Coloured Cubes

Task 185 ... Years 2 - 10

Summary

In recent mathematics history, professional mathematicians have worked on this problem. So it is in our collection because it confirms that a mathematician's work begins with an interesting problem and because it is easy for students to understand what is required and to get started. However, it is not easy to solve and it is perfectly all right to leave the problem without completing it. It can always be revisited. The task uses four cubes, each with its six faces coloured using four colours in a unique way. The challenge is to line them up, or make a tower, so that the four different colours appear on each side of the tower.

Materials

• Four cubes with faces coloured as shown on the card

Content

• concept of proof
• graph theory
• history of mathematics
• reasoning
• spatial perception, 2D or 3D

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

With a history stretching back to at least the early 1900s, this task became famous in the 1960s and 70s as a commercial puzzle, called Instant Insanity, made from four plastic cubes. Everyone - even yours truly - had one in the same way as a generation later everyone had a Rubik's Cube. According to Wikipedia, over 12 million of these coloured cube puzzles were sold. In earlier generations the equivalent puzzle sold under other names.

Anyone can start this task because it only requires the user to line up the cubes and check whether there are four different colours along each side. It is a little easier if you decide to make a tower rather than a line, because then you can move around the tower to check the colours, rather than trying to roll the cubes over to see if the ones on the bottom work. Rearranging the four cubes in a genuinely random way has some chance of producing a correct result. In fact, again according to Wikipedia, 2 chances out of 41,472! (Other mathematicians argue for 82,944 ways of stacking the cubes.)

A little reasoning can help to improve the chances:

• There are 24 faces, but only 16 of them that matter - the ones that show along the sides.
• So, 8 faces have to be 'removed' to find a solution.
• The faces used must be four of each colour - one for each side.
• Checking the colouring on the nets shows that there are 7 red (so 3 have to be removed), 6 blue (so 2 have to be removed), 6 green (so 2 have to be removed) and 5 yellow (so 1 has to be removed).
• Faces can be removed by using them on the outer ends or 'hiding' them where faces touch.
• Opposite faces are either both removed or both used.
Since 3 reds have to be removed, it might be sensible to begin with the cube that has three red faces and use one of those on an end. This can be done so that a red is also showing on each of two sides, or, because two of the reds are opposite, it can be done so that one is at the outer end and the other is ready to be hidden when touched by the next cube.

Now we have a starting point, well two starting points actually. It might not be the right one - we might have to work our way through all the possibilities from here to discover that neither start will work and then come back and begin with a different cube - but it is worth trying.

From here it takes organised trialing of possibilities and, most importantly, developing a system for recording what has been trialled. It is not necessary, perhaps not even possible, to solve this problem in one sitting, therefore journal recording is going to be important to prevent unnecessarily repeating effort when the problem is revisited.

When it becomes too frustrating for students there is one more clue to solution supplied on the card. It will lead to an answer, but be careful about giving it away, because doing so means the students haven't really solved the problem themselves.

The nets on the card are in the correct order for a solution.
It still takes a certain amount of spatial perception to discover the solution from here. However, once again, let's confirm that finding the solution to this problem is the work of mathematicians. F de Carteblanche presented a solution to The Coloured Cubes Problem in Eureka 9 (April 1947), pages 9 - 11. Eureka is The Archimedeans' journal and The Archimedeans is the name of The Cambridge University Mathematical Society. The Society was established in 1935 and Eureka, which first appeared in 1939, is still published. The Archimedeans web site is https://www.archim.org.uk/. (Thanks to Ron Haller-Williams, life member, The Archimedeans, for corrections to the original text of this paragraph.)

Interestingly, F de Carteblanche, is likely to be a pseudonym for a group of mathematicians, possibly even students. In the same issue of Eureka there is a paper titled The Three-colour Problem by Blanche Descartes and that name is known to be a pseudonym for the mathematicians R. Leonard Brooks, Arthur Harold Stone, Cedric Smith and W. T. Tuttle who met as Cambridge undergraduates in 1935.

In more recent times (1999), Associate Professor Josep M. Basart Muņoz, Department of Information and Communications Engineering at Barcelona University has generalised the problem and published, with P. Guitart, a solution for any number of cubes and colours. His proof begins:

Let C = {c1, c2, c3, ..., cq} be a set of q cubes in which every face in each one of them has been coloured using one colour in a set K = {k1, k2, k3, ..., kq} of q colours. The problem we are dealing with is to raise, if it is possible, a pile with the q cubes in such a way that every one of the q colours will appear once in every one of the four sides of the pile. This problem was introduced and solved by De Carteblanche in [1} for the case q = 4. We aim to follow the same analysis of the problem in the case q 'greater than or equal to' 4, and ...
Graph Theory is the tool used to solve the four coloured cubes problem and Gresham College, London, has mounted a video (5:42) on YouTube of Professor Robin Wilson explaining this approach to students. Senior students who have been trying to solve Coloured Cubes would benefit from viewing this short video. But a mathematician asks, Can I check it another way? and Professor Wilson is featured in another video (5:07) finding a solution to the same problem using number theory which begins by simply assigning a number to each colour.

 (This process was also used in Markus Bucher's work developing Task 112, Coloured Squares. Markus is a teacher in Tasmania.) Both solutions rely on the principles listed above and students often realise many of them as they play with the problem. (Another of Professor Wilson's publications might also be of interest to students. Lewis Carroll was, of course, Alice in Wonderland's 'godfather'. The book is available in most university and major public libraries and is generally reviewed as very readable. It has more meaning the more mathematics you know.)

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

Coloured Cubes is a great investigation to link with Task 31, Cube Nets. Each table of four needs a collection of 3d Geoshape squares, or the smaller MiniGeofix, in four colours (the same four colours across the class). The nets of the coloured cubes are displayed - a quick sketch on the board using letters for colours will achieve that - and each person in the group makes one of the cubes. Now set the challenge.

It is possible to stack your cubes so all four colours show on each side of the stack. Let's see which is the first group to achieve this.
It is not likely that any group will achieve it, but it will begin discussion which brings out student observations of some of the key points. List these on the board as they develop and invite groups to continue working using new information when it is listed. The aim of the lesson is to highlight the work of a mathematician, so its conclusion will be to review the lesson in the light of the Working Mathematically process. The solution of the problem plays a lesser role and there is always the possibility of solving it using the clue on the card. Teachers might like to use the references above to develop to highlight that the professionals do use the same key points to solve the problem.

A second way to begin the lesson is to arrange for groups to construct a paper version of each cube from the nets.

At this stage, Coloured Cubes does not have a matching lesson on Maths300.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

Coloured Cubes is not in any MWA kit. However it can be used to enrich the Space & Logic kit at Years 5/6 and the Space & Logic kit at 9/10.