Earlier today I have set these two puzzles for you. Here they are again with answers.

**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 another shop at a price of 5 for £1.

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

Note: please do not fall into the trap.

**Solution**

This is one of the problems when you are deliberately led to give the wrong answer. At first one might think that the average number of apples is 4 for a £1, but this is wrong in the scenario described.

Imagine Adam buys 15 apples in each store. He would spend £5 in shop 1, and £3 in shop 2. In total he spent £8 and has 30 apples. So for every pound spent, he bought 30/8, or 3.75 apples.

If Adam spent an equal number of pounds per store, the average number of apples per pound spent would be 4. But the question makes it clear that what is equal is the number of apples bought, not the money spent.

**2. I can’t get any satisfaction**

1+ *x* + *x*^{2} = *y*^{2}

Show that there are no positive integers *x *and *y *which satisfies the above equation.

**Solution**

You prove it by contradiction. You also need to remember how to multiply terms in parentheses.

As *x *is then a positive integer *x*^{2} < (*x* + 1)^{2}

The term (*x* + 1)^{2} is equal to (*x* + 1)(*x* + 1), which expands to *x*^{2} + 2*x *+ 1.

Since *x *is a positive integer, we also know that

*x ^{2} < x*

^{2}+

*x*+ 1

*x*

^{2}+ 2

*x*+ 1

From the equation in the question

*x ^{2} 2* <

*x*

^{2}+ 2

*x*+ 1

*x ^{2} 2* <(

*x*+ 1)

^{2}

In other words the square of *y *must be between the squares of *x *and *x *+ 1. So *y *must be a positive integer between *x *and *x *+ 1, which is impossible. QED.

*Thanks to Owen O’Shea, author of* *The Call of Chance: Mathematical Gems, Peculiar Patterns, and More Tales of Numerical Serendipitythe source of today’s mysteries.*

*I’ve been doing a puzzle here on alternate Mondays since 2015. I’m always on the lookout for great puzzles. If you want to suggest one, email me.*