Mutational changes were induced in a single codon in the gene of T4 phage.

also looking at this scratching my noggin, what did you do to obtain those?

Okay, so we already know what Mutant A is UGG, because it was given.  According to the flow chart, MA derived from the normal, AA Serine. Since only one base is changing, the only codon that could mutate from Serine to Tryptophan is UCG.
 For Mutant B, both the normal and Mutant A affect it's possible outcomes.  Giving us, UUG. (The C in the normal, and the G in MA mutate to U giving us Leucine.) Since the flow chart says that is what MB is, we have no other choice. 
Mutant C which is Arginine, from Mutant A, has to be CGG. 
Mutant D, which is Glycine, from Mutant C, has to be GGG. 
Mutant E, which is Valine is from both Mutant B and Mutant D, and this is the one I am not so certain about, but I can only surmise that it is GUG. 

Basically, I just did a process of elimination based on the information I had.  Each three base codon mutated only one base.  Since we know what Amino Acid it mutated to, we need only find which codons represent which AAs.
Would love it if someone could validate my process, or tell me I am a moron, and explain what is really going on. 

Why is tryptophan a mutant if UGG is the actual code that represents it?

Because is was, in its normal form Serine, and mutated to Tryptophan. 

UCG(Normal)--->UGG(Mutant A).

Okay... So UCG (normal) to UGG (mutant A) to CGG (mutant C) to GGG (mutant D)?

That's what I have.  Accuracy is debatable...lol.

So you're right, it all comes down to guessing and checking which ones would fit. This isn't science lol

Couldn't Mutant C be AGG OR CGG?
If not, please let me know why.

Yes, it can.  Thanks for pointing that out.  The rest of the answers are correct, per the instructor.

is this question from the UC Berkley online genetics class?

yes