Ten. Though, by chance, she might
have taken down four matching plates, consider the possibilities
if the queen had taken down the following
number of plates:
4 might have resulted in 2, 1, 1
5 might have resulted in 2, 2, 1
6 might have resulted in 2, 2, 2
7 might have resulted in 3, 2, 2
8 might have resulted in 3, 3, 2
9 might have resulted in 3, 3, 3
Only with ten, would it be inevitable that she would
have at least four of any one pattern: 3, 3, 4.
In order to ensure matching plates, the queen had to
bring down three extra plates, one for each pattern, for
each person more than two.
The formula: N + 1 + N(X) (with N representing the
number of patterns or types and X representing the
number of people more than two).
The queen had three patterns of plates and so N = 3. In
addition to the king and queen, there were two guests,
and so X = 2. 3 + 1 + 3(2) = 10.
Try it and you will find that for three persons the queen
would have needed to take down seven plates, and for five
she would have had to take down 13 plates.
If her children had not been out ogre-fighting, the royal
queen might have had to mix and match!