Alcuin's Problems to Sharpen the Young Mind

Oh, this is just too much fun to pass up. Call it the SAT-10(th Century)

Alcuin was a student of Egbert (who was a student of Bede), Charlemagne's chief advisor, organizer of his university and intellectual equal. He was personally involved with educating Charlemagne's children and courtiers. He came up with a list of math problems all his students should be able to solve (actually, he stole them from Bede). They look distressingly familiar....
The latin isn’t that hard to translate, so I went to PROPOSITIONES ALCUINI DOCTORIS CAROLI MAGNI IMPERATORIS AD ACUENDOS JUVENES

These are only the first 21 questions. There are 57 total.

About the units:
Alcuin's students would have known the original units, so my translation has modern equivalents. That'll give us a closer feel to what it was like to sit through a Carolingian lecture.

The original latin units:
The denominations of money were the libra, solidus and denarius. I’m using penny, dime and dollar.

A leuca (the predecessor of league) is 1500 paces- about a mile and a half.

The units of weight were the libra (now pound) and uncial (now ounce) with 12 unical to a libra.

A pace is about a foot.
One metretis is about 9 gallons. One sextarius is a pint.
One cubit is 1.5 feet.



1. A limace (A snail)
Limax fuit ab hierundine invitatus ad prandium infra leucam unam. In die autem non potuit plus quam unam unicam pedis ambulare. Dicat, qui velit, in quot diebus ad idem prandium ipse limax perambulat?

A leech invited a snail to lunch a mile and a half away. But he snail can only walk an inch a day. Tell me, if you will, how long will he have to walk for his meal?

2. Viro ambulante in via (A guy was walking down the street)
Quidam vir ambulans per viam vidit sibi alios homines obviantes, et dixit eis: Volebam, ut fuissetis alii tantum, quanti estis; et medietas medietatis; et hujus numeri medietas; tunc una mecum fuissetis. Dicat, qui velit, quanti fuerunt, qui in primis ab illo visi sunt?

When a guy walking down the street met others coming toward him, he said "I wish there were others with you, as many as you are, plus a quarter of the sum that would be plus half of that last amount. Then with me as well as thee there would be 100 all together". Tell me, if you will, many people did he see on the road [probably crossing rapidly to the other side as soon as he started talking]

It took me a couple of seconds to figure this out, but I used algebra (2x+1/2x+1/4x+1=100). I don't think that they had access to that kind of symbology. I *think* they would have broken it down into quarters in their minds (or on a scratch board)- 4/4 + 4/4 + 2/4 + 1/4 = 100 - 1
11/4 = 99
1/4 = 9
and everybody knows 1/4 of 36 is 9.

That's the beauty of thinking in 360 degrees- it's easy to simplify things into fractions. I’m beginning to suspect they visualized all their problems as pie charts. .

3. Duobus proficiscentibus visis ciconiis (Two Travelers See Storks)
Duo homines ambulantes per viam, videntesque ciconias, dixerunt inter se: Quot sunt? Qui conferentes numerum dixerunt: Si essent aliae tantae; et ter tantae et medietas tertii, adjectis duobus, C essent. Dicat, qui potest, quantae fuerunt, quae imprimis ab illis visae sunt?

Two men were walking down the road and saw storks. The first guy asked said “How many are there?” After talking it over, they decided there were all totaled one part and one part again and half of a third of those parts plus 2 more there would be 100. Tell me, if you can, how many storks did they see? [I wonder what the latin is for “pink elephants”]

Alcuin writes half of a third, which is an interesting way of putting it because a sixth was just as accessible. I think he may have been getting cute here because I can’t see an advantage into breaking the pie chart down that way. 6/6 + 6/6 + (6/6 + 6/6)/3 + 2 = 100

It never occurred to me before, but I think they must have been balancing equations, at least implicitly.

4. Homine at equis in campo pascentibus (A man watches horses graze in the field)
Quidam homo vidit equos pascentes in campo, optavit dicens: Utinam essetis mei, et essetis alii tantum, et medietas medietatis; certe gloriarer super equos C. Discernat, qui vult, quot equis imprimis vidt ille homo pascentes?

A man watches horses graze in a field, wishing “If only they were mine and they were the same number again and half of half of that number, I would have 100 horses to brag about.” Discover, if you will, how many horses does the man see in the pasture?


5. Emptore denariorum (The customer and the dollar)
Dixit quidam emptor: Volo de denariis C porcos emere; sic tamen, ut verres X denariis emantur; scrofa autem V denariis; duo vero porcelli denario uno. Dicat, qui intelligit, quot verres, quot scorfae, quotve porcelli esse debeant, ut in neutris numerus nec superabundet, nec minuatur?

The customer said “I have 100 dollars to spend on pigs. If a total of 10 dollars will buy a boar, 5 dollars a sow, and 2 piglets for 1 dollar, tell me, who that understands, how many boars, how many sows and how many piglets must I buy in order that the number of pigs equals the number of dollars?

