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Question

# Three couples arrive at the bank of a river. Each of the wives is jealous and does not trust her husband when he is with one of the other wives (and perhaps with other people), but not with her. How can six people cross to the other side of the river using a boat that can hold no more than two people so that no husband is alone with a woman other than his wife? Use a graph theory model.

Solution

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Given: Three husbands $H_1,H_2,H_3$ and three wives $W_1,W_2,W_3$

Husband cannot be with other women without his wive.

Boat can only cross at most 2 people (and requires as least one person).

Let $(x,y)$ represents the people on either side of the river: $x$ the left bank and $y$ the right bank. $B$ represents the position of the boat

Initial position:

$(H_1W_1H_2W_2H_3W_3B,\text{none})$

We first cross one couple to the right bank (doesn't matter which one).

$(H_2W_2H_3W_3,H_1W_1B)$

Then the wive goes back to the left bank with the both.

$(W_1H_2W_2H_3W_3B,H_1)$

Next the wive cannot go back with another husband (as his wive would be jealous) and the wive cannot go back with another wive (as that other wive will be jealous as her husband remains with another women on the island) Thus the six people cannot cross.

$\text{\color{#4257b2}Note: There will ALWAYS be a situation where a husband with be left with another wive and thus this exercise simply does not have a solution.}$

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