### Tennis

A tennis ball is served from directly above the baseline (assume the ball travels in a straight line). What is the minimum height that the ball can be hit at to ensure it lands in the service area?

### Nine Colours

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

### Take Ten

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3 cube?

# Soma - So Good

##### Stage: 3 Challenge Level:

 Static, printable figure Click and drag on this figure to move it around!

It is said that a Danish mathematician, Piet Hein, posed the following problem to himself during a particularly boring lecture in 1929. Without pencil and paper, he was able to find all seven pieces. He is credited as being the inventor of the puzzle that became known as the SOMA cube, the 7 pieces being known as the SOMA set.

Can you solve the following puzzle in your mind, by visualizing the pieces? It might be easier if you close your eyes and get someone to read out the instructions - fairly slowly. See how far you can get using your imagination.

"Imagine you have 27 small cubes which will stick to each other or fit together in some way, something like Multilink. Fit 3 of the cubes together to make a three-dimensional 'L' shape. Now, make another 6 'L' shaped solids, each made of 3 cubes.

Are all of your 'L' shaped solids the same, even if you imagine turning them around or over?

You now have 6 single cubes unused and 7 'L' shaped solids. Imagine putting one of the 'L' shaped solids to one side and working with the other 6. Add a single cube to each of these 6 'L' shapes to make 6 different solids - be careful, two of them are mirror images of each other; we count them as different because it's impossible to turn one over or around to get the other.

You should now have a mental picture of 7 pieces - 1 made of 3 cubes and 6 made of 4 cubes, with no cubes left over we hope!

Now comes the BIG ONE! (I am not sure that Piet Hein did this bit!) - Can you mentally fit these 7 pieces together to make a cube?

Can you do it in more than one way?"

If you gave up imagining after the first paragraph don't worry - it is more difficult for some people than others. However, you can improve your skills of imagination with practice.

Of course, it would probably be much more fun actually to make the pieces and try to find as many ways as possible of fitting them together to make a cube. There are 240 solutions, excluding reflections and rotations! Have a go.

You might want to draw the 7 SOMA pieces on triangular dotty (isometric) paper. You can see the seven pieces here . Experiment with the 7 SOMA pieces to see what other 3-D shapes you can make. Here are two examples to start you off.

There is a set of 60 variations available from the " Binary Arts Corporation " in their "block by block" puzzle, which includes the seven SOMA pieces and a pack of cards with hints.