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Challenge 21: Six Villages

  1. Nov 3, 2014 #1
    There are six villages along the coast of the only perfectly round island in the known universe. The villages are evenly distributed along the coastline so that the distance between any two neighboring coastal villages is always the same. There is an absolutely straight path through the jungle connecting every pair of villages. These paths create thirteen crossings in the interior of the island, one of which is in the middle of the island where paths from every village meet.

    The island has a strange courtship custom. Before a father will give permission for his daughter to marry, her suitor must bring the father a fish each day until he has traveled by every route from his village to the father's village. The young man only travels along routes where he is always getting closer to his destination. The young man may visit other villages along the way.

    On April first a father's three sons come to tell him of their intent to woe a bride, each from a different village. The brides' villages are the first three villages encountered when traveling clockwise around the island.

    If the sons begin their courtship today and the couples are married on the day following each son's last trip, what are the three wedding dates?

    Bonus Question: If the coastline of the island is ten miles long, how long is the longest route that any of the sons takes to reach their betrothed's village?

    Please make use of the spoiler tag
     
  2. jcsd
  3. Nov 3, 2014 #2

    LCKurtz

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    Surely the father would respond "Woe is me!" My sons have gone astray..."
     
  4. Nov 3, 2014 #3
    Spoiler Tag ??
    Where does one find the tag?

    ssss
     
  5. Nov 3, 2014 #4
    In the editor, click on the + button
     
  6. Nov 3, 2014 #5
    Thank You

    Weddings:April 2nd, 3rd, 5th. Longest trip = 5miles
     
    Last edited: Nov 3, 2014
  7. Nov 3, 2014 #6
    That's not correct.
    the last son will be married on
    April 10th
     
  8. Nov 3, 2014 #7
    You're close on all fronts, but no cigar :)
     
  9. Nov 5, 2014 #8
    I reckon the first village has 5 possible paths, the second has 41 paths and the third village - directly across the island - has 121 paths.
    So the wedding dates are:
    6 April
    12 May
    31 July
     
  10. Nov 5, 2014 #9
    This one is very close
     
  11. Nov 5, 2014 #10
    "Very close"... hmm, please check my working...

    Diagramming the number of ways to get from the start village (red circle with the "1" inside) to each feasible path junction on the way to the destination village (red circle at the bottom). The blue arrows show the permitted direction along each path segment. Possible routes to each junction is counted by summing across all feed arrows from previous junctions

    Path1.png
    Path2.png
    Path3.png
     
  12. Nov 5, 2014 #11
    Joffan good work! Can you get the bonus question?
     
  13. Nov 5, 2014 #12
    I reckon this (along with its mirror image) is the longest path:
    Path long.png
    and the length is D(1+2/√3) = 6.8586 miles
     
  14. Nov 5, 2014 #13
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