### Two Cubes

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]

### Rationals Between...

What fractions can you find between the square roots of 65 and 67?

### Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

# Square Mean

##### Age 14 to 16Challenge Level

Well done Freddie from Packwood Haugh School and Danny from Milliken High School, Canada.

Is the mean of the squares of two numbers greater than, or less than, the square of their means? Let the two numbers be $p$ and $q$.

$$\text{Square of mean } = {( p + q )^2\over 4} = {{p^2 + 2pq + q^2}\over 4}$$

$$\text{Mean of squares } = {{p^2 + q^2}\over 2}$$.

$${{p^2 + q^2}\over 2} - {{p^2 + 2pq + q^2}\over 4} = {{p^2 -2pq + q^2}\over 4}= {(p - q)^2\over 4}\geq 0.$$.

Note that this difference, ${(p - q)^2\over 4}$, is zero if $p=q$ and positive for all other choices of $p$ and $q$. So if the numbers are equal then the mean of the squares is equal to the square of the mean. Otherwise the mean of the squares is greater than the square of the mean.