Enigmatic Code

Programming Enigma Puzzles

Enigma 1622: Chances for a business meeting

From New Scientist #2787, 20th November 2010 [link]

Jack will enter the office today at a random time between 2pm and 3pm. He will stay in the office for 15 minutes. John has a similar schedule, with the entry time chosen independently.

They have a business to transact and it requires that they are together in the office for 5 minutes before 3pm.

What are the chances that the business is transacted today?

[enigma1622]

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2 responses to “Enigma 1622: Chances for a business meeting

  1. jimrandell 20 December 2011 at 11:07 pm

    This doesn’t require any programming. By drawing a graph we can determine that the time when it is possible to have a successful meeting is when 0 < x < 55/60, 0 < y < 55/60, |x-y| < 10/60, which is the diagonal hexagonal area in the graph.

    This has area (55/60)² – 2 × 1/2 x 45/60 × 45/60 (the area of the whole square less the two triangles on the NW and SE corners).

    Solution: The chances of business being transacted today is 5/18.

    • jimrandell 20 December 2011 at 11:22 pm

      The following code runs a simulation of the problem. It runs a million random trials by default (in around 780ms) and gets an answer of around 0.278, close to the expected answer of 0.2777…

      import sys
      
      N = eval(sys.argv[1]) if len(sys.argv) > 1 else 1e6
      
      from itertools import count
      from random import random
      
      m55 = 55.0 / 60.0
      m10 = 10.0 / 60.0
      
      s = 0 # number of successes
      for n in count(1):
      
        j1 = random()
        j2 = random()
      
        if j1 < j2:
          if j2 < m55 and j2 < j1 + m10:
            s += 1
        else:
          if j1 < m55 and j1 < j2 + m10:
            s += 1
      
        if n == N:
          print("{f} [{s}/{n}]".format(f = float(s) / float(n), s=s, n=n))
          break
      

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