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Six Villages

Math brain teasers require computations to solve.


Puzzle ID:#44101
Fun:*** (2.39)
Difficulty:**** (3.45)
Submitted By:javaguru*us**
Corrected By:javaguru




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?

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Feb 14, 2009

OK, I made a significant mistake on the answer to the bonus question.

A side of the smaller equilateral triangle is not 5*sqrt(5)/2Pi. It is 5/(sqrt(3)*Pi). This makes the longest path

10/Pi + 20/(sqrt(3)*Pi) ~ 6.86 miles
Mar 04, 2009

Dear javaguru-
You've mixed your use of route and path. The puzzle is a confusion until one realizes the boy is travelling all routes which with every step take him closer to the girl's village.
I initially figured the boy would take just enough routes to use every acceptable path or path fragment at least once.
Mar 04, 2009

OK. I was trying to keep the prose consistent with tropical-island-village-speak and "route" didn't seem like the term they'd use, but I can see how "path" can be misinterpreted.

When I saw you had posted a message here I was assuming it was to point out the error in the answer. There is a mistake in the assignment of values in the longest path (sorry, "route" ).

M = Ai + EFi + BCi = 13

There is no path from EFi or BCi to M. The value for M should instead be:

M = Ai + Fi + Bi = 11

Once you propagate the correct value for M through the rest of the routes you end up with 121 routes, not 131. This gives a wedding date of August 1st.

At the speed that teaser corrections get approved, the fix for this and the error in the bonus question and the wording change for "route" should make it into the teaser by Christmas.
Mar 06, 2009

The hint and the solution have us record, mark, or set a number by villages and crossings. This charts the number of ways a point can be reached. The number of ways to reach a point is equal to the sum of the numbers of ways in which adjacent points which are visitable but more distant (and hence already scored) can be reached.
The only way to be at the first village, the point most distant from the goal, is to live there; score 1. There is only way to reach the second most distant village or intersection, so it always scores 1. The sum/score grows when more than one of a point's immediate neighbors has a score to contribute to the sum. The number of permissible routes is the sum at the goal.
Mar 11, 2009

I really liked this teaser. At first it seemed difficult, then it seemed tedious, but once I figured out the pattern to solving it, it was just plain fun!

Although it's not an easy teaser, I don't think it's as hard as the rating suggests.
May 07, 2009

javaguru, thank you for taking the time to come up with and post these amazing teasers. I was one of the few people that actually liked math in school and I'm very glad to have a place where I can maintain (and possibly increase) my math skills. I havn't been on braingle long, but so far the best workout my brain has gotten, and therefore the most fun I've had, is working on your math teasers. I currently have a logic-grid teaser pending and one more to be submitted soon, but I want to try my hand at making a good math puzzle. So be on the lookout for one from me

Aug 20, 2009

Ok after reading the teaser I decided it was way above my head, but who in their right mind has time to come up with this and write it all out!? I mean Come on! get a life! lol great job tho
Sep 04, 2009

Isn't there anyone her who isn't completely stufedd from the first paragraph of the answer??? How bothered must you be to make that teaser??
Sep 13, 2009

This is one of the hardest brain teasers I've ever seen. I get confused just looking at the explanation.
Jun 25, 2010

Has no-one else noticed? I think there is a mistake in the answer for the third son (the one whose bride is in the third village along the coastline (directly across) from his home village). I first only looked at the date in the answer which says Aug 1. I was getting Jul 31 and could not figure out where my mistake was. Then I checked further down the number of days - 121. I have that too so the problem must be incorrect adding of 121 days to Apr 1 either in the answer or on my side. I tried various ways, asked my mother () and used GNumeric. Everything said Jul 31. I am not forgetting anything, am I? The first 2 sons are correct though.
Jul 08, 2010

@ kunk: You are correct, the answer should be July 31. If you want to submit the correction you may. If you don't in the next week or so I will.
Mar 31, 2011

This is the hardest teaser on Braingle!
May 01, 2011

hi there, just to clarify, so travelling from A to B, one wouldn't head towards B in the opposite direction along the coast? ie.leave A towards F, then to village E, to D, to C, then to B? Or,from A, walk towards FAi, then turn towards F at the intersection, then back towards Fi, then, EFi, etc. and so on and so forth?
Jun 25, 2012

By my Logic, the weddings can all be 'tomorrow' (April 2) if the sons take a very fast mode of transport. - it says they must 'travel' doesn't specify on foot, or how long travelling any specific route takes.

each son completes all the possible path's to the village (heading directly back to home village between each) in one day and chucks the old man a Fish then heads home for their triple-wedding the next day.

Sep 10, 2012

Interesting problem. Some programmers will recognize this as dynamic programming.

I also wanted to add that you can also do the computation in reverse. Meaning that you begin with the village you are traveling to. The rule for choosing the next vertex (crossing) will then be to choose the one closest to any of the one already calculated. The reason this makes more sense in a way is that you will get all three answers at ones.

An image of what I mean /photo/my-images/854 /6villages.jpg/
(For some reason you cant have words longer than 50 letters so the link has got two spaces in it)
Note that in this image. You are traveling towards A and the numbers at the crossings denote how many ways this can be done in from that point.
Nov 01, 2012

Excellent observation Patrick!

That's exactly what I like to do when solving puzzles: find a simpler or more elegant means of solving it. Good job!
Nov 10, 2012

Very good teaser =) I thought A-B-ABi-Bi-M-Ci-CDi-Di-D was the longest path, but on calculating the length of your path, I found mine lacked by ((2/sqrt(3)) - 1)r... Do you have a proof to show that your path is indeed the longest?
Aug 06, 2013

i...didnt even bother to read this one. too long and im tired. ima go sleep! sorry!
Jun 26, 2014

Nice problem to solve
Jul 08, 2014

I used the same approach as PositivaPatrik. The only difference is that I considered D to be my farthest destination rather than A. I will admit that I'm concerned that a father would have three sons that would want to cause woe to their intended brides, rather than woo their brides.
Dec 02, 2014

There are no weddings, they were all playing April Fools jokes (you did say April 1, right?)

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