Propositiones ad Acuendos Juvenes/Problems/17 - De Tribus Fratribus Singulas Habentibus Sorores/Solution 2

Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $17$

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?

Solution

The constraint here is that at no time may a man be on the same bank as (or in the boat with) the sister of another, without the brother of that sister being present as well.

Let $M_1$, $M_2$ and $M_3$ denote the men, and $S_1$, $S_2$ and $S_3$ denote their respective sisters.

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

As follows:

$M_1$ and $S_1$ cross from $A$ to $B$.

$M_1$ crosses back alone from $B$ to $A$.

$S_2$ and $S_3$ cross from $A$ to $B$.

$S_1$ crosses back alone from $B$ to $A$.

$M_1$ and $S_1$ cross from $A$ to $B$.

$M_1$ crosses back alone from $B$ to $A$.

$M_2$ and $M_3$ cross from $A$ to $B$.

$S_1$ crosses back alone from $B$ to $A$.

$M_1$ and $S_1$ cross from $A$ to $B$.

$\blacksquare$

Sources

• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): 'Propositions to Sharpen Up the Young': $81$