### 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 56 and 58?

### Square Mean

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

# Inside Outside

##### Stage: 4 Challenge Level:

A good start by Jia and Jeremy from Raffles Institution, Singapore.

To balance a bar balance, equal weights are required on each side of the pivot.

Taking $a$ as the distance of the $3$ weight from the pivot, and $b$ as the distance of the $2$ weight from the pivot. And using each column from the pivot as a distance of one, the weight on the left of the pivot has a moment of $40$.

So $3a+2b$ must equal (that is, balance) $40$.

Matching pairs of $a$ and $b$ could be:

$$a = 5 \quad\mbox{and}\quad b = 12.5$$
or
$$a = 4 \quad\mbox{and}\quad b = 14\;.$$

And some good algebra reasoning from Joan in Edinburgh.

If we want $a$ less than $b$ it would be good to find where $a = b$, that means that the two weights are together at the same place.

If $a = b$ then $a$ and $b$ have to be $8$ (from $5a = 40$ or $5b = 40$ whichever you prefer).

But we want $a$ less than $b$, so a can't be more than $8$.
The $3$ weight can go as near to the pivot as we like and $b$ will just have to get bigger to keep the balance. If the $3$ weight does go to zero $b$ will have to be $20$.

$20$ is the furthest out from the pivot that the $2$ weight goes if it keeps balanced with the $4$ weight on the other side.