The Perfect Square Dance!

Oct 20, 2008

RAY: This was from my ballroom dancing series and it was sent in by Dennis Vogel.

Sally invited 17 guests to a dance party at her estate in the Hamptons. She assigned each guest a number from 2 to 18, keeping 1 for herself.

At one point in the evening when everyone was dancing, Sally noticed the sum of each couple's numbers was a perfect square. Everyone was wearing their numbers on their clothing.

The question is, what was the number of Sally's partner?

Here's a reminder: a perfect square is attained by squaring, or multiplying by itself, an integer. So four is a perfect square of two. Nine is a perfect square of three. Sixteen is a perfect square of four. So these numbers are adding up to either 4, 9, 16, 25, etc. And the question is, with the information you have available to you, what's the number of her partner?

RAY: Here's the answer. We know the sum of each pair has to be a perfect square. And we know that 4, 9, 16, and 25 actually are the only perfect squares available to us.

So let's look at dancer 18. Dancer 18 must be paired up with dancer number 7 because that adds up to 25. Dancer 17 must be paired up with dancer number 8. And 16 must be paired up with 9. All those three pairs of dancers have 25 as their total.

So who's left? We have 1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, and 15. Well let's look at our hostess Sally and her possible dance partners. Remember she's number 1. So she could be dancing with number 3, or number 8. Or she could dance with 15.

But she can't dance with 8 because 8's already been taken by 17. If she chooses 3, then dancer number 6 must pair up with 10. And now there's no one left for dancer number 15, because 15's only choice would have been either 1 or 10.

So the only way it's going to work is that Sally has to dance with 15. 10 gets paired with 6, 13 with 3, and then you can figure out the rest.

TOM: Wow.

RAY: Who's our winner this week?

TOM: The winner this week is Jerry Darmond from Madison, Wisconsin, and for having his answer selected at random from among all the correct answers that we got, Jerry's going to get a $26 gift certificate to the Shameless Commerce Division at, with which he can get our new book, Ask Click and Clack.

RAY: Congratulations, Jerry!

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