Propositiones ad Acuendos Juvenes/Problems/18 - De Lupo et Capra et Fasciculo Cauli

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Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $18$

De Lupo et Capra et Fasciculo Cauli
A Wolf, a Goat and a Bunch of Cabbages
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?


Solution

The constraints here are that:

$(1): \quad$ The wolf, if left alone with the goat, would eat it, so these two may not be left alone together.
$(2): \quad$ The goat, if left alone with the bunch of cabbages, would eat it, so these two may not be left alone together.


Let $M$, $W$, $G$ and $C$ denote the man, wolf, goat and cabbages respectively.

Let them be crossing from $A$ to $B$.

As follows:


Solution 1

$M$ takes $G$ from $A$ to $B$.
$M$ returns to $A$.
$M$ takes $W$ from $A$ to $B$.
$M$ returns to $A$ with $G$.
$M$ takes $C$ from $A$ to $B$.
$M$ returns to $A$.
$M$ takes $G$ from $A$ to $B$.
"By this procedure there would be some healthy rowing, but no lacerating catastrophe."


Solution 2

$M$ takes $G$ from $A$ to $B$.
$M$ returns to $A$.
$M$ takes $C$ from $A$ to $B$.
$M$ returns to $A$ with $G$.
$M$ takes $W$ from $A$ to $B$.
$M$ returns to $A$.
$M$ takes $G$ from $A$ to $B$.


Also presented as

The objects being transported can be seen presented in various ways, for example:

fox, duck and grain
African leopard, goat and cassava leaves

and so on.


$4$ or more objects may be transported in this way, each being incompatible with its two neighbours, but in this case one of the objects needs to be able to row.


Sources

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