6. Duobus negotiatoribus C solidos communis habentibus (Two dealers have 100 dimes)
Fuerunt duo negatiatores, habentes C solidos communes, quibus emerunt porcos. Emerunt autem in solidis duobus porcos V, volentes eos saginare, atque iterum venundare, et in solidis lucrum facere. Cumque vidissent tempus non esse ad saginandos porcos, et ipsi eos non valuissent tempore hiemali pascere, tentavere venundando, si potuissent, lucrum facere, sed non poterunt; quia non valebant eos amplius venundare, nisi ut empti fuerant, id est, ut de V porcis duos solidos acciperent. Cum hoc conspexissent, dixerunt ad invicem: Dividamus eos. Dividentes autem et vedentes, sicut emerant, facerunt lucrum. Dicat, qui valet, imprimis quot porci fuerunt; et dividat ac vendat et lucrum faciat, quod facere de simul venditis non valuit.

There were two dealers who, between them, had 100 dimes. They used the money to buy pigs. They bought 2 pigs for 5 dimes each, intending to fatten them up, then sell them again, increasing their money. However, it was the wrong season for fattening pigs and they didn’t want to feed them through winter because they would only be able to sell them for as much as they bought them for the next year and not making any profit. The decided to divide the pigs and sell them, each making a profit on the 5 dimes. Tell me, observant one, how did they divide them? How could they, by dividing them, gain a profit? Tell me how many they sold and how did they make the profit [Two words- Jimmy Dean].

I have to put in the solution from the Mathematical Gazette because no one in the right mind would ever figure this out.

Solution. Firstly there were 250 pigs bought with 100 shillings at the
above mentioned rate, for five 50s are 250. On division each merchant
had 125. One sold the poorer quality pigs at three for a shilling; the
other sold the better quality pigs at two for a shilling. The one who sold
the poorer pigs received 40 shillings for 120 pigs; the one who sold the
better quality pigs received 60 shillings for 120 pigs. There then
remained 5 of each sort of pig, from which which they could make a
profit of 4 shillings and 2 pence.

7. Disco pensante libras XXX (The dish that weight 30 pounds)
Est discus qui pensat libras XXX sive solidos DC, habens in se aurum, argentum, aurichalcum, et stannum. Quantum habet auri, ter tantum habet argenti. Quantum argenti, ter aurichalci. Quantum aurichalci, ter tantum stanni. Dicat, qui potest, quantum in unaquaque specie pensat?

Now we discuss the dish that weighs 30 pounds, made of gold, silver, brass and tin. It has gold, three times as much silver as gold. It has brass, three times as much as of silver. It has tin, three times as much as of brass. Tell me, if you can, how much of each type is there?

8. Cupa (Barrel) 
[Oddly, enough, cupa also means dancing girl or tavern wench]
Est cupa una, quae C metretis impletur capientibus singulis modia tria; habens fistulas III. Ex numero modiorum tertia pars et VI per unam fistulam currit: per alteram tertia pars sola: per tertiam sexta tantum. Dicat nunc, qui vult, quot sextarii per unamquamque, fistulam cucurissent.

There is a barrel with 100 metretis capacity, filled by three openings. One opening will carry one third and one sixth of the total flow. One opening will carry on third of the flow. One opening will carry one sixth of the flow. Tell me, if you will, how many sextarii flow through each opening.


9. Sago (Cloak)
Habeo sagum habentem in longitudine cubitos C, et in latitudine LXXX. Volo exinde per portiones sagulos facere, ita ut unaquaeque portio habeat in longitudine cubitos V, et in latitudine cubitos IIII. Dic, rogo, sapiens, quot saguli exinde fieri possint?

I think that cloak would fit Minimal…

There is a cloak 100 cubits long and 80 wide. I wish to make from this cloak smaller cloaks each with a length of 50 cubits and a width of 30 cubits. Say, my rational inquirer, how many small cloaks can be made?

10. Linteo (Linen) 
Habeo linteamen unum longum cubitorum LX, latum cubitorum XL. Volo ex eo portiones facere, ita ut unaquaeque portio habeat in longitudine cubitos senos, et in latitudine quaternos, ut sufficiat ad tunicam consuendam. Dicat, qui vult, quot tunicae exinde fieri possint? 

I have a piece of linen 60 cubits long and 40 cubits wide. I want to sew tunics 6 cubits wide and 4 cubits long. Tell me, if you will, how many tunics can I make? [With the way I sew, one- but it won’t get finished before 12th Night.]

11. Duobus hominibus sorores accipienibus (Two men who receive each other’s sisters)
Si duo homines ad invicem, alter alterius sororem in conjugium sumpserit; dic, rogo, qua propinquitate filii eorum pertineant?

If two men mutually marry each other’s sister, I ask what is the relationship of the sons? [Insert obligatory Aethelmearc joke].

12. Quodam paterfamilias et tribus filiis Rius (The father and his three sons)
Quidam paterfamilias moriens dimisit haereditatem tribus filiis suis, XXX ampullas vitreas, quarum decem fuerunt plenae oleo. Aliae decem dimidiae. Tertiae decem vacuae. Dividat, qui potest, oleum et ampullas, ut unicuique eorum de tribus filiis aequaliter obveniat tam de vitro, quam de oleo.
A father was dying and gave his three sons 30 glass jars. 10 were full of oil, 10 were half full and the third 10 were empty. Tell me, if you can, how can the oil and jars be divided equally among the sons?

