"Gedanken" Experiment

• Scenario:
  • You are going to a train station.
  • The station has 16 tracks.
  • You have no idea which track your train is leaving from.
  • With five minutes to spare, you turn to the ``Information'' booth.
  • There is a monosyllabic attendant in the booth who only answers yes-or-no questions.
  • Furthermore, he does not answer more than one question per minute.
• Question: Will you make your train?
• Problem: Think of the shortest (on average) sequence of yes-or-no questions that allows you to find your track.

• Solution:
  • Assume the train leaves from track 9.
  • The following sequence of questions yields the answer with 4 questions:
    • Is the track number greater than 8? -- Answer: yes
    • Is the track number greater than 12? -- Answer: no
    • Is the track number greater than 10? -- Answer: no
    • Is the track number equal to 9 -- Answer: yes
• Notice:
  • The process of constructing questions is very similar to A/D conversion.
  • If we set yes=1 and no=0, then we could represent the sequence of answers as 1001.
  • This is equal to the binary coded decimal representation of the number 9.
  • This strategy will always find the track with 4 questions.
  • This is because 2⁴=16 or $\log_2 16 = 4$.