The four villages.?

Question

Four villages... A,B,C and D form the shape of a square whose dimentions are 10 miles. One day, the coucillors of the villages get together and decide to build a road network that connects all of the villages in a manner that it is possible to go from any village to any other. Unfortunately, cost is a prime factor so they dissmiss the idea of building 4 roads ie. from A to B, B to C, C to D and finally, D to A since this would be 40 miles of road.

They then realise tha by eliminating one of the roads it would still be possible to get to any village (although some poor sod will now have to do a thirty mile trip instead of what would have been 10) and this would make the overall length of the three roads 30 miles.

Then they came up with the idea of an 'X' shape which would also solve the problem. Applying the basic Pythagoras theorem most of you will come up with an answer of about 28.3 miles. So this is surely the best answer.....isn't it?

Answer 1

This is not the correct answer. While the shortest between two points is a straight line, the distance between 4 points is not. Actually, there is a principal which is referred to as Steiner Points. This is applicable in this situation. Two Steiner points (P) are placed inbtween points ABC and D.

Imaging a Y, except with a triangle, similar to the upper portion of the Y attached to the bottom of the Y. Essentially, you would have a triangle connecting points A and B and one connecting C and D. However, their height is only 2.9 Miles as opposed to 5 miles. The remaining 4.2 mile gap would be connected with a straight line between each triangles point.

The two diagonal lines of the triangles would be 5.78 miles, or 23.12 miles for all four. Now you add in the 4.2 mile straight line, for a total ROAD distance of 27.32 Miles.

Answer 2

Fireman is right. Here's a relevant blurb from the below source link. My first thought was that the two diagonals would do the best job but the study of "steiner points" and minumum paths is quite extensive and obviously useful. You learn something new every day.

Blurb
Let's connect the corners of a unit square. Without steiner points, the spanning tree has length 3, with 3 of the 4 sides drawn in. Your next impulse is to connect the four corners to the center, introducing one steiner point. This gives an edge length of 2.828, a definite improvement over 3. Next, split the steiner point in two and pull the two points apart, towards the left and right sides of the square, until the angles are 120°. This gives an edge length of 2.732. This is the best steiner tree for the square

Footnote: so for a square with 10 mile sides the answer is 27.32 mi.

Answer 3

Sounds like the right work, but the wrong number. A right triangle with two legs of length 10 has a hypotenuse of 14.1.

Answer 4

28.3 is the correct answer for the total miles of road.

The hyp comes out to 14.16 or so and then it will take 2 to connect all the cities. 14.16(2)=28.3

Answer 5

What firemanbj says is correct. This is what he's talking about:
http://www.maa.org/mathland/mathland_4_8.html

What it says is you get a shape like this:

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(Ignore the dotted line, it's there coz Answers doesnt like blank spaces)

I think firemanbj deserves credit for the best answer!

Answer 6

yes. it would be the same distance to each village.

Answer 7

very interesting story