Recreational problems Alcuin 1 Albrecht Heeffer
Alcuin
Alcuin (c800) Propositiones alcuini doctoris caroli magni imperatoris ad acuendos
juvenes. Beati Flacci Albini seu Alcuini Abbatis Caroli Magni Regis ac Imperatoris
Magistri Opera post primam editionem, viro clarissimo D. Andrea Quercetano
curatam, de novo collecta, multis locis emendata, et opusculis primum repertis
plurimum aucta, variisque modis illustrata cura ac studio Frobenii, S.R.I. Principis et
Abbatis ad S. Emmeramum Ratisbonae M.DCC.LXXVII
Patrologia Latina Database Vol. 101 Main volume text [Opera Omnia Alcuini]
Prob Transcription
ALC01 propositio de limace
A snail was invited by a swallow to lunch a league away. However, it
could not walk further than one inch per day. How many [years and]
days did it take for the snail to walk to that lunch?
ALC02 propositio de viro ambulante in via
A certain man walking in the street saw other men coming towards him,
and he said to them: "O that there were so many [more] of you as you are
[now]; and then half of half of this [were added]; and then half of this
number [were added], and again, a half of [this] half. Then, along with
me, you would number 100 [men]." How many men were first seen by
the man?
ALC03 propositio de duobus proficiscentibus
Two men were walking in the street when they noticed some storks.
They asked each other, "How many are there?" Discussing the matter,
they said: "If [the storks] were doubled, then taken three times, and then
half of the third [were taken] and with two more added, there would be
100." How many [storks] were first seen by the men?
ALC04 propositio de homine et equis
A certain man saw some horses grazing in a field and said longingly: "O
that you were mine, and that you were double in number, and then a half
of half of this [were added]. Surely, I might boast about 100 horses."
How many horses did the man originally see grazing?
ALC05 propositio de emptore denariorum
A certain buyer said: "I want to buy 100 pigs with 100 denarii in such a
way that a mature boar is bought for 10 denarii; a sow for five denarii;
and two small female pigs for one denarius." How many boars, sows,
and small female pigs should there be so that there are neither too many
nor too few of either [pigs or denarii]?
ALC06 propositio de duobus negotiatoribus c solidos habentis
There were two businessmen who had 100 solidi between them, with
which they bought some pigs. For two solidi, they bought five pigs,
wishing to fatten them and to sell them again at a profit. But when they
saw that the time was not right to fatten the pigs, and being unable to
pasture them over the winter, they tried to make a profit by selling them.
However, they were unsuccessful because they could only sell the pigs
for what they had paid (i.e., five pigs for two solidi). When they realized
this, they said to each other, "We shall divide the pigs." But by dividing
Recreational problems Alcuin 2 Albrecht Heeffer
and selling the pigs for as much as they had paid, they made a profit.
How many pigs were there at first, and how did the men divide and sell
for a profit that which they could not do together?
ALC07 opositio de disco pensante libras xxx
There is a plate weighing 30 pounds or 600 solidi. In it, there is gold,
silver, brass and tin. It has three times are much silver as gold, three
times as much brass as silver, and three times as much tin as brass. How
much does each type of metal weigh?
ALC08 propositio de cupa
There is a cask which has three cracks in it. It is filled with 100
metretae, each holding three modia. Of the modia, a third and sixth part
run out through one crack. Through another [crack], only a third part
runs out. Only a sixth part runs out of the third crack. How many sextarii
ran out through each crack.
ALC09 propositio de sago
I have a material for cloaks which is 100 cubits long, 80 cubits wide.
From it, I wish to make smaller cloaks from portions in such a way that
each portion is five cubits in length and four cubits wide. How many
smaller cloaks can be made from [the material]?
ALC10 propositio de linteo
I have a single linen cloth which is 60 cubits long, 40 cubits wide. I
wish to make it into smaller portions, each being six cubits in length,
four cubits in width, so that each piece is ample for making a tunic. How
many tunics can be made [from the larger piece]?
ALC11 propositio de duobus hominibus sorores accipientibus
proposition concerning the two men marrying [one another's] sister.
If two men should marry one another's sister. What will be the sons'
relations to each other?
ALC12 propositio de quodam patrefamilias et tribus filiis ejus
A certain father died and left as an inheritance to his three sons 30 glass
flasks, of which 10 were full of oil; another 10 were half full, while
another 10 were empty. Divide the oil and flasks so that an equal share of
the commodities should equally come down to the three sons, both of oil
and glass.
ALC13 propositio de rege
A certain king ordered his servant to gather an army from 30 villages as
follows: He should bring back as many men [from each successive
village] as he had taken there. Thus, [the servant] came to the first village
alone; he came with one other person to the next; three people came to
the third. How many men were collected from the 30 villages.
ALC14 propositio de bove
How many footprints in the last furrow does an ox make which has been
plowing all day?
