# Royal Dinner

To enlist the help of the other kingdoms, the king talked to the queen about inviting neighboring royalty to dinner. This put the queen into a royal snit. Theirs was not a very wealthy kingdom and the royal dinnerware was in a disgraceful condition. Apart from ordinary dishes for everyday use, all that the royal pantry contained were a few dinner plates of three different

dinner patterns:

1. five silver ones with birds

2. six crystal with seashells

3. seven gold with the royal crest

They were all stored in disarray on a very dark top shelf of the royal pantry. Only those would be suitable for entertaining other royalty.

If the queen didn’t want to climb up to the top shelf twice, how many dinner plates would she have to take down to be sure she had matching dinner plates for herself, her royal spouse, and for the neighbouring king and queen?

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!

### Your Answer

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