##### Dec 08, 2008

**RAY:**This puzzler is from my mathematical series. Every two-digit number can be represented as AB, where B is the ones digit and A is the tens digit. Right? So for example the number 43, A is 4 and B is 3.

Imagine then that you took this two-digit number and you squared it, AB x AB, and when you did that the result was a three-digit number, CAB.

Here's the question: What's the value of C? So, for example if AB is 43, CAB might be 943. Of course this is a totally bogus answer, but you get the idea.

So again, what is the value of C, so that AB(squared)= CAB?

Answer:

**RAY:**Here's the answer. Let's start by looking at what we know. For C-A-B and A-B to end with the same digit B, B must be zero, one, five, or six.

Six sixes are 36, five fives are 25. One times one is one. And since C-A-B is less than a thousand, A-B must be less than 32 because 32 squared is one thousand twenty-four.

So there are only nine possibilities then for A-B. And we narrowed it down from 99 to 9. It can be 10, 15, 16, 20, 21, 25, 26, 30, or 31. Those are the only nine possibilities, and we eliminate a bunch of them right away. Ten, twenty, and thirty we eliminate because 10 squared is a hundred, it would have to be 110. Twenty squared is 400 and 30 squared is 900. Fifteen is out because it's 225.

The only one that works is 25. If A-B is 25, then C is 6. 625 is the square of 25. Not bad, huh?

**TOM:**Wow, yeah, that's not bad.

**RAY:**Who's our winner?

**TOM:**The winner this week is Matthew Phelps from San Clemente, California, and for having his answer selected at random from among all the correct answers that we got Matthew's going to get a $26-gift certificate to the Shameless Commerce Division at cartalk.com with which he can get our latest four-CD set featuring four of our all-time favorite shows, The Field Guide to the North American Wacko.

**RAY:**Congratulations, Matthew!