ALC15 propositio de homine
How many furrows might a man have in his field if he shall have made
three turns at each head of the field?
ALC16 propositio de duobus hominibus boves ducentibus
Two men were leading oxen along the road when one said to the other,
"Give me two oxen, and I shall have as many oxen as you." Then the
Recreational problems Alcuin 3 Albrecht Heeffer
other said, "You give me two oxen, and I shall have twice as many as
you." How many oxen there were, and how many each man had.
ALC17 propositio de tribus fratribus singulas habentibus sorores
There were three men, each having an unmarried sister, who needed to
cross a river. Each man was desirous of his friend's sister. Coming to the
river, they found only a small boat in which only two persons could cross
at a time. How did they cross the river, so that none of the sisters were
defiled by the men?
ALC18 propositio de homine et capra et lupo
A certain man needed to take a wolf, a she-goat and a load of cabbage
across a river. However, he could only find a boat which would carry
two of these [at a time]. Thus, what rule did he employ so as to get all of
them across unharmed?
ALC19 propositio de viro et muliere ponderantibus [plaustri pondus onusti]
A man and his wife, each the weight of a loaded cart, who had two
children each the weight of a small cart, needed to cross a river.
However, the boat they came across could only carry the weight of one
cart. Devise [a way] of crossing in order that the boat should not sink.
ALC20 propositio de hirtitiis
A masculine and feminine [....] who had two children weighing a pound
wished to cross a river.
ALC21 propositio de campo et ovibus in eo locandis
There is a field which is 200 feet long, 100 feet wide. I want to put
sheep in it as follows: Each sheep should have [an area] five feet long
and four feet wide. How many sheep can be put in such a place?
ALC22 propositio de campo fastigioso
There is a slanting field which is 100 perticae on each side, 50 perticae
on one front, 60 perticae in the middle, and 50 perticae on the other front.
How many aripennae does [this field] enclose?
ALC23 propositio de campo quadrangulo
There is a field which is 30 perticae on one side, 32 perticae on another,
34 perticae in the front, and 32 perticae on the remaining side. How
many aripenni are contained in such a field?
ALC24 propositio de campo triangulo
There is a field which is 30 perticae on one side, 30 perticae on another,
and 18 perticae in the front. How many aripenni must be contained [in
such a field]?
ALC25 propositio de campo rotundo
There is a round field which contains 400 perticae in its circle. Tell me,
How many aripenni ought it to hold?
ALC26 propositio de cursu cbnks. bc. fvgb. lfp:rks
There is a field which is 150 feet long. At one end stood a dog, at the
other, a hare. The dog advanced behind [the hare], namely, to chase the
hare. But whereas the dog went nine feet per stride, the hare went [only]
seven. How many feet and how many leaps did the dog take in pursuing
the fleeing hare until it was caught?
ALC27 propositio de civitate quadrangula
There is a quadrangular city which has one side of 1100 feet, another
side of 1000 feet, a front of 600 feet, and a final side of 600 feet. I want
Recreational problems Alcuin 4 Albrecht Heeffer
to put some houses there so that each house is 40 feet long and 30 feet
wide. How many houses ought the city to contain?
ALC28 propositio de civitate triangula
There is a triangular city which has one side of 100 feet, another side of
100 feet, and a third of 90 feet. Inside of this, I want to build a structure
of houses, however, in such a way that each house is 20 feet in length, 10
feet in width. How many houses should be contained [within this
structure]?
ALC29 propositio de civitate rotunda
There is a city which is 8000 feet in circumference. , How many houses
should the city contain, such that each [house] is 30 feet long, and 20 feet
wide?
ALC30 propositio de basilica
There is a basilica which is 240 feet long, 120 feet wide. One tile of the
tiled basilica is 23 inches long, that is, one foot, 11 inches, while being
12 inches wide, i.e. one foot. How many tiles are needed to cover the
basilica?
ALC31 propositio de canava
There is a wine cellar which is 100 feet long and 64 feet wide. How
many casks can it hold, given that each cask is seven feet long and four
feet wide, and given that there is an aisle four feet wide in the middle [of
the cellar]?
ALC32 propositio de quodam patrefamilias
A certain head of household had 20 servants. He ordered them to be
given 20 modia of corn as follows: The men should receive three modia;
the women, two; and the children, half a modium. How many men,
women and children must there have been?
ALC33 propositio de alio patrefamilias erogante suae familiae annonam
A certain head of household had 30 servants whom he ordered to be
given 30 modia of corn as follows: The men should receive three modia;
the women, two; and the children, a half modium. How many men,
women and children were there?
ALC34 propositio altera de patrefamilias partiente familiae suae annonam
A certain head of household had 100 servants. He ordered that they be
given 100 modia of corn as follows: The men should receive three
modia; the women, two; and the children, half a modium. Thus how
many men, women, and children were there?
