Here we will look at the sofa moving problem. I assume that you voluntarily clicked on the link to this note and thus you are of adventurous nature, so some more adventurous math will not scare you. Therefore we will address the question here in full generality.

A corridor A meters wide turns at right angle into a corridor B meters wide. We want to find the maximal length such that an object of that length and w meters wide (w < A) can be passed through this turn without lifting it from the floor (it lies flat).

We will try to follow the basic idea of the simpler problem with w = 