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Shuffle Shriek

Stage: 3 Challenge Level: Challenge Level:1

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

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:

Shuffles of order 2 that swap two balls
Shuffles of order 2 that swap two pairs of balls
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.
Shuffles of order 3
Finally, he found the shuffles of order 4.
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!