# Russian Cubes

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

## Problem

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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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James (using blue) and Cath (using red) paint the faces of a cube in turn so that the six faces are painted in order "blue then red then blue then red then blue then red".

Having finished one cube, they begin to paint the next one.

Show that, even though she always goes second, Cath can choose the faces she paints in a way that will ensure that both cubes are identical.

## Student Solutions

Well done to Alex from Stoke-on-Trent Sixth Form College, and Anurag from Queen Elizabeth's Grammar School, Horncastle, for their solutions to this problem.

Here is Alex's explanation for the first part of the problem:

Consider one of the sides painted blue, and rotate the cube so that this side is on top. Out of the four sides that touch it (and are rotationally symmetrical), at most three of those are red because there are three red and three blue, therefore at least one of these is blue. Rotate the cube so that the top and a selected blue side is facing. All combinations must be rotatable to this position, so considering the uncoloured sides is sufficient to deduce the number of combinations. Out of the four remaining sides, one must be coloured blue and the other three coloured red. This reduces the problem to four possible combinations, each found by colouring a different remaining face blue. Two of these result in three coloured faces that share a vertex, and the other two form a lines of colour that bends around the cube with another line of the other colour in the same shape. Both of these are the same if the colours are swapped over, so there are exactly two combinations.

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This diagram shows the two possibilities. The left cube has three blue faces meeting at a vertex (and three red faces on the other side of the cube). The right cube has blue faces on the front, top, and back of the cube.

Anurag explained very clearly how Cath can ensure the two cubes are the same:

James has only one possible move first. This is because the cube could be rotated so the bottom cube is always blue. If Cath now wants the shared vertex cube, she colours the opposite side to whichever James paints. This will always produce the desired result as, in this cube, no opposite sides are the same colour. By painting the opposite side red, Cath removes James's ability to colour it blue, thus forcing him to paint a cube where similar sides meet at a vertex.

However, Cath cannot force a cube in which similar sides are in a strip. In a strip there must be two similar sides opposite each other (they share no vertices, and so whatever the third side is, a strip is formed). Cath must avoid painting any side opposite to a blue. This is so, on James's last move, his only two options are the only sides opposite his other, blue coloured side. So, on her first move, Cath must paint a side adjacent to the side James painted blue side. But if James now paints the sides opposite to Cath's, Cath cannot make two of her reds opposite each other, which she too must need. Which means Cath cannot force the second type of cube, so to ensure both were the same she would have to make sure the first cube was the shared vertex type.

## Teachers' Resources

### Why do this problem?

This
problem invites learners to consider a familiar geometrical
setting (the cube) and think more deeply about how you can utilise
its properties to analyse a problem, and develop and share their
ideas and deductions.

### Possible approach

Invite the learners to colour nets of cubes (using just two
colours) so that they can be made into different cubes. Ask them
how they decide whether two cubes are identical and allow time for
the group to investigate. This process could be shortened by using
linking squares of two colours.

Allow groups to share their findings and encourage others to
challenge them so that further reflection can take place.

- How do they know they have them all?

The surprising result that there are just two, helps with the
final part of the problem and here you might wish to encourage the
sharing of different approaches to providing a convincing argument.
Learners may chose to do this practically (several cubes each
partially painted to show the possibilities at each stage) or more
theoretically by defining sides and edges and talking about the
freedoms and constraints arising at each stage.

### Key questions

How do you know you have all the possibilties?

### Possible extension

Consider using three colours instead of two. How many more
possibilities does that open up?

The problem Proximity
might be tackled next.

### Possible support

Making nets and creating different cubes may be essential for
learners to be able to viusalise and remove redundant examples.
Focus on the discussions around what you could do to one cube to
make it look the same as another before actually doing
it.