How to win University Challenge

In the final of University Challenge, Jeremy Paxman asked the following question:

If the difference between two positive numbers is 5 and the difference between their squares is 55, what is the sum of the two numbers?

If you know the formula for the difference of two squares, this question is easy. Indeed, you can answer it without needing to work out (or even knowing) what the two numbers are.

The difference of two squares formula is hugely useful and often arises, yet in my experience students rarely know it. Alex Guttenplan, the captain of Emmanuel College, Cambridge – the winning team – certainly did when he gave the correct answer.

What is the sum of the two numbers? And how did you work it out? What are the two numbers themselves?

Comments

[TQ]

I watched Alex answer this question on UC almost without missing a beat - after all, he had to wait for this longish question to unfold. Yet I was astonished to find myself there with him as he answered, although I'm no mathematician.

Once JP had said "the difference between their squares is 55" I immediately thought, what's first square number above 55? It is 64, whose root is 8. Subtracting 55 from the square 64 produced a difference of nine - which itself is a square, with root 3. Add 8+3 = 11.

Of course, a second path to the answer could have occurred at my "whose root is 8" moment. Frankly I hadn't absorbed the first piece of information, that "the difference between two positive numbers is 5" because that would have simply prompted the subtraction of five from eight to produce 3 as the second starting number. Ergo, 8+3 = 11.

Now you show me an equation for reaching the same result!

[G]

a^2 - b^2 = 55
a - b = 5

a + b = (a^2 - b^2)/(a-b) = 55/5 = 11

After that working out the numbers themselves is trivial. A pair of numbers whose difference is 5 and sum is 11. Got to be 8 and 3.