# The Perfect Square Dance!

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 cartalk.com, with which he can get our new book, Ask Click and Clack.

RAY: Congratulations, Jerry!

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