Problem 1:

For each part, 1 pt for correct answer, 0.5 pts for reasonable
attempt.  If they have not given explanations that is fine.

Problem 2:

For each part, 2 pts for correct answer, 1pt for partially correct 
reasonable attempt.

Problem 3:

For each part, 2 pts for correct answer. 1pt for partially correct or
reasonable incorrect attempt.

Problem 4:

Split the pts as 3+1.  For first part, 3 for all correct words, 2 if
there is a small error, and 1 for a reasonable attempt but several
mistakes.  For part (b), 1 for correct answer, 0.5 for close answer.

Problem 5:

They do not need to prove this part, just demonstrate it.  3 pts for
the calculation.  Full pts, if correct.  Deduct a point if the answer
shows they do not understand orthogonality well.

1 pt for the second part.  0.5 pts for partly correct answer.  (Note
this is a question that has multiple correct answers.)

Problem 6:

5 points for each part.  For part(a), deduct a point if any of the
states are incorrect, 1 point if any of the edges have incorrect
labels and 1 point if any of the edges are incorrect.  If the edges
labels are all of length 2 instead of 3, deduct 2 points instead of
the one mentioned above.

Problem 7:

Deduct a point or two if their explanation is unclear.

Problem 8:

For each part, 2 pts for correct answer, 1 for reasonable incorrect attempt.