Poll

4 votes (44.44%)
No votes (0%)
5 votes (55.55%)
No votes (0%)
No votes (0%)
1 vote (11.11%)
1 vote (11.11%)
1 vote (11.11%)
No votes (0%)
3 votes (33.33%)

9 members have voted

Wizard
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Wizard
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March 1st, 2019 at 7:41:32 AM permalink
You are in charge of transportation of a square state with side length of 200 kilometers. There is a town in each corner of the state. Your goal is to connect the four towns with roads. How can you do this with the least total miles of roads?

Let the record show that I can't prove my answer is optimal. For the beer, you will have to match my answer. For a fancy cocktail, like sex on the beach, beat my answer. I will submit what I believe to be the best answer to OD by PM in the interests of being above reproach.

As usual, anyone who has won a beer is on a 24-hour hold.

The question for the poll is what is the basic shape of the roads?
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SkittleCar1
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March 1st, 2019 at 8:07:33 AM permalink
I've recently returned to this forum. Is bigotry no longer allowed in polls?

I would imagine a 90 degree square or diamond shape would be the shortest. Closer to the center of the four borders.
I dunno...someone smarter can answer.
Joeman
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March 1st, 2019 at 8:18:37 AM permalink
I'm going to go with the obvious (which means it's probably not optimal) and say connect the cities in the opposite corners, creating an 'X.' This would require 400 * 21/2 or roughly 566 miles of road.
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beachbumbabs
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March 1st, 2019 at 8:34:28 AM permalink
It's going to look like an "H" with bent sides, either upright or on its side.. Sort of 》=《. That's because the crossbar will be less road paved overall than needed by 2 hypotenuses.

With sides 200, an unbent "H" is 600 km roads paved. A full "X" is approx. 565.68 km of paving. But the shape I described is going to be less than either for some series of answers because of the crossbar duplication. I'm just not sure how to calculate the maximum saving of road length.

For example, if the bent sides represent 4 30-60-90 triangles, the length of each bent side is

a = x = road not needed.
b = (x*sqrt3) = 100 = half the distance to the center
c = 2x = top or bottom leg of diagonal

so a = x = 57.735

115.470 x 2
x2 sides
= 461.88 km.

The remainder crossbar is
200 - (2x57.735) = 200 - 115.47
= 84.53 k.

Total paved length is 546.41 km

But maybe optimum is with a top angle of 31°, or 29°, or some fraction of those or another angle. There's going to be a bell curve of better answers, starting with 1° off "X", ending somewhere short of the savings of paving an "X", in the direction of straight sides. What the top point of that bell is, I don't know.

Edit: Doing spot checks on some other angles seems to indicate that the arm angles should be 30° from each corner, leaving the answer above, corrected to two places. No other edits. So that's my final answer.

Last edited by: beachbumbabs on Mar 1, 2019
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DJTeddyBear
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March 1st, 2019 at 8:45:47 AM permalink
You can build a road that is U shaped, 200km from A to B, another 200km from B to C, and another 200 from C to D.

That's fine if you happen to be going from A to B, or B to C, or C to D. Unfortunately, it's a 600km trip from A to D along this U shaped highway.

Or you can ...
... build two highways, X shaped. Since it's (( 200^2 ) + ( 200^2 )) ^ 0.5 = 282.8km from corner to corner, the total road surface being built is about 565.6km. 34.4km less than the U shape, and far more efficient.

With this highway, it's 141.4km from any city to the intersection of the two roads, or 282.8km from any city to any other.

(I suggest being the guy who owns the gas / rest stop at the intersection.)


Mind you, this assumes that the four cities have no area to speak of, and that there's nothing else in the state.

Of course, having grown up in New Jersey, this is not far from what we thought of the "square" states out west before venturing out to see them. :D
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unJon
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March 1st, 2019 at 9:12:55 AM permalink
Quote: Wizard

You are in charge of transportation of a square state with side length of 200 kilometers. There is a town in each corner of the state. Your goal is to connect the four towns with roads. How can you do this with the least total miles of roads?

Let the record show that I can't prove my answer is optimal. For the beer, you will have to match my answer. For a fancy cocktail, like sex on the beach, beat my answer. I will submit what I believe to be the best answer to OD by PM in the interests of being above reproach.

As usual, anyone who has won a beer is on a 24-hour hold.

The question for the poll is what is the basic shape of the roads?

Sounds like typical local government incompetence. Minimizing road miles rather than travel time between towns.
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DJTeddyBear
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March 1st, 2019 at 9:55:34 AM permalink
I submitted my answer before reading anyone's spoiler.

Sigh, now having done so, I think Babs is on the right track, although she provided a concept only, with no math.

Therefore, here's some math...
Babs' idea of the bent H, or 》=《 seems right. Her example of a straight H uses the same 600km as my U shape, but the bent H is a good compromise to my X shape.

My knee-jerk thinking says that if all 5 segments are the same length, you'll get the optimal, smallest amount of road to build.

After playing with Excel, I find that that is not the case. Not even close.

In the 》=《 design, the optimal length of the crossbar is 84.50km, while the 4 legs are 115.48km, producing a road with a total length of 546.410km.

I'm sure there's a trig type function that would have made this easier, but I used columns in Excel and brute force to get to the estimate:

Col A = 75 thru 95 in .5 steps.
Col B = (200-A_)/2 to find how far from each border the two intersections will be.
Col C = B^2 This is one side of the leg's triangle, squared.
Col D = sqrt(C_+10000) Note 10000 is the height squared. This produces the length of each leg.
Col E = D_ * 4 + A_ for the total road length.

A short trip, one that doesn稚 need the crossbar, is only 230.95km and a long trip is 315.46km.



Of course, this will create a rivalry between the owners of the two gas/rest stops at the intersections. The more successful one will be the one with a casino that allows slot points to be used for gas, and has NO PARKING FEES! (but I digress...)

Edit: Babs originally didn稚 have any math.
Last edited by: DJTeddyBear on Mar 1, 2019
I invented a few casino games. Info: http://www.DaveMillerGaming.com/ 覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧 Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
Wizard
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March 1st, 2019 at 10:56:31 AM permalink
Thanks for the responses so far. I'd like to say that to get full credit I want to see a solution that for the given type of road shape, the details are optimal. Not just using goalseek in Excel, or some similar product.
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OnceDear
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March 1st, 2019 at 11:30:20 AM permalink
Quote: Wizard

You are in charge of transportation of a square state with side length of 200 kilometers. There is a town in each corner of the state. Your goal is to connect the four towns with roads. How can you do this with the least total miles of roads?

Let the record show that I can't prove my answer is optimal. For the beer, you will have to match my answer. For a fancy cocktail, like sex on the beach, beat my answer. I will submit what I believe to be the best answer to OD by PM in the interests of being above reproach.

As usual, anyone who has won a beer is on a 24-hour hold.

The question for the poll is what is the basic shape of the roads?

Presumably roads can be joined at any place they touch?
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Chuckleberry
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March 1st, 2019 at 11:37:02 AM permalink
Here's a wrong answer:

I tried to use the H shape, but rounded the heights, so the roads looked like this:
)--(

I used excel, inputting chord length, chord height, radius, arc angle in radians, and used that to compute the arc length. (I pulled all the formulas off of google.) Using every possible chord height (in .5km increments), the optimal amount of total road used was 561.39km. The heights of the rounded 'H' are each 220.69km, and the horizontal cross road is 120km.

Less than an X shape, but not the best answer so far.
Just trying to stay positive

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