### 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?

### 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:

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The bar balances with a weight of 4 units on one side and weights of 2 units and 3 units on the other.

The 2 is on the inside and the 3 is on the outside. The 3 weight and the 4 weight are exactly the same distance from the pivot, and the 2 weight is half way between the 3 weight and the pivot.

Can you find an arrangement with the 4 staying exactly where it is, balanced again by the 2 and 3, but this time with the 3 on the inside and the 2 on the outside?

There is more than one way to do that, so first find at least two arrangements that work, then try to describe, using algebra, the connection between the 2 and 3 positions.

You can use the lines in the background to help you gauge distances when you are looking for the relationship between the positions of the 2 and 3 unit weights.