13. Rege et de eius exercitu (The king and his army)
Quidam rex jussit famulo suo colligere de XXX villis exercitum, eo modo ut ex unaquaque villa tot homines sumeret quotquot illuc adduxisset. Ipse tamen ad villam primam solus venit; ad secundam cum altero; jam ad tertiam tres venerunt. Dicat, qui potest, quot homines fuissent collecti de XXX villis.

Once there was a king who ordered his slave to collect from his 30 estates an army so that each estate would give a total of however many men there had been collected up until then. He went to the first estate and took one. At the second estate he took two. At the third estate he took four. Tell me, if you can, how many men he collected from the villas.
[Alcuin thought he was so smart, but he got the answer wrong :-P]
 
14. Bove (Ox) 
Bos qui tota die arat, quot vestigia faciat in ultima riga?

An ox ploughs all day, how many footprints does he leave in the last furrow?

15. Homine (Man)
Quaero a te ut dicas mihi quot rigas factas habeat homo in agro suo, quando de utroque capite campi tres versuras factas habuerit?

One said to me “I saw a man plow his field. In what direction was he facing after the third turn on each side?”

That was my translation, but the answer is 7.

16. Duobus hominibus boves ducentibus (Two men lead an ox)
Duo homines ducebant boves per viam, e quibus unus alteri dixit: Da mihi boves duos; et habeo tot boves quot et tu habes. At ille ait: Da mihi et tu duos boves, et habeo duplum quam tu habes. Dicat qui vult, quot boves fuerunt, quot unusquisque habuit.

Two men leading oxen down a road. One man asks the other “Give me two oxen, then I’ll have as many as you”. The other responds “Then you give me two oxen and I’ll end up with twice as much as you have”. Tell me, if you will, how many oxen there were and how many each man had.

17. Tribus fratribus singulus habentibus sorores (Three brothers each had a sister)
Tres fratres erant qui singulas sorores habebant, et fluvium transire debebant (erat enim unicuique illorum concupiscientia in sorore proximi sui), qui venientes ad fluvium non invenerunt nisi parvam naviculam, in qua non potuerunt amplius nisi duo ex illis transire. Dicat, qui potest, qualiter fluvium transierunt, ne una quidem earum ex ipsis maculata sit?

There were three brothers each with a sister who needed to cross a river (each knew that the others wanted his sister [I think this goes along with Problem 11- either that or fratribus translates to friend, but that’s not as interesting]). They found a boat which could only carry two. Tell me, if you can, how to transit the river without a sister being dishonored.

18. Homine capram et lupum (Man, Goat and Wolf)
Homo quidam debebat ultra flavium transferre lupum, capram, et fasciculum cauli. Et non potuit aliam navem invenire, nisi quae duos tantum ex ipsis ferre valebat. Praeceptum itaque ei fuerat ut omnia haec ultra illaesa omnino transferret. Dicat, qui potest, quomodo eis illaesis transire potuit.

A man needed to transfer a goat, a wolf and a bunch of cabbages over a river. He could obtain a boat that would carry two and he could not leave any two behind. He had been instructed to transfer all across the river. Tell me, if you will, in what manner did he accomplish this transit.

 19. Viro et muliere ponderantibus (A husband and his wife are weighty)
De viro et muliere, quorum uterque pondus habebat plaustri onusti, duos habentes infantes inter utrosque plaustrali pondere pensantes fluvium transire debuerunt. Navem invenerunt quae non poterat ferre plus nisi unum pondus plaustri. Transfretari faciat, qui se putat posse, ne navis mergatur.

A husband and a wife each weighed as much as a laden cart. They had two children, both of whom weighed as much as the cart. They desired to transit a river. The came upon a boat which could hold only the weight of a cart. How can they all get across, do you think, without the boat sinking?

21. Hirtiis (The hairy ones) 
De hirtiis masculo et femina habentibus duos natos libram ponderantibus, flumen transire volentibus.

A hairy male and female have 2 babies who each weigh a pound. They wish transport over a river.

I think something got lost over the years.

21. Campo et ovibus in eo locandis (the field in which sheep are located) 
Est campus qui habet in longitudine pedes CC, et in latitudine pedes C. Volo ibidem mittere oves; sic tamen ut unaquaeque ovis habet in longo pedes V, et in lato pedes IV. Dicat, rogo, qui valet, quot oves ibidem locari possint?

There is a field which is 200 paces long and 100 paces wide. Someone wishes to release sheep into the field in such a way that each sheep has an area 5 paces long and 4 paces wide. Tell me, smartypants, if you wish, how many sheep is it possible to locate there?


* Problems to Sharpen the Young
John Hadley; David Singmaster

The Mathematical Gazette, Vol. 76, No. 475, The Use of the History of Mathematics in the Teaching of Mathematics. (Mar., 1992), pp. 102-126.

http://links.jstor.org/sici?sici=0025-5572%28199203%292%3A76%3A475%3C102%3APTSTY%3E2.0.CO%3B2-W