Propositiones ad Acuendos Juvenes/Problems

Problems from Propositiones ad Acuendos Juvenes by Alcuin of York

$1$: De Limace

A leech invited a slug for lunch a leuca away.

But he could only walk an inch a day.

How many days will he have to walk for his meal?

$2$: De Viro Ambulante in Via

A man walking down the road saw others coming towards him, and he said to them,

"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 there would be $100$ altogether."

How many did he see on the road?

$3$: De Duobus Proficistentibus Visis Ciconiis

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 a half and a third of the sum that would make,

plus two more,

that would make $100$.

How many storks were seen?

$4$: De Homine et Equis in Campo Pascentibus

A man saw some horses at pasture and wished they were his,

and that there were others with them that were his,

the same number again,

plus a quarter of the sum that would result,

for then he would glory in $100$ horses.

How many did the man see at pasture?

$5$: De Emptore in $\text C$ Denariis

A merchant wanted to buy $100$ pigs at $100$ pence.

For a boar he would pay $10$ pence;

and 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?

$6$: De Duobis Negotiatoribus $\text C$ Solidos Communis Habentibus

$2$ wholesalers with $100$ shillings between them bought some pigs with the money.

They bought at the rate of $5$ pigs for $2$ shillings, intending to fatten them up and sell them again, making a profit.

But when they found that it is not the right time of year for fattening pigs,

and they were not able to feed them through the winter,

they tried to sell them again to make a profit.

But they couldn't, because they could only sell them for the price they had paid for them, $2$ shillings for $5$ pigs.

When they saw this, they said to each other, "Let's divide them.

By dividing them, and selling them again at the rate they had bought them for, they made a profit.

How could they be divided to make a profit, which could not be made by selling them all at once?

$7$: De Disco Pesante Libras $\text {XXX}$

A dish weighing $30$ pounds is made of gold, silver, brass, and lead.

It contains $3$ times as much silver as gold;

$3$ times as much brass as silver;

and $3$ times as much lead as brass.

How much is there of each?

$8$: De Cupa

A cask is filled with $100$ metretae through $3$ pipes.

One third plus one sixth of the capacity flows through one pipe,

one third of the capacity flows through another,

but only a sixth of the capacity flows through the third.

How many sextarii flow in through each pipe?

$9$: De Sago

I have a cloak, $100$ cubits long and $80$ cubits wide.

I wish to make small cloaks with it, each small cloak $5$ cubits long and $4$ cubits wide.

How many small cloaks can I make?

$10$: De Linteo

I have a piece of linen, $60$ cubits long and $40$ cubits wide.

I want to cut it into portions, each of which will be $6$ cubits long and $4$ wide.

How many portions can be made of it?

$11$: De Duobus Hominibus Singulas Sorores Accipientibus

If two men take the other's sister in marriage,

what is the relationship between their sons?

Variant 1

De Duobus Hominibus Singulas Matres Accipientibus: Two men marrying each other's mother

If two men take the other's mother in marriage,

what is the relationship between their sons?

Variant 2

De Patre et Filio et Vidua Eiusque Filia: A father and son and a widow and her daughter

If a relict or widow and her daughter take a father and son in marriage,

so that the son marries the mother and the father marries the daughter,

what is the relationship of their sons?

$12$: De Quodam Patrefamilias et Tribus Filiis Eius

A father, when dying, gave to his sons $30$ glass flasks,

of which $10$ were full of oil,

$10$ were half full,

and the last $10$ were empty.

Divide the oil and the flasks so that each of the $3$ sons receives equally of both glass and oil.

$13$: De Rege et de Eius Exercitu

A king ordered his servant to collect an army from $30$ manors,

in such a way that from each manor he would take the same number of men as he had collected up till then.

The servant went to the first manor alone;

to the second he went with one other;

to the next he took $3$ with him.

How many were collected in all?

$14$: De Bove

An ox ploughs a field all day.

How many hoofprints does he leave in the last furrow?

$15$: De Homine

How many furrows has a man made,

when he has made $3$ turnings at each end of the field?

$16$: De Duobus Hominibus Boves Ducentibus

Two men were leading oxen down the 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?

$17$: De Tribus Fratribus Singulas Habentibus Sorores

Three friends, each with a sister, needed to cross a river.

Each one of them coveted the sister of another.

At the river, they found only a small boat, in which only two of them could cross at a time.

How did they cross the river, without any of the women being defiled by the men?

