# Does it make sense?

## Example 3

## Activity 8

Look carefully at Example 3 and try to make some sense of it. Notice that there are four bullet points: try understanding each of these points separately, but don't worry if you find it difficult to follow. However, do make a note of the point when it becomes very difficult to understand.

No doubt you will have found parts of this mathematical writing puzzling. In some places, the writer does not make clear exactly what he means, so as a piece of mathematical communication it does not work. However, the author was not intending to communicate with anyone other than himself. It was a means for him to record his solution to a problem he had set himself and as such it may have been useful and successful.

An explanation of the mathematics goes as follows:

The first bullet point tackles the problem of what is meant by a ‘solution’. Imagine all seven pieces have been combined to make a cube. Now imagine taking a single piece out and seeing whether that piece can be re-inserted in any different ways. The first piece pictured can be positioned in only one way, whereas the piece next to it could be turned round and put back upside down. This is to do with the symmetry of the piece (see ‘*Preparatory Resource Book B*’, Module 7). The number of ways each piece can be inserted in any cube has been noted: 1, 2, 2, 2, 2, 2 and 3. These numbers are then multiplied to give 96, the total number of different ways of putting the pieces together to form any single cubic arrangement.

The second bullet point tackles the problem of how to record a particular cubic arrangement. The three-dimensional picture of the cube is easy to visualize but difficult to draw. It also does not record the position of pieces on the far side of the cube. The author is experimenting with a notation where the three horizontal layers of the cube are drawn side by side but there are then problems visualizing the individual pieces.

In the third bullet point the author realizes that any complete cubic arrangement of the seven pieces can be turned around in various ways (because of the symmetry of the cube). He reckons there are 24 ways of positioning each complete cubic arrangement.

Finally he begins to record some of what he thought originally were different solutions. Alongside the drawings he has written ×4, ×2, ×1, ×2, presumably indicating that he thinks there are 4, 2, 1 and 2 similar cubic arrangements to those illustrated but it is not clear from the writing what these are. The 1+4+2+2+1=10 may be the total of these cubic arrangements (including 1 from the second bullet point). He then combines the 96 ways of putting the pieces together, with the 24 ways of positioning each complete cubic arrangement and multiplies by 10 the number of cubic arrangements. Counting like this, he seems to have found about 2,400 ways of completing the Soma Cube. Only another 14,000 or so to go!