Charlemagne's Math Tutor

We’ve been working on the middle ages this year. While I would have said that I knew quite a bit about the middle ages, I’m realizing that I do have quite a few blank spots. One of those blank spots is Charlemagne. I found that all I really knew about him was that his name meant Charles the Great and that he was king in what is now France and Germany.

One of the cool things I’ve learned this year came from a German education magazine, Focus Schule. This month in their Lernatlas is a discussion of the Carolingian monk Alcuin. Alcuin had been at the cathedral school in York. He met Charlemagne and later became head of the palace school that Charlemagne founded in Aachen in 782. His Propositiones ad acuendos iuvenes is a series of logic problems that give a great math workout.

Here are a few of the problems from Propositiones. (I have translated these from a German translation of the Latin. I’m looking forward to trying to figure out the original Latin problems in a couple years.)

1. A snail was invited to brunch by a swallow who lived one mile away. The snail could only travel one inch in a day. How many days will it take the snail to arrive at brunch? (1 Gallic mile = 1500 paces, 1 pace = 5 feet, 1 foot = 12 inches)

2. A group of people met a wanderer, who said to them, “If there were another group as large as you and another half as big and another a quarter as big, along with me there would be exactly 100. How many met the wanderer?

3. A merchant wants to buy 100 pigs with 100 denarii. A boar costs 10 denarii; a sow costs 5 denarii; 2 piglets cost 1 denaris. How many boars, sows and piglets are can he purchase using exactly 100 denarii?

4. I have a large piece of linen, 100 yards long and 80 yards wide. I want to divide it into smaller pieces, each 5 yards long and 4 yards wide. How many smaller pieces will there be?

5. A father left his three sons 30 small glass bottles as their inheritance. Ten were full of oil, another 10 were half full of oil. The rest were empty. How can the oil and bottles be divided equally amonth the 3 sons. (There are at least two good answers for this one.)

6. A king ordered his retainer to raise an army from 30 villages in the following manner. From each village he should recruit as many men as enter the village. At the first village he goes in alone. At the second, he has one companion. At the third he has 3 companions. How many men will there be after the 30th village? (Please note that I've corrected this question. I think it was a poor translation rather than just poor math. That's my story and I'm sticking to it.)

7. An ox plows all day long. How many tracks does he leave in the last furrow?

8. How many furrows does a man plow in a field if he turns three times on each side of the field?

9. A man must cross a river with a wolf, a goat and a head of cabbage. He can only find a boat that will hold himself and one of the three other things. The wolf cannot be left alone with the goat, and the goat cannot be left alone with the cabbage. How can he get all three things across the river unharmed? (I didn’t realize this one was so old.)

10. A man and a woman each weigh the same as a cart load. Their three children together weigh the same as a cart load. They need to get across a river but can only find a boat that can carry the weight of one cart load. How can they all get across the river without swamping the boat?

11. A ladder has 100 rungs. On the first rung is a pigeon. On the second rung are two pigeons. On the third rung are three pigeons and on the fourth rung are four pigeons. And so on up to the 100th rung which has 100 pigeons. How many pigeons are on the ladder?

I didn’t have much luck finding the complete Propositiones online in English. I found a few here . Maybe this is another argument for learning to read Latin.

Solutions:
1. 1500 x 5 x 12 = 90,000 days
2. x + x + x/2 + x/4 + 1
3. He buys one boar (b), 9 sows (s) and 90 piglets (p). b + s + 2p = 100; 10b + 5s + p = 100. Therefore 19b + 9 s = 100 . b = 1 and s = 9 (I have to push the I believe button on this on. I came pretty close when I set the problem up but then messed up the arithmetic somewhere in the middle. Maybe I should pull out the Friendly Frogs math counters and try a more concrete example.)
4. 400 = 100/5 x 80/4
5. One son gets the 10 half full bottles. The other two each get 5 full and 5 empty. OR Pour half of one of the full bottles into an empty bottle. Then each son gets 3 full bottles, 4 half full bottles and 3 empty bottles.
6. The number of recruits doubles at each village. 2 raised to the 30th power = 1,073,741,824 men.
7. None, the plow erases all the ox’s tracks.
8. Seven.
9. He takes the goat first, then the wolf. He takes the goat back with him, gets the cabbage and leaves the goat. Then he brings the goat across again.
10. The three children go across, one rows the boat back. The man goes across. The second child rows the boat back. The wife goes across. The third child rows the boat back, gets his siblings and rows back to the far side.
11. There are 5050 pigeons on the ladder.