Discover Smith Mountain Lake
WINTER 2015/16
26
PROBLEM:
SOLUTION:
Edited by Gerald Muus
As Assistant Editor of Discover Smith
Mountain Lake, “Gremmie” loves to get mail.
Here she is, busy at work, looking over the
entries in our puzzle competition.
If you care to drop her a line, send it to:
Gremlin the Wonder Cat,
Assistant Editor
Discover Smith Mountain Lake Magazine
40 Village Springs Dr., Suite 25
Hardy, VA 24101
Or via e-mail at:
Your diagram will look like this:
Using your knowledge of right triangles, you can calculate
which routes will take each son farther away from the target
village. Remember that although it may be longer than a
direct route, it may still be closer (or at least not farther
away) than the distance “as the crow flies”.
From this diagram, we can see that there are five qualifying
routes to an adjacent village, since we must count every
route that does not take us further away from the next
village. Remember that mirror images of routes must also
be considered. So that son will make his last trip on April
5th and the wedding will be the next day on April 6th.
To get to the second village along the circle, there are 41
possible routes. Again, remember that any route that does
not take the son farther from the village than the direct route
must be considered, including mirror images. So that son
will make his last trip on May 11th and the wedding will be
the next day on May 12th.
There are 121 qualifying routes from the first village to the
one directly opposite on the circle, so that son will make his
last trip on July 31st and the wedding will be the next day
on August 1st.
BONUS: If the coastline is 10 miles long, how long is the
longest route that any of the sons takes to his bride’s
village?
The longest route taken is a route to the opposite village.
If the circumference is 10 miles, then the diameter is 10/Pi
miles and the radius is 5/Pi miles. Since a regular hexagon is
formed from six equilateral triangles, the length of a side of
the hexagon is equal to its radius.
Three of the paths taken are all sides of equilateral triangles,
with a height of one half of the radius.The remaining
two paths are each half of the side of one of these same
triangles.The side of one of these equilateral triangles is
5/(sqrt(3)*Pi)
So the total distance traveled is
10/Pi + 4*(5/(sqrt(3)*Pi)) = 10/Pi + 20/(sqrt(3)*Pi) ~ 6.86 miles
Gremmie’s
Pages
LAST ISSUE’S BRAIN TEASER:
27
Last issue’s puzzle described a perfectly round island, with six villages evenly
spaced along its coastline. Every pair of villages is connected by a perfectly
straight path through the jungle. Courtship custom on the island dictates
that a prospective bridegroom must deliver a fish to his intended’s father
along a different route. Every route must bring the young man closer to his
destination at all points.
On April 1st, a father’s three sons tell him of their intent to marry. Each
bride is from a different village, and they are the next three villages going
clockwise around the island. If the sons each start their courtship today, and
they are married the day following completion of the ritual, what will the
wedding dates be?
There were no correct solutions submitted, and the complex nature of
the puzzle makes it difficult to present the solution in detail due to space
limitations. Here is the thought process that you would need to use:
This is a lot easier to solve if you create a diagram. Draw a regular hexagon
with lines connecting each pair of villages.You’ll end up with three
intersections created along each of these connecting lines, with the three
lines connecting opposite villages meeting in the middle.
