# Puzzle: leaning a ladder at $45^\circ$ to a wall using only yourself

This question was asked to my friend in an interview. You are provided a ladder and are led to a wall. The ladder must be kept against the wall making an angle of $45^\circ$ with the floor. You are given no measuring instruments whatsoever. All you have is yourself and the ladder.Nothing else. You are not given the value of the ladder's length or anything else of that sort.

How will you go about doing it?

If you think this involves physics rather than just math, please do mention if I must move the question elsewhere. And in case you're thinking, the interviewer has measuring instruments and will check your work after you're done.An exact value of $45^\circ$ is expected. And I am not sure if "not possible" or "insufficient data" is an accepted answer.

I really don't know what tags to use for this. I request you to edit my question with the appropriate tags.

• You could use the spacing between two rungs as a unit of measurement... Jun 1, 2015 at 19:25
• @Demosthene Assuming that the rungs are equally spaced and that there isn't any unusual spacing at the ends Jun 1, 2015 at 19:26
• Did your friend ask if the ladder was telescoping? If it was, you could stand the ladder vertically and horizontally at the foot of the wall, mark the endpoints on the floor and the wall, then extend the ladder until it was just sufficient to meet those two marks. Jun 1, 2015 at 19:27
• ...were we to be given a pencil, anyway. :)
– Ken
Jun 1, 2015 at 19:32
• @Ken: True. :-) Although I can't see how to do it without some kind of marking device. Jun 1, 2015 at 19:52

1. Stand the ladder up vertically against the wall.
2. Now lie straight on your front on the ground perpendicular to the wall (it shouldn't matter if you aren't perpendicular) with your arms outstretched above your head and touching the feet of the ladder with your hands.
3. Keeping your feet in the same position, stand up straight.
4. rock the ladder towards you, keeping its feet in the same place until you are holding it by the sides (rungs are not guaranteed to be available) with your arms outstretched directly above your head
5. Somehow pick the ladder up, walk away from the wall and turn around so that the ladder is standing on its feet with its end suspended in the air but not touching the wall. You should be between the feet of the ladder and the wall
6. Walk (perpendicular to the wall) towards the wall, allowing the ladder to slide along the ground, pivoting about your hands until it touches the wall.
7. By similar triangles, the ladder is now touching the wall.

• Clever! But how do you determine that you're walking perpendicular to the wall in Step 6? I see that it doesn't matter in Step 2, but it sure seems to matter in Step 6. Jun 1, 2015 at 19:57
• @BrianTung If you aren't perpendicular to the wall then you won't get all four ends of the ladder on a surface at the end. move from side to side to adjust. Jun 1, 2015 at 20:01
• Can we count on that? (Maybe this is part of the back-and-forth between the interviewer and interviewee.) Jun 1, 2015 at 20:02
• If you can count on that, by the way, then you can use the ladder itself to determine the perpendicular from the wall, and use that as a gauge for further actions. Jun 1, 2015 at 20:04
• I think it is unnecessary to stand the ladder against the wall. With the ladder lying beside you, touch the wall instead of the ladder in step 2, do step 3 as written, then pick up the ladder so you can do step 4. (But I like this answer as written, too.) Jun 1, 2015 at 20:35

Assuming you have some kind of marking device (chalk, or the like):

If the ladder is telescoping (building on my answer in the comments), you could stand the ladder vertically and horizontally at the foot of the wall, mark the endpoints on the floor and the wall, then extend the ladder until it was just sufficient to meet those two marks.

If it's just an ordinary single-length ladder, pick two points $A, B$ at the foot of the wall, closer together than half the length of the ladder. Use the ladder as a rusty compass to construct the perpendicular bisector of $\overline{AB}$ that meets the wall at $C$, and is one ladder length distant from the wall at $D$. Then draw a circular arc, centered at $C$, from $D$ to the wall. Construct the bisector of that arc at $E$. Place one end of the ladder at $E$ and let it fall against the wall. It will be at a $45$-degree angle (for ideal ladders and walls, but this is an imaginary problem, anyway).

• One might recall the "marking" scene from Never Cry Wolf, though if you use that particular device you might fail the interview for reasons other than the angle of the ladder. Jun 1, 2015 at 20:28

Why not use yourself as the measuring device? (If you know your exact height, which is mostly the case, that's perfect!). Measure the ladder length as $x$ units of your height. Now you know that when the ladder is up against the wall, the base and the altitude formed by the ladder (with its own length as the hypotnuse of the supposed triangle) are equal to $x \cdot cos(\theta)$. Just incline the ladder such that the base is equal to what you calculated. The method is prone to errors of measurement but given it is an interview question and the fact that what I said is logically correct, you ace the interview. :)