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I'm standing on the roof of my building with nothing but a laser range-finder, and I want to know the height of my neighbor's home. Who can tell me how to do it with the least number of laser measures?

The range-finder only tells me the distance to things in a straight line.

Some ascii art for illustration:

          O    <- Me
         -|-
  _______/_\_
 |          |
 |          |          ___________
 |          |         |           |   \
 |          |         |           |    }
 |          |         |           |    } measure this
 |          |         |           |    }
_|          |_________|           |   /
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    $\begingroup$ As none of the answers that take 3 measures is accepted so far, do we have additional information like the angle of measurement (which somehow doesn't count as an additional measure) or the height of our own building? $\endgroup$ – A. P. Dec 1 '17 at 19:10
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    $\begingroup$ I can do it in zero measurements, it's 5 lines high! $\endgroup$ – Lord Farquaad Dec 1 '17 at 20:45
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    $\begingroup$ The simplest answer is to climb down from the roof, jump the fence into the neighbor's yard, and aim the rangefinder upwards at the neighbor's gutters before he spots you. $\endgroup$ – chif-ii Dec 1 '17 at 20:47
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    $\begingroup$ Do you know your own height? $\endgroup$ – Acccumulation Dec 2 '17 at 5:19
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    $\begingroup$ Zero measurements: go to your neighbour and offer her your snazzy laser rangefinder if she'll tell you how tall her house is. $\endgroup$ – David Richerby Dec 2 '17 at 12:21
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Let the two buildings be $ABCD$ and $EFGH$ clockwise, from the bottom right corner

   B-C
   | |
   | I    F-G
   | |    | |
---A D----E H---

First measure $CD$ and $CE$, then calculate $DE = \sqrt{CE^2-CD^2}$.

Let $I$ be the point on your building so that $IF = DE$ ($I$ is at the same height as $F$) and $ID = FE$.

Then measure $CF$ and you can calculate $CI = \sqrt{CF^2-IF^2} = \sqrt{CF^2-DE^2}$

You know $CD$, you know $CI$: $FE = ID = CD-CI$

So the final formula is $$FE = CD - \sqrt{CF^2-CE^2-CD^2}$$ You only need three measurements: $CD, CE$ and $CF$.

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    $\begingroup$ If you can confidently get to point I on your own building then you only need one measurement -- straight down to ID. $\endgroup$ – A. I. Breveleri Dec 1 '17 at 21:35
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    $\begingroup$ @A.I.Breveleri: None of the measurements in this method actually involve measuring a distance to point I; it's just there to illustrate the construction. $\endgroup$ – Michael Seifert Dec 1 '17 at 21:57
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Note: It is assumed that we don't know the height or width of any building or the relation inbetween them.

I have illustrated the distances you need to measure below:

enter image description here

$a,b,c$ are the distances we needed. As a result

We will able to find $|DE|$ distance by using $|DE|=\sqrt{b^2-a^2}$.

After that,

we can find the $|CI|$ distance by using $|CI|=\sqrt{c^2-|DE|^2}$

At the end, the height of the next building will be:

$|FE|=|CD|-|CI|$

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Like this:

First find the distance along your own building's right wall, and then the distance to the bottom of your neighbour's. Pythagoras then tells you the horizontal distance between the "inner" walls of the two buildings. Then find the distance to the top of the other building's left wall. The square of this equals (horizontal distance)^2 + (difference in heights)^2. You know the former, so now you know the latter, and the first thing you measured was the height of your own building, and you're done.

That takes

three measurements. It doesn't look to me as if you can do it in two

unless

you already know the height of your own building, in which case you can skip the first measurement.

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    $\begingroup$ Darn it typing on mobile is hard, completely sniped. $\endgroup$ – Quintec Dec 1 '17 at 15:09
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This can easily be accomplished with two distance measurements.

  __________ [A]
 |          |
 |          |      [C] ___________
 |          |         |           |   \
 |          |         |           |    }
 |          |         |           |    } measure this
 |          |         |           |    }
_|          |[B]___[D]|           |   /

First, come down off the roof.

Stand at the base of your building (B). Measure the distance to the base of your neighbor's building BD. Measure the distance to the top of your neighbor's building BC. As any 6th-century-BCE Greek academician could tell you, the height of your neighbor's building is the square root of (BC x BC - BD x BD).

Of course, if you feel that the puzzle requires that you take the measurements from the top of your building, you should have said so instead of merely saying that's where you're standing now.

Actually I've thrown away only one assumption. I'm still assuming you're not willing to trespass on your neighbor's property. Otherwise, as @chif-ii indicates, you can measure the height directly from point D.

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I have no formal proof, but I know it is possible with 3 measurements:

  1. Measure the height of the building you are on (point straight at ground)
  2. Measure distance between two buildings (point laser at base of building in front of you, can be calculated using trigonometric formula)
  3. Measure distance between you and the closest edge of the roof of the second building. You now have 2 of 3 edges of the triangle that allows you to calculate the height difference between the two buildings
  4. Use the height difference between the buildings and the height of your building (point 1) to calculate height of 2nd building
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It can be done in 2 measurements this way:

  1. Jump off the building (the question statement never says measurements had to be taken from the roof) and measure the distance to the neighbour's house from the base of your house.
  2. DON'T MOVE. Now measure the distance from the base of your house to the roof of your neighbour's house.
  3. Do math. Height of neighbour's house is
$$\sqrt{\text{(distance to roof)}^2-\text{(distance between houses)}^2}$$

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    $\begingroup$ Unfortunately, this is exactly the same as A. I. Breveleri’s answer, which was posted seven hours earlier. $\endgroup$ – Peregrine Rook Dec 2 '17 at 6:15
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    $\begingroup$ @PeregrineRook This means we need only 2 more copies of this answer to make it even with the 4 copies of the other one. By the way, I think the all caps "don't move" isn't really required after you jumped off the building. $\endgroup$ – Sleafar Dec 2 '17 at 6:33
  • $\begingroup$ @PeregrineRook I thought I had scrolled down to the bottom. And the spoilers thing is new to me. $\endgroup$ – tlhIngan Dec 2 '17 at 7:54
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As A. I. Breveleri commented, you can do this with one measurement:

1) Take the stairs down until you reach the height of of the opposing roof. (the correct position is easy to find if the roof is level, as pictured.)

2) Stick your head out the window and use the laser to measure the distance down.


The traditional solution which requires zero measurements:

Take the stairs down to your office, phone the relevant city officials / the architect who designed the opposing building / the manager of the opposing building and ask/bribe/cajole them into telling you the height.


Alternative way with three measurements, without moving off the roof and without annoying the doorman of the opposite building:

          O B   <- Me
         -|-
  _______/_\A
 |          |
 |          F         C___________
 |          |         |           |   \
 |          |         |           |    }
 |          |         |           |    } measure this
 |          |         |           |    }
_|          D_________E           |   /

Stand at the corner of your roof, and measure the distances from your feet and from the top of your head to the opposing corner. (i.e. AC and BC)

Knowing your own height (AB), you can get the height difference (AF) of the buildings from the two triangles.

Then measure the distance from your feet to the ground, i.e. height of your own building (AD), and get the opposing building height CE = FD = AD - AF

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Climb down your building few floors of your building until you see the roof of your neighbour's building. Let's say this is point 'A'. Now climb down from this point till the ground of your building. Let's say this is point 'B' . Now calculate the distance between point 'A' & 'B'. That's it. You got the height of your neighbour's building.

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