pdf.

Muse, March 2006, p. 42.