Can you solve it? How do you like them apples?

Matt Damon in Good Will Hunting (1997)
Photograph: Miramax/Sportsphoto/Allstar

Here’s a curious fact taken from The Call of Coincidence, a book by Owen O’Shea about serendipity in maths.

The triangle with sides 45, 97 and 56 has an area of √459.756.

Bazinga!

And here are two puzzles taken from the same tome.

1. Adam’s apples

Adam buys a number of apples in one shop at the rate of 3 for £1.

He then buys the same number of apples in a different shop at a rate of 5 for £1.

What was the average number of apples bought for each pound spent?

Note: please don’t fall into the trap.

2. I can’t get no satisfaction

1 + x + x² = y²

Show that there are no positive whole numbers x and y that satisfy the above equation.

I’ll be back at 5pm UK with the answers. PLEASE NO SPOILERS. Please discuss your favourite mathematical coincidences.

UPDATE: Read the answers here.

Thanks to Owen O’Shea, author of The Call of Coincidence: Mathematical Gems, Peculiar Patterns and more Stories of Numerical Serendipity

I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.