$18$: De Lupo et Capra et Fasciculo Cauli

A man takes a wolf, a goat and a bunch of cabbages across the river.

The only boat he could find could take only two of them at a time.

But he had been ordered to transfer all of these to the other side in good condition.

How could this be done?

$19$: De Viro et Muliere Ponderantibus Plaustrum

A man and a woman, each the weight of a cartload,

with two children who together weigh as much as a cartload,

must cross a river.

They find a boat which can only take one cartload.

Make the transfer if you can, without sinking the boat.

$20$: De Ericiis

About a male and female hedgehog with two young,

having weight,

wanting to cross a river.

The Latin text appears to be defective, and the question does not lend itself to rational interpretation.

However, it appears to be another example of $19$: De Viro et Muliere Ponderantibus Plaustrum (A Very Heavy Man and Woman).

$21$: De Campo et Ovibus in Eo Locandis

There is a field $200$ feet long and $100$ feet wide.

I want to put sheep in it, so that each sheep has $5$ feet by $4$.

How many sheep can I put in there?

$22$: De Campo Fastigioso

There is an irregular field,

measuring $100$ perches along each side,

and $50$ perches along each end,

but $60$ perches across the middle.

How many acres does it contain?

$23$: De Campo Quadrangulo

A four-sided field measures $30$ perches down one side

and $32$ down the other;

it is $34$ perches across the top

and $32$ across the bottom.

How many acres are included in the field?

$24$: De Campo Triangulo

A triangular field is $30$ perches along $2$ sides

and $18$ perches along the bottom.

How many acres must be enclosed?

$25$: De Campo Rotundo

A round field is $400$ perches in circumference.

How many acres does it contain?

$26$: De Campo et Cursu Canis ac Fugo Leporis

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?

$27$: De Civitate Quadrangula

A four-sided town measures $1100$ feet on one side

and $1000$ feet on the other side;

on one edge $600$

and on the other edge $600$.

I want to cover it with roofs of houses,

each of which should be $40$ feet long

and $30$ feet wide.

How many dwellings can I make there?

$28$: De Civitate Triangula

A triangular town measures $100$ feet along one side,

$100$ feet along another side

and $90$ feet along the front.

I want to build houses there,

each house being $20$ feet long

and $10$ feet wide.

How many houses can there be?

$29$: De Civitate Rotunda

There is a round town, $8000$ feet in circumference.

I want to build houses there,

each house being $20$ feet long

and $10$ feet wide.

How many houses must it contain,

each house being $30$ feet long

and $20$ feet wide?

$30$: De Basilica

A basilica is $240$ feet long and $120$ feet wide.

It is paved with paving stones one foot $11$ inches long

and $12$ inches, that is one foot, wide.

How many stones are needed?

$31$: De Canava

There is a wine cellar $100$ feet long and $64$ feet wide.

How many casks will it hold if each cask is $7$ feet long and $4$ feet wide across the middle,

and there is one path $4$ feet wide?

$32$: De Quodam Patrefamilias Distribuente Annonam

A gentleman has a household of $20$ persons and orders that they be given $20$ measures of grain.

He directs that:

each man should receive $3$ measures,

each woman $2$ measures,

and each child $\frac 1 2$ a measure.

How many men, women and children must there be?

$33$: De Alio Patrefamilias Erogante Suae Familiae Annonam

A gentleman has a household of $30$ persons and directs that they be given $30$ measures of grain.

He directs that:

each man should receive $3$ measures,

each woman $2$ measures,

and each child $\frac 1 2$ a measure.

How many men, women and children must there be?

$34$: Altera de Patrefamilias Partiente Familiae Suae Annonam

A gentleman has a household of $200$ persons and orders that they be given $100$ measures of grain.

He directs that:

each man should receive $3$ measures,

each woman $2$ measures,

and each child $\frac 1 2$ a measure.

How many men, women and children must there be?

$35$: De Obitu Cuiusdam Patrisfamilias

A dying man left $960$ shillings and a pregnant wife.

He directed that:

if a boy is born he should receive $\frac 9 {12}$ of the estate and the mother $\frac 3 {12}$.

However if a daughter is born she would receive $\frac 7 {12}$ of the estate and the mother $\frac 5 {12}$.

It happened however that twins were born -- a boy and a girl.

How much should the mother receive, how much the son, how much the daughter?

$36$: De Salutatione Cuiusdam Senis ad Puerum

An old man greeted a boy as follows:

"May you live long --

as long as you have lived so far,

and as long again as your age would be then,

and then to three times that age,

and let God add one year more

and you will be $100$.

