Guy solves over 200 CS exam problems in 4 hours

The person's logic is wrong: Clearly

  (0*10*10*)*10*

can generate two ones next to each other (choose * = 0 except for the parenthetical, where you add pairs of 1s).

However the problem also seems completely broken.

  (0*10*10*)*10*

can't generate the string 01, which is clearly in the set of binary strings with an odd number of 1s. It misses the entire set of strings that match 0*1, in fact...