16 replies to this topic

[bonanova]

A classic puzzle notes that 3 miles of road is sufficient to connect four cities situated on the corners of a square one mile on a side. Yes, the cities are really small; point-like, for our purposes. The question is then asked: what length of road(s) is necessary to connect them? I.e. the shortest distance of road(s) required.

[Thalia]

Spoiler for

You could have 2 roads in an X. Each road would be sqrt(2 miles) so the total would be 2*sqrt(2 miles) = 2.83 miles.

[Ryo]

Spoiler for 1.78

Essentially, the roads would look like this: )—(

The ends of each curve meets with the cities and the horizontal line connects each 'pair' of cities

The curved roads are from unit circles.

To get the curved roads I simply did: (1/3 * pi) = 1.0472 The 1/3 comes from the two curves combined

For the horizontal line, I needed to know the distance between the furthest point of each curve to the side of the imaginary box

This required pythagoras theorem and the previous unit circle. I did: sqrt of (1^2 - 0.5^2) which resulted in sqrt of 0.75 = 0.8660. However, this only took into account one curve. So for two curves the horizontal line is: 1 - 0.8660 = 0.1340 * 2 = 0.2680, 1 - 0.2680 = 0.7320

0.7320 + 1.0472 = 1.7792 miles

[fabpig]

Spoiler for I could be missing something.....

but as there is a mile between each city, there must be 2 miles using straight lines joining, say, A to B, and C to D. With curves it must be more. So the 1.78 doesn't look right.

[bushindo]

A classic puzzle notes that 3 miles of road is sufficient to connect four cities situated on the corners of a square one mile on a side. Yes, the cities are really small; point-like, for our purposes. The question is then asked: what length of road(s) is necessary to connect them? I.e. the shortest distance of road(s) required.

Here's an attempt,

Spoiler for attempt

Let the cities be arranged on a grid with coordinates (0,0), (1,0), (0,1), and (1,1). We draw the lines with the following shape: >-<. Essentially, we make two way points, each joining two cities. We then draw a straight line between the two waypoint.

If we let the coordinates of the waypoints be ( sqrt( 1/12), 1/2 ) and ( 1 - sqrt( 1/12), 1/2 ), then the total distance used is about 2.732.

[Yoruichi-san]

Here's an attempt,

Spoiler for attempt

Let the cities be arranged on a grid with coordinates (0,0), (1,0), (0,1), and (1,1). We draw the lines with the following shape: >-<. Essentially, we make two way points, each joining two cities. We then draw a straight line between the two waypoint.

If we let the coordinates of the waypoints be ( sqrt( 1/12), 1/2 ) and ( 1 - sqrt( 1/12), 1/2 ), then the total distance used is about 2.732.

I think you're right

Spoiler for Yay, calculus

Calling the straight line distance x, the total distance is (x+2sqrt(2-2x+x^2). If I minimize that function (taking the derivative and setting to 0), I get x=(1-1/sqrt(3)), which is equivalent.

The case of drawing two diagonals is a special case of this, where x=0.

[WitchOfSecrets]

Spoiler for You can't get any shorter than this.

I can connect them with a road that is slightly over 1 mile long.

How?

The road is exactly one mile wide.

[bonanova]

Spoiler for You can't get any shorter than this.

I can connect them with a road that is slightly over 1 mile long.

How?

The road is exactly one mile wide.

Very clever. I like this solution so much that I submitted it to the combined Town Boards of the ABCD Municipality for approval. They texted me back and said they were willing to proceed, even to bear the cost of all that additional asphalt, provided that

Spoiler for we also submit

the location of the necessary six-inch-wide yellow divider line. The Board is extremely safety conscious, it seems. By the way, the Town Cartographer chimed in that the length of the road will be recorded as the total length of the yellow line.

The Board awaits our specification.

[Yoruichi-san]

Very clever. I like this solution so much that I submitted it to the combined Town Boards of the ABCD Municipality for approval. They texted me back and said they were willing to proceed, even to bear the cost of all that additional asphalt, provided that

Spoiler for we also submit

the location of the necessary six-inch-wide yellow divider line. The Board is extremely safety conscious, it seems. By the way, the Town Cartographer chimed in that the length of the road will be recorded as the total length of the yellow line.

The Board awaits our specification.

Spoiler for Roundabout soln...

...is a giant roundabout. So what if Americans have some weird difficulty with them? They're safe and efficient and if you're going to pave the entire square, then stick a sign in the middle indicating it's a roundabout and paint white dividers for the lanes. Length of road->0

[WitchOfSecrets]

bonanova: No problem!

Spoiler for Divider line

The road and divider line will run east-west.

Of course, we'll also want crosswalks so pedestrians can move between the northern and southern towns.

Edited by WitchOfSecrets, 13 May 2012 - 06:33 AM.