ALC35 propositio de obitu cujusdam patrisfamilias
A certain father died and left behind children, a pregnant wife, and 960
solidi from his estate. [However, on his deathbed], he stipulated that if a
son should be born to her, then the son should receive three fourths of the
inheritance -- that is, nine twelfths. The mother should get a quarter [of
the estate], that is, three twelfths. However, if a daughter were born, she
should receive seven twelfths, and the mother, five twelfths. But as it
happened, she gave birth to twins -- both a boy and a girl. How much did
the mother, son and daughter each receive?
ALC36 propositio de salutatione cujusdam senis ad puerum
A certain old man greeted a boy, saying to him: "May you live, boy, may
you live for as long as you have [already] lived, and then another equal
Recreational problems Alcuin 5 Albrecht Heeffer
amount of time, and then three times as much. And may God grant you
one of my years, and you shall live to be 100." How many years old was
the boy at that time?
ALC37 propositio de quodam homine volente aedificare domum
A certain man, wanting to build a house, found six workmen, of whom
five were masters and one an apprentice. It was agreed between the man
who wanted to build and the workmen that 25 denarii should be given to
them per day as pay, and that the apprentice should receive half what the
masters receive. How much did each of them receive per day?
ALC38 propositio de quodam emptore in animalibus centum
A certain man wanted to buy 100 various animals for 100 solidi. He
wished to pay three solidi per horse, one solidus per cow, and one solidus
per 24 sheep. How many horses, cows and sheep were there?
ALC39 propositio de quodam emptore in oriente
A certain man wished to buy 100 assorted animals for 100 solidi in the
East. He ordered his servant to pay five solidi per camel, one solidus per
ass, and one solidus per 20 sheep. How many camels, asses and sheep
were obtained for 100 solidi?
ALC40 propositio de homine et ovibus in monte pascentibus
A certain man saw from a mountain some sheep grazing and said, "O
that I could have so many, and then just as many more, and then half of
half of this [added], and then another half of this half. Then I, as the 100th
[member], might head back to my home together with them." How many
sheep did the man see grazing?
ALC41 propositio de sode et scrofa
A certain head of household set up a new [quadrangular] enclosure in
which he placed a sow. The sow gave birth to seven piglets in the middle
of the sty. The offspring, along with the mother, the eighth pig, each
gave birth to another seven piglets in each corner [of the sty]. Then, in
the middle of the sty, the mother and all her offspring [each] gave birth
to seven more. How many pigs were there [in the end], including the
mother?
ALC42 propositio de scala habente gradus centum
There is a ladder which has 100 steps. One dove sat on the first step, two
doves on the second, three on the third, four on the fourth, five on the
fifth, and so on up to the hundredth step. How many doves were there in
all?
ALC43 propositio de porcis
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. What odd number of pigs out of 300
or 30 were to be killed in three days? (This ratio is indissoluble and was
composed for rebuking.)
ALC44 propositio de salutatione pueri ad patrem
A certain boy addressed his father, saying, "Greetings, father!" The
father responded, "May you fare well, my son, and may you live three
times twice your years. Then, adding one of my own years, you will live
to be 100." How many years was the boy at the time?
ALC45 Propositio
Recreational problems Alcuin 6 Albrecht Heeffer
A dove sitting in a tree saw some other doves flying and said to them, "O
that you were doubled, and then tripled. Then, along with me, you would
number 100." How many doves were initially flying?
ALC46 propositio de sacculo ab homine invento
A certain man walking in the street found a small bag containing two
talents. Some other people saw this and said to him: "Brother, give us a
portion of your discovery." But the man shook his head and did not want
to give them any. The others then rushed at him and tore apart the sack,
each obtaining for himself 50 solidi. And when the man saw that he
could no longer resist [their attack], he grabbed 50 solidi for himself. ,
How many men were there?
ALC47 ropositio de episcopo qui jussit xii panes dividi
A certain bishop ordered 12 loaves of bread divided amongst the clergy.
He stipulated that each priest should receive two loaves; a deacon, half a
loaf; and a lector, a quarter part. Hence, it should turn out that the
number of clerics and loaves is the same. How many priests, deacons and
lectors must there have been?
ALC48 propositio de homine qui obviavit scholaribus
A certain man met some students and asked them, "How many of you are
there in school?" One of [the students] responded to him: "I do not want
to tell you [except as follows]: double the number of us, then triple that
number; then, divide that number into four parts. If you add me to one of
the fourths, there will be 100." How many [students] first met the man?
ALC49 propositio de carpentariis
Seven carpenters [each] made seven wheels. How many carts did they
build?
ALC50 propositio de vino in vasculis
I ask so that one who wishes might respond: How many sextarii do 100
metra of wine contain, and how many meri do 100 metra have?
