### Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

### Permute It

Take the numbers 1, 2, 3, 4 and 5 and imagine them written down in every possible order to give 5 digit numbers. Find the sum of the resulting numbers.

### Flagging

How many tricolour flags are possible with 5 available colours such that two adjacent stripes must NOT be the same colour. What about 256 colours?

# Shuffle Shriek

##### Stage: 3 Challenge Level:

Robin sent us some pictures of his work on this problem. First of all, he found the only shuffle of order 1:

Then he found the shuffles of order 2. He noticed that there were two types, those that just swap two balls, and those that swap two pairs of balls. Here they are, together with the result of doing each of them twice so you can see that they have order 2:

Can you see how he was systematic, so we know that he found them all?
Next, he found the shuffles of order 3. Again, he was careful to make sure that he'd found them all.
Finally, he found the shuffles of order 4.
Robin counted these. There was 1 shuffle of order 1, then 9 of order 2, 8 of order 3 and 6 of order 4. That makes a total of 24 shuffles with four balls. Here's what Robin said:
I know that these must be all of the shuffles, because I know that there are 24 to find. That's because each different shuffle puts the balls in a different order. There are four possibilities for the first ball (because it could be any of them), then three for the second ball (because I've already picked one), then two for the third one and then the fourth ball is fixed, so there are 4 x 3 x 2 x 1 = 24 possible shuffles.
Good work, Robin!