It's probably too late for general comments, but here are some
specific ones:

Problem 2:

I don't like the (*) notation.  I think it's confusing and vague.  Can
we make it more precise, perhaps by putting the entire contents of the
rule in a box and calling it something (say, procedure A), and then
saying, "applying Procedure A again."

Part b: Again, if we call the two steps either PROCEDURE A1 and
PROCEDURE A2 (for instance, we could write something a bit more clear
by writing something like:

"To apply Procedure A2, the following condition must be true of the
current state S"

I think starting with "Transitions S->S'..." and then saying how it
has to do with the card switch rule is more confusing.

Problem 3:

a. People might be confused here -- what are they allowed to write in
the box?  I mean, they might think that they can simplfiy 2^k i when
it's larger than 53.  I guess that might be a lack of understanding of
congruence, but I can see students thinking too hard about this one.

b. I like this problem.  But maybe a hint is in order -- "What is the
position of an arbitrary card after 52 shuffles?"  I'm almost certain
some students will absolutely bomb this problem.

The other problems seem just fine to me.