ALC51 propositio de vini in vasculis a quodam patre divisione
A certain dying father left four small vessels of wine to his four sons. In
the first vessel, there were 40 modia [of wine]; in the second, 30; in the
third, 20; and in the fourth, 10. Calling his house treasurer, he said:
"Divide these four vessels containing wine amongst my four sons in such
a way that each son receives an equal portion of wine and vessels." How
must the vessels have been divided so that all [the sons] received an
equal amount from this?
ALC52 propositio de homine patrefamilias
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, and that [the camel] eats
one modium per league, how many modia were left over [at the end of
the transport]?
ALC53 propositio de homine patrefamilias monasterii xii monachorum
A certain Father of a monastery had 12 monks. Calling the treasurer of
his chapter, he gave them [the priests] 204 eggs, and he ordered that [the
treasurer] should give an equal portion to each individual. He further
stipulated that [the treasurer] give 85 eggs to the five priests,[68 to the
four deacons, and 51 to the three lectors]. How many eggs did each
Recreational problems Alcuin 7 Albrecht Heeffer
[monk] receive as his portion, so that no one received too many, nor too
little, but so that as we stated above, he will take an equal portion to all?
Compiled from Burkholder (1993). Burkholder gives a literal translation which I
simplified without losing the original meaning.
Index in the Singmaster classification
Class Description Prob Proposition
ALC17 propositio de tribus fratribus singulas
habentibus sorores
ALC18 propositio de homine et capra et lupo
ALC19 propositio de viro et muliere ponderantibus
[plaustri pondus onusti].
5.B Crossing problems
ALC20 propositio de hirtitiis
5.N Jeep problem ALC52 propositio de homine patrefamilias
7.G Posthumous twins ALC35 propositio de obitu cujusdam patrisfamilias
7.L.1 ALC41 propositio de sode et scrofa
7.L.2
Geometric
progression ALC13 propositio de rege
ALC05 propositio de emptore denariorum
ALC32 propositio de quodam patrefamilias
ALC33 propositio de alio patrefamilias erogante
suae familiae annonam
ALC34 propositio altera de patrefamilias partiente
familiae suae annonam
ALC38 propositio de quodam emptore in animalibus
centum
ALC39 propositio de quodam emptore in oriente
7.P.1 Hundred fowls
ALC53 propositio de homine patrefamilias
monasterii xii monachorum
7.R If I had one from
you
ALC16 propositio de duobus hominibus boves
ducentibus
7.Y Selling apples ALC06 propositio de duobus negotiatoribus c solidos
habentis
7.AF.1 Collecting stones ALC42 propositio de scala habente gradus centum
7.AN Three odds make
an even
ALC43 propositio de porcis
7.AU Number of cuts to
make n pieces
ALC15 propositio de homine
10.A Overtaking ALC26 propositio de cursu cbnks. bc. fvgb. lfp:rks
References
Burkholder, Peter (1993) “Alcuin of York's Propositiones ad acuendos juvenes:
Introduction, Commentary & Translation,” History of Science & Technology
(HOST) Bulletin, Vol 1, number 2.
Folkerts, Menso (1978) “Die älteste mathematische Aufgabensammlung in
lateinischer Spräche: Die Alkuin zugeschriebenen Propositiones ad Acuendos
Recreational problems Alcuin 8 Albrecht Heeffer
Iuvenes” Denkschriften der Österreichischen Akademie der Wissenschaften
athematische-naturwissenschaftliche Klasse 116:6 pp13-80.
Folkerts, Menso and Gericke, Helmuth (1993) “Die Alkuin zugeschriebenen
Propositiones ad Acuendos Juvenes (Aufgaben zur Schärfung des Geistes der
Jugend).” In: Science in Western and Eastern Civilization in Carolingian
Times ed. by P. L. Butzer & D. Lohrmann; Birkhäuser, Basel, 1993, pp. 283-
362.
Pressman, Ian; Singmaster, David (1989) “The jealous husbands and the missionaries
and cannibals”. Mathematical Gazette 73 (1989), no. 464, 73-81.
Singmaster David (1992) “Problems to sharpen the young. An annotated translation of
‘Propositiones ad acuendos juvenes’ The oldest mathematical problem
collection in Latin attributed to Alcuin of York”. Translated by John Hadley,
annotated by David Singmaster and John Hadley. Mathematical Gazette 76
(No. 475) (Mar 1992) 102-126. (Extended and revised version of 7 November
2002. Electronic copy from the author)
Singmaster, David (1998) “The history of some of Alcuin's Propositiones.” In:
Charlemagne and his Heritage 1200 Years of Civilization and Science in
Europe: Vol. 2 Mathematical Arts; ed. by P. L. Butzer, H. Th. Jongen & W.
Oberschelp; Brepols, Turnhout, 1998, pp. 11-29.