Propositiones ad Acuendos Juvenes/Problems/16 - De Duobus Hominibus Boves Ducentibus

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Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $16$

De Duobus Hominibus Boves Ducentibus
Two Men Leading Oxen
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?


Solution

There were $12$ oxen:
The first man had $4$
The second man had $8$.


Proof

Let $a$ and $b$ be the number of oxen owned at the start by the first and second man respectively.

It is assumed, by use of the word now in the question, that the suggested transaction actually went ahead, and the $2$ oxen actually changed hands.


We have:

\(\text {(1)}: \quad\) \(\ds a + 2\) \(=\) \(\ds b - 2\) after the first transaction
\(\text {(2)}: \quad\) \(\ds 2 a\) \(=\) \(\ds b\) after the second transaction, when they were back where they started
\(\ds \leadsto \ \ \) \(\ds a + 4\) \(=\) \(\ds b\)
\(\ds \) \(=\) \(\ds 2 a\)
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds 4\)
\(\ds \leadsto \ \ \) \(\ds b\) \(=\) \(\ds 8\)

$\blacksquare$


Historical Note

This is a less common variant of the traditional Donkey and Mule problem, in which the suggested exchange is not actually perceived as going ahead.

Hence, in the usual case, when the second man makes his suggested exchange, the assumption is that they are starting with their original distribution of goods.

In this case, the second suggestion happens after the first exchange has actually been done.


Sources

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