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!