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.]
What fractions can you find between the square roots of 65 and 67?
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
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
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.