[Prof. Templeton]

Prof. Templeton had purchased a 1 Km square plot of land. He was told by the Utility Company that their lines pass through his property somewhere, but they didn’t know exactly where they were and if the Prof. wanted to tap into the lines, he was responsible for paying to have them dug down to and determining where they ran through the property. All the Utility Company would tell him for certain was that the lines ran parallel to the ground and traveled in a straight line. So Prof. Templeton asked Doug the digger to come over and give him an estimate on finding the utility lines.

“The cost to dig is $1.00 per meter and I’ll have to dig around the perimeter to find those lines, so that’s four grand”, said Doug.

“I don’t think you need to do that”, said the Prof.

“You’re right. I can just dig around three sides and find those lines”, replied Doug

“You can still do better than that”, said the Prof.

What optimal digging path to find the utility lines did the Prof. have in mind and what is the least it will cost to have Doug dig?

[Prof. Templeton]

I'm totally stumped. I've been looking for a while, but I can't find anything better than the two adjacent sides + opposite half-diagonal. I think a hint might be in order.

Sometimes nature provides the optimal solutions. Think about bees and their homes.

Let's leave the opposite half diagonal and the point on the square it comes from alone and focus on the remaining three points of the square. If you wanted the shortest lines from those three outside points to a central point, what would the three angles around that central point look like?

[Prof. Templeton]

I’ve been looking at this for a while and I think Imran has the best answer possible.

Being a landscape designer, this type of problem comes up every once in a while but the utility companies usually have a pretty decent grasp on where their lines are. If a client suggested one of the multiple broken and dotted lines solutions, I’d probably charge them twice as much per meter due to the increased labor costs of mapping the whole square kilometer off.

I better get back to work now.

The solution is only optimal for Prof. Templeton not Doug the Digger.

[Guest]

I get

2.651

[Prof. Templeton]

I get

2.651

I can do a little better than this, but can you explain how or post a picture?

[Guest]

Alright, I think I see what you're going for.

2.639km

Label the square WXYZ. Center is O. Draw line segment ZO, length=sqrt(1/2)=.707.

Draw another dot in the middle of the square, label it P (it will be closest to corner X). Draw line segments WP, XP, and YP. WP and YP have the same length (call it A), XP has a shorter length (call it B).

I did some calculus to minimize 2A+B. I found this was least when A=.816, B=.299 (the exact exp​ressions would be really annoying to type out). This happens when angle WPX is 60 degrees.

This gives a total of 2.639, which is slightly better than the two-sides + half-diagonal solution.

[Prof. Templeton]

Alright, I think I see what you're going for.

2.639km

Label the square WXYZ. Center is O. Draw line segment ZO, length=sqrt(1/2)=.707.

Draw another dot in the middle of the square, label it P (it will be closest to corner X). Draw line segments WP, XP, and YP. WP and YP have the same length (call it A), XP has a shorter length (call it B).

I did some calculus to minimize 2A+B. I found this was least when A=.816, B=.299 (the exact exp​ressions would be really annoying to type out). This happens when angle WPX is 60 degrees.

This gives a total of 2.639, which is slightly better than the two-sides + half-diagonal solution.

Good job, Chuck

Inside the square are two trianges whose obtuse angle is 120 degrees, the short sides of the triangles bisect the corner so those angles must be 45 degrees and the other angles must be 15 degrees. I used the law of sines to get the length of the sides at .816 and .299. So all the lines are

1 @ .707

2 @ .816

1 @ .299

or a total of 2.639 km

Here's a picture

[Guest]

That's what I had, but I was doing it quickly and so rounding errors resulted. What can you do.

[Guest]

Good job, Chuck

Inside the square are two trianges whose obtuse angle is 120 degrees, the short sides of the triangles bisect the corner so those angles must be 45 degrees and the other angles must be 15 degrees. I used the law of sines to get the length of the sides at .816 and .299. So all the lines are

1 @ .707

2 @ .816

1 @ .299

or a total of 2.639 km

Here's a picture

I dont think this will work! unless I misunderstood the question

Edited September 4, 2008 by taliesin

[soop]

basically, from a birds-eye view, he's got a pipe going through that square in a straight line.

to prove it works, draw that pattern yourself, and then try and draw a straight line from one side to the other, without crossing the pattern he's drawn.

[Prof. Templeton]

I dont think this will work! unless I misunderstood the question

No straight line can pass through the square without crossing one of the lines drawn in the picture above.

[Guest]

No straight line can pass through the square without crossing one of the lines drawn in the picture above.

Ok, i though (for some reason) you had to cross 2 so you know where line was.

[Guest]

a diagonal ditch

[soop]

Even though I didn't get the answer, I really liked this puzzle. 5* IMO.

[Guest]

had a solution that didn't work

Edited September 5, 2008 by hmmmm...

[Prof. Templeton]

_______.........
| .
| .
| .
| _____
| .
| .
| .
|_______.......

I believe this is 2500 m

It looks like I could pass a line through there.

[Prof. Templeton]

Even though I didn't get the answer, I really liked this puzzle. 5* IMO.

Thank you. I appreciate that.

[Guest]

Prof. Templeton had purchased a 1 Km square plot of land. He was told by the Utility Company that their lines pass through his property somewhere, but they didn’t know exactly where they were and if the Prof. wanted to tap into the lines, he was responsible for paying to have them dug down to and determining where they ran through the property. All the Utility Company would tell him for certain was that the lines ran parallel to the ground and traveled in a straight line. So Prof. Templeton asked Doug the digger to come over and give him an estimate on finding the utility lines.

“The cost to dig is $1.00 per meter and I’ll have to dig around the perimeter to find those lines, so that’s four grand”, said Doug.

“I don’t think you need to do that”, said the Prof.

“You’re right. I can just dig around three sides and find those lines”, replied Doug

“You can still do better than that”, said the Prof.

What optimal digging path to find the utility lines did the Prof. have in mind and what is the least it will cost to have Doug dig?

1414.21 the diagonal

[Guest]

1414.21 the diagonal

Problem with just one diagonal is that the utility lines don't have to go through opposite sides of the plot, just any two sides.