How old was the boy at the time?

$37$: De Quodam Homine Volenti Aedificare Domum

A man wanting to build a house contracted with $6$ builders,

$5$ of whom were master builders.

and the $6$th an apprentice,

to build it for him.

He agreed to pay them a total of $25$ pence a day,

the apprentice to get half the rate of a master builder.

How much did each receive a day?

$38$: De Quodam Emptore in Animalibus Centum

A man wanted to buy $100$ assorted animals for $100$ shillings.

He was willing to pay $3$ shillings for a horse,

$1$ shilling for an ox,

and $1$ shilling for $24$ sheep.

How many horses, oxen and sheep did he buy?

$39$: De Quodam Emptore in Oriente

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,

$1$ shilling for a donkey,

and $1$ shilling for $20$ sheep.

How many camels, donkeys and sheep did he buy?

$40$: De Homine et Ovibus in Monte Pascentibus

A man looked at some sheep grazing on the hillside, and said,

"I wish I had these,

and as many again,

plus half of half that total,

plus half of that last amount;

then, counting myself, I would take $100$ to my home.

How many sheep did he see?

$41$: De Sode et Scrofa

A farmer created a new yard in which he put a breeding sow,

which produced a litter of $7$ piglets in the centre,

which with their mother makes $8$.

They all bear litters of $7$ piglets each in the first corner of the yard,

then all of them bear litters of $7$ piglets each in the next corner

and so on for all corners.

Finally they all bear litters of $7$ in the centre pigsty.

How many pigs are there now altogether, including the mothers?

$42$: De Scala Habente Gradus Centum

A stairway consists of $100$ 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 every step up to the hundredth.

How many pigeons are there altogether?

$43$: De Porcis

A man has $300$ pigs, and orders that the pigs must be killed,

an odd number each day,

in $3$ days.

How many pigs should be killed each day?

$44$: De Salutatione Pueri ad Patrem

A son greeted his father: "Hello Father;"

to which the father replied: "Hello Son.

May you live long, as much as you have lived.

If you triple that number and add one of my years,

you will have lived $100$ years."

How old was the boy at the time?

$45$: De Columba

A pigeon sitting in a tree saw others flying, and said to them:

"I wish there were others with you --

the same number again and a third time.

Then with one of me, there would be $100$."

How many pigeons were there at first?

$46$: De Sacculo ab Homine Invento

A man walking along a road, found a purse containing $2$ talents.

Others saw this and said: "Friend, give us a portion of your find."

He refused to give any to them.

So they set upon him and took the bag from him,

and each one took $50$ shillings.

Seeing that he could not stop them, he reached out and snatched $50$ shillings for himself.

How many men were there?

$47$: De Episcopo qui Jussit $\text {XII}$ Panes in Clero Dividi

A certain bishop ordered that $12$ loaves be divided among his clergy.

He ordered that:

each priest should receive $2$ loaves,

each deacon one half

and each reader one quarter.

There were as many loaves as clergy.

How many priests, deacons and readers must there be?

$48$: De Homine qui Obviavit Scolaris

A man met some scholars and said to them: "How many are there in your school?"

One of them answered: "I don't want to tell you that.

Count us twice and multiply by $3$.

Then divide into $4$ parts.

If you add me to this $4$th part, you will have $100$."

in the second $30$,

in the third $20$,

and in the fourth $10$.

Calling his steward, he said:

"Divide these $4$ flasks among my $4$ sons so that each of their portions shall be equal,

both in wine and in flasks."

$52$: De Homine Patrefamilias

A man ordered that $90$ measures of grain were to be moved from one of his houses to another, $30$ leucas away.

One camel was to carry the grain in $3$ journeys, carrying $30$ measures on each journey.

The camel eats one measure for each leuca.

How many measures will remain?

$53$: De Homine Patrefamilias Monasterii $\text {XII}$ Monachorum

An abbot had $12$ monks in his monastery.

Calling his steward he gave him $204$ eggs and ordered that he give equal shares to each monk.

Thus he ordered that he give:

$85$ eggs to the $5$ priests,

and $68$ to the $4$ deacons

and $51$ to the $3$ readers.

How many eggs went to each monk, so that none had too many or too few,

but all received equal portions as above?

Propositiones ad Acuendos Juvenes/Problems

Problems from Propositiones ad Acuendos Juvenes by Alcuin of York

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