The King and his Four Cities.

TheTick · 2013-01-31 13:05:41

There is a King who has just built four cities. This king is a very peculiar king as he built these four cities so that each individual city made up the corner of a perfect square. In addition all four cities are on perfectly level ground. Though there is a slight problem; the king forgot to build the roads. This king, in all of his peculiarity, decided he wanted all of his roads to be straight, and he wanted to be able to get from one city to any other city using these roads. How can you design the king's roads so that you use the least amount of road possible, while still meeting his criteria?

anonimnystefy · 2013-01-31 18:42:09

This is a well known problem and is easily solved using soap bubbles. Interestingly enough, if you dip two frames connected by four pegs in a square position into soap watter, the soap film that forms around the pegs will try and minimize its surface so you'll get something that looks like this: >-<

This result can also be proved rigorously, but I thought it might be interesting to mention this.

MathsIsFun · 2013-01-31 18:49:51

I love it when nature solves mathematical puzzles for us!

Now if we could just get soap bubbles to solve traveling salesman problems ... hang on, is this remotely possible? Maybe being only 3-dimensional is a limiting factor.

anonimnystefy · 2013-01-31 18:56:43

Hi MIF

Unfortunately, the soap will get you only a path which connects all the roads and that has minimal length and which probably doesn't go from any city to any other with a direct road, ie. the cities are not the only nodes int the "graph" formed by the soap film.

pRo9aMeR · 2013-02-04 21:54:03

I think I'm oversimplifying it, but wouldn't a direct road to and from each city suffice? The roads would make the shape of a square with an "X" in the middle.

Bob · 2013-02-04 22:23:57

hi pRo9aMeR

Close, but you can get shorter.

Oh yes, in case you have misunderstood: it is not required that every city is connected independently to each of the others. You jsut have to be able to get from one to another by road(s)

Bob

anonimnystefy · 2013-02-04 22:49:35

Hi pRo9aMeR

The solution looks something like in the picture below.

pRo9aMeR · 2013-02-05 19:56:57

I did slightly misunderstand the question...But now I'm confused. Why wouldn't a simple "X" be shorther than the ">-<"? I 'looks' like it takes longer to travel that way...

Bob · 2013-02-05 20:16:46

hi pRo9aMeR

You have to calculate the total distance along all the roads. With a simple x that should be easy enough with Pythag.

With the more complex Stefy-x it will help to know the angle between the lines is 120 degrees.

Bob

Math Is Fun Forum

#1 2013-01-31 13:05:41

The King and his Four Cities.

#2 2013-01-31 18:42:09

Re: The King and his Four Cities.

#3 2013-01-31 18:49:51

Re: The King and his Four Cities.

#4 2013-01-31 18:56:43

Re: The King and his Four Cities.

#5 2013-02-04 21:54:03

Re: The King and his Four Cities.

#6 2013-02-04 22:23:57

Re: The King and his Four Cities.

#7 2013-02-04 22:49:35

Re: The King and his Four Cities.

#8 2013-02-05 19:56:57

Re: The King and his Four Cities.

#9 2013-02-05 20:16:46

Re: The King and his Four Cities.

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