How do I complete the sultan's challenge?

Before you do anything, take careful note of the Vizier's clothes.

The colors are very important.

A nice massage might relax you.

So sit down.

Watch the girls closely while you have the chance.

To go anywhere, you'll need a cover.

Veils make a nice cover.

Wait until the girls all leave and take one.

Of course, to get anywhere, you'll need the right one.

Do you think there's a connection between the Vizier's clothes and
the body parts each of the girls?

Of course.  The unfaithful wife's favorite body part (the one she
massages) is covered with clothing of her color on the Vizier.

So take her veil and go downstairs.

You might want to take those figs, just in case.

Now you'll need to get the Vizier to meet the proper wife.  With
what you have from the palace, how can you signal him?

There's not much.  Give him a fig.

Now it's time for the "meeting."  Head up to the proper bedroom.

The Vizier isn't going to do anything in your presence.