****Problem 1.

(a)
In the recursive definition, we say a string "is a BMU".
But after that, we say  a string "is in BMU" and use BMU as the set.  


(b)
Seems that the concatenation operator \cdot is not defined in the problem.


Justifications:

A bit concerned that it's too easy to obtain the solution by elimination.
I'd suggest adding these:

structural induction hypothesis for t'
definition of P(t')


****Problem 2

(b)

"... to be a transition corresponding to removing two cards from the deck"

Maybe we can say

"... to be a transition corresponding to removing two cards from the deck and 
reshuffling"  

Or we can label the two cases and say that it's the second case.


****Problem 3

(b)
I wonder if we need to give the number "52" in the question...
Since this is a 15 points part, would it be reasonable if omit
giving away "52"?

Hmm... it's because the question will be too difficult if we ask 
for a proof that 52 is the minimum number of perfect suffles? 

****Problem 4
The sentence:

"Thus, if Sammy's interest rate is what he considers a reasonable 5% per day,
then p=1.05."

I'd like to suggest something like this instead...

"For example, if Sammy chooses an interest rate of 5% per day, then $p=1.05$"

I'm a bit concerned that some students might actually use the number 1.05
in their solutions.

We can also consdier writing
"For example, if Sammy chooses an interest rate of r% per day, then $p=1+r/100$"
But, this will let the students use the varible $r$ in their solutions,
which is somewhat undesirable to.

(c)

Not sure what's meaning of "without ellipsis (three dots)"...
Since we already say closed form gets full credit, can we omit this?


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I'm also concerned if problem 4 would be too easy? 
Seems to be the easiest problem (to me) :P
(a) and (b) are almost give-aways..

Two ideas for making this harder:

1.
Instead of adding $f$ dollars every evening, we can add $k$ dollars
in the evening on the $k$-th day. So 1 dollars first day, 2 dollars second
day, and so on...

The formula will beocome mp^d + \sum kp^k . 
The closed form is more complicated. But it's seen in the problem set.

2.
Well, k%  interest rate on the k-th day.
So p=1.01 on day 1, 1.02 on day 2, 1.03 on day 3...
Formula: (without f)

  m \prod_k^d (1+k/100)   which is around   m e^{d(d+1)/200}   for small d

Well, probably asking too much then :P