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A wolf, a goat and a cabbage. You have one boat and need to get all three across safely. Many will be familiar with this river crossing puzzle or one of its variants. The earliest known instance of this puzzle dates back to the early Middle Ages and is found in a peculiar work attributed to Anglo-Saxon monk and scholar Alcuin of York. This blog post calls attention to Alcuin’s mathematical puzzles and invites you to a game of medieval brain training!
Alcuin of York (c. 735-804) and the Propositiones ad acuendos iuvenes
Possibly the most knowledgeable scholar of his age, the Anglo-Saxon Alcuin of York (c.735–804) made a career at the Carolingian court of Charlemagne (d. 814). He was an advisor to Charlemagne himself and taught the royal children at the Palace School. This Northumbrian monk quickly became recognised as one of the court’s foremost scholars and he wrote many poems, letters and books on various topics, including grammar, orthography, theology and hagiography.
Deep in his heart, Alcuin always remained a teacher, as he made clear in a letter to Charlemagne:
I shall not be slow to sow the seeds of wisdom among your servants in these parts, as far as my poor talent allows. … In the morning, at the height of my powers, I sowed the seed in Britain, now in the evening, when my blood is growing cold, I still am sowing in France, hoping both will grow, by the grace of God. (trans. Allott 1974, let. 8)
As a result, many of his works were of a didactic nature and perhaps the most peculiar of these bears the title Propositiones ad acuendos iuvenes [Propositions to Sharpen the Young]. This text is a collection fifty-three mathematical puzzles, which were intended to make mathematics palatable for students (a noble cause!). The collection includes, as noted above, the earliest attestation of the famous wolf-goat-cabbage-river cross problem (which is so famous that the Internet features several games that allow you to solve this problem, such as this one featuring ‘Sailor Cat’). Aside from various river crossing problems, the collection includes puzzles that involve mathematical and geometric calculations. To make the puzles relatable for his students, Alcuin wrote them in the form of little stories, involving men sharing oxen, pigeons sitting on staircases, families crossing rivers and dogs chasing hares (Yes, teaching mathematics through story has a long history, indeed!). Alcuin even included some trick questions, such as “An ox ploughs a field all day. How many footprints does he leave in the last furrow?”. The answer to that last question, of course, is ‘none’, since the ox precedes the plough and, therefore, all of its footprints are erased!
Below follow seven of Alcuin’s brain teasers. Some may test your wits, while others may require a piece of paper. If you want the real medieval experience, do not use a calculator! The solutions are provided at the end of this blog post. The translation of the Propositiones ad acuendos iuvenes is that of John Hadley (Hadley & Singmaster 1992):
i) Don’t sink the boat!
A man and woman, each the weight of a cartload, with two children who together weigh as much as a cartload, have to cross a river. They find a boat which can only take one cartload. Make the transfer if you can, without sinking the boat.
ii) A dog chasing a hare
There is a field 150 feet long. At one end is a dog, and at the other a hare. The dog chases when the hare runs. The dog leaps 9 feet at a time, while the hare travels 7 feet. How many feet will be travelled by the pursuing dog and the fleeing hare before the hare is seized ? [i.e., how long will it take the dog to overtake the hare which has a 150 feet head start?]
iii) Buying camels, sheep and asses
A man in the east wanted to buy 100 assorted animals for 100 shillings. He ordered his servant to pay 5 shillings for a camel, one shilling for an ass and one shilling for 20 sheep. How many camels, asses, and sheep did he buy ?
iv) A stair case of pigeons
A staircase has 100 steps. On the first step stands a pigeon; on the second two; on the third three; on the fourth 4; on the fifth 5. And so on, on every step to the hundredth. How many pigeons are there altogether?
v) A flock of storks
Two walkers saw some storks and wondered how many there were. Conferring, they decided: if there were the same number again, and again, and then half of a third of the sum that would make, plus two more, that would be 100. How many storks were seen ?
vi) Evening out the oxen
Two men were leading oxen along a road, and one said to the other: “Give me two oxen and I’ll have as many as you have”. Then the other said: “Now you give me two oxen and I’ll have double the number you have.” How many oxen were there, and how many did each have?
vii) Slaughtering pigs
A certain man had 300 pigs. He ordered all of them slaughtered in three days, but with an uneven number being killed each day. He wished the same thing to be done with 30 pigs. Let him say, he who can, What odd number of pigs out of 300 or 30 were to be killed in three days?
- i) The kids do most of the rowing! First: the two kids go to the other side, one rows back. Next, one of the parents rows to the other side. Then, the kid who had stayed on the other side goes back alone and picks up the other child and goes to the other side again. Now, the two children and one of the parents are on the other side, while one parent has stayed behind. One child rows to the missing parent, the missing parent rows to the other side – the child who is now with two parents rows back to pick up his sibling and -hurray- the family is reunited again!
- ii) The hare has a head start of 150 feet, but the dog goes 2 feet per leap faster. In other words, it will take the dog 150/2=75 leaps to catch up to the hare. The dog will have travelled 75*9 = 675 feet, while the hare would have travelled 75*7 = 525 feet.
- iii) For 5 shillings you have 100 sheep and 95 shillings to spare! But that would be cheating, of course. The trick is to buy 80 sheep for 4 shillings, 1 ass for 1 shilling and then spend the other 95 shillings on 19 camels.
- iv) The answer is 5050. Alcuin outlines an easy way of calculating this total:
We count them as follows. Take the single one on the first step and add it to the 99 on the ninety-ninth step, making 100. Taking the second with the ninety-eighth likewise gives 100. So for each step, one of the higher steps combined with one of the lower steps, in this manner, will always give 100 for the two steps. However the fiftieth step is alone, not having a pair. Similarly the hundredth remains alone. Join all together and get 5050 pigeons.
- v) 100-2 = 98. 98 is three and a half times the original number; 98 divided by 3.5 = 28. So they originally saw 28 storks!
- vi) There were 12 oxen. The first man had 4 oxen and the other 8 – if the first man received 2 oxen from the other, they would both have 6. If the other would get his 2 oxen back, we would be back to the initial situation: 4 against 8.
- vii) Aha! Alcuin is trying to trick us here. His own solution runs as follows “This is a fable. No-one can solve how to kill 300 or 30 pigs in three days, an odd number each day.” He then notes that this trick question is to be given to (misbehaving) children. That would teach them! (I am not sure whether Alcuin’s reasoning is didactically sound here!)
Works referred to:
- S. Allott (trans.), Alcuin of York: His Life and Letters (York, 1974)
- John Hadley (trans.) & David Singmaster, ‘Problems to Sharpen the Young’, The Mathematical Gazette 76 (1992), pp. 102–126.
Post Scriptum: As @AlcuinsLibrary has rightly pointed out to me, most of the puzzles in the Propositiones stem from a long tradition and were certainly not made up by Alcuin (although some occur here for the first time). In fact, Alcuin’s authorship itself is a matter of conjecture rather than fact (some manuscripts attribute the puzzles to Bede). Alcuin’s authorship is suggested (but not proven) by a letter that he wrote to Charlemagne in which he talks of having sent “certain subtle figures of arithmetic, for pleasure” (Hadley & Singmaster 1992). These pleasurable figures of arithmetic may or may not have been the Propositiones…