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I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

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Picture Story

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Long Short

Age 14 to 16 Challenge Level:

Why do this problem?

This problem is all about geometrical reasoning and proof. It gives learners the opportunity to play with a problem and come up with a conjecture from practical demonstrations, and then justify their findings. It also opens up discussion of when it is necessary to write something as a strict inequality.

Possible approach

This problem lends itself to investigation with a dynamic geometry tool such as Geogebra.
The learners could start by constructing a quadrilateral on a unit circle, and displaying the lengths of each side. The challenge is to make the shortest side as long as possible. Once the length has been found, learners need to come up with a convincing argument that it is not possible for the shortest side to be any longer.
Learners could then discuss in pairs the best way of making the second side as large as possible. What happens to the shortest side in this process? The question says "Side $b$ must be less than a certain value" so there is the opportunity to discuss why it has to be strictly less than that value and can never actually reach it.
The last part of the question encourages discussion along similar lines. It is good to bring out of the discussion that the third side can be as close to $2$ units as you wish but can never actually be exactly $2$.

Key questions

If I move the points on the circumference to increase one of the sides, what will happen to the adjacent side?
What shape should I make in order to make the shortest side as long as possible?

Possible extension

The same problem could be tackled as a hexagon on the circumference of a circle, rather than a quadrilateral.

Possible support

Start by considering a triangle on the circumference of a circle and calculate the maximum and minimum side lengths.