Shouldn't the friend also communicate the time he actually finished the exam (which can be different depending on the difficulty of the questions)? In other words, it took him 23 minutes to answer, left after 53, which means that the information about the answers is encoded in "30" (53-23). If it would have taken him 17 min to solve the questions, he would have left after 17+30=47 min.
This also means that the more difficult the questions, the less information he can provide (because the "time" allocated to coding the answers is smaller (as in smaller keyspace)
This also means that the more difficult the questions, the less information he can provide (because the "time" allocated to coding the answers is smaller (as in smaller keyspace)