How do I complete the sultan's challenge?

Hint 1 of 20:

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

Hint 2 of 20:

The colors are very important.

Hint 3 of 20:

A nice massage might relax you.

Hint 4 of 20:

So sit down.

Hint 5 of 20:

Watch the girls closely while you have the chance.

Hint 6 of 20:

To go anywhere, you'll need a cover.

Hint 7 of 20:

Veils make a nice cover.

Hint 8 of 20:

Wait until the girls all leave and take one.

Hint 9 of 20:

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

Hint 10 of 20:

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

Hint 11 of 20:

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