Propositiones ad Acuendos Juvenes/Problems/52 - De Homine Patrefamilias

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Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $52$

De Homine Patrefamilias
A Lord of the Manor
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?


Solution

$20$ measures.


Proof

On the $1$st journey, the camel carries $30$ measures over $20$ leucas, eating $1$ measure for each leuca, leaving $10$ leucas.

On the $2$nd journey, the camel likewise carries $30$ measures over $20$ leucas, eating $1$ measure for each leuca, leaving $10$ leucas.

On the $3$rd journey, the camel again carries $30$ measures over $20$ leucas, eating $1$ measure for each leuca, leaving $10$ leucas.

Now he is at the point $10$ leucas away, with $30$ measures to take over $10$ leucas.

During that journey the camel eats $10$ measures, leaving $20$ measures.


Historical Note

David Singmaster suggests that the camel does in fact make $4$ journeys, not $3$.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$ contend that the $3$rd journey to carry $30$ measures $20$ leucas, then $30$ leucas the remaining $10$ leucas, is in fact one journey, just broken into $2$ parts.

It is suggested that there may be a further unspoken constraint given here: that a camel may not be able to carry more than $30$ measures at one time.

Note also that the camel does not require food when he is not carrying a load.


There exists a better solution if more than $3$ trips are allowed:

The camel makes $3$ trips carrying $30$ measures to $10$ leucas, there now being $60$ measures at $10$ leucas.
The camel makes $2$ trips carrying $30$ measures to a point another $15$ leucas further, there now being $30$ measures $25$ leucas on.
The camel makes $1$ further trip of $5$ leucas carrying those $30$ measures to the destination, eating $5$ and so carrying $25$ measures in total.


This may be the earliest appearance of a desert crossing problem.


Sources

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