### Square Mean

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

### AMGM

Can you use the diagram to prove the AM-GM inequality?

### Cubic Rotations

There are thirteen axes of rotational symmetry of a unit cube. Describe them all. What is the average length of the parts of the axes of symmetry which lie inside the cube?

# The Mean Problem

##### Stage: 4 Challenge Level:

A lot of solutions were sent in for this which was excellent - keep them coming!

Some told us the answer, and even showed that it worked, but didn't tell us how to find an answer in the first place.

Some solutions used algebra and systems of equations, but the best solutions were neat and direct . . .

. . . like Samantha's here, from Hamlin.

I knew that to find the mean, you add up the numbers, and divide by the quantity of numbers.

If the first two numbers had a mean of four, then they had to add up to eight, because eight divided by two is four.

If the first three numbers had a mean of nine, they had to add up to 27 because 27/3 is 9.

The third number then had to be 19, because 27 - 8 = 19

The sum of the four numbers had to be 60, since 60/4 = 15.

The third number had to be 33 because 60 - 27 = 33.

If one of the numbers is 2, and the sum of the first two numbers is eight, then the second number must be 6.

The answer to the problem is 2, 6, 19, 33.

Easy really! Well done Samantha.