In Search of a Magic Number

Dec 19, 2005

RAY: This next puzzler is mathematical in nature. Imagine two numbers, such that their product is equal to their sum. I'll give you two numbers: 5, and 1 and one-fourth. You add them, and you get 6 and 1/4. You multiply them, and you also get 6 and 1/4.

TOM: You do?

RAY: You do! Now, I'm going to put on one additional condition. Both numbers have to be the same. For example, let's take 3/7ths as the number. 3/7ths and 3/7ths add up to 6/7. But, unfortunately, they multiply to 9/49ths. But if it worked, then, well, that would be the kind of answer I'm looking for.

TOM: So you're looking for whole numbers as well as fractions?

RAY: Any number you can think of. Fractions, irrational numbers, you name it, we'll use it.

Is there such a number that if you multiply it by itself the product is equal to adding it to itself?

If so, what are those numbers?

Of course zeros are not allowed, because everybody knows zero plus zero is equal to zero times zero. Zero is disqualified.
RAY: I've been waiting a long time to use this answer. It only works if the number is two!

This puzzle was the kind of puzzle where you thought you'd be able to come up with some answer like four ninths. But, the truth of the matter is you come up with two equations, which is Y equals X times X, and the other one is Y equals X plus X. If you draw those two graphs, the first one, Y = X squared, is a parabola. The second is a straight line with a slope of two.

TOM: And of course they only intersect at two places: zero and X = two.

RAY: Who's our winner this week?

TOM: Well, the winner this week is Ben Poole from Omaha, Nebraska, and for having his answer selected at random from among all the correct answers that we got, Ben is going to get from us a 26-dollar gift certificate to the Shameless Commerce Division at

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