Professor Kafka tells proudly:

  • Today I gave a fascinating lecture.
  • There were five students sitting in the first row of the lecture room, and each of them was awake as I started my lecture.
  • Each of the five fell asleep exactly twice during my lecture.
  • Furthermore, every pair of these five students was simultaneously asleep at some point in time.
  • At any moment in time, at least three of these five students were awake and listening closely to my lecture.

Is it possible that professor Kafka remembers the situation correctly?

  • 4
    $\begingroup$ No, Kafka does not remember the situation correctly. Students would not have fallen asleep if his lecture were actually fascinating. $\endgroup$
    – Kevin
    Jan 21, 2015 at 19:52

2 Answers 2


Professor Kafka is

probably wrong


We have 10 pairs of students. Each of these pairs must be sleeping separately, or we would fail the 'at least 3 awake at once' condition. We require a student to fall asleep to change the sleeping pair, and 5x2 students allows us 10 state changes. Unfortunately, to get from our presumed (hence the 'probably' above) initial state of 'all students awake' to the first pair we require 2 students to fall asleep, meaning we can only fulfill 9 pairs in total. If a student may start the lecture asleep the solution is trivial.

  • 2
    $\begingroup$ NVM - you are right, it is trivial - AB, BC, CD, DE, EA, AC, CE, EB, BD, DA. All fall asleep twice, but A is asleep three times, starting the lecture asleep. $\endgroup$
    – Trenin
    Jan 20, 2015 at 15:21

Professor Kafka is probably wrong because "every pair of these five students was simultaneously asleep at some point in time" is impossible if "at any moment in time there were at least three of these students awake and listening to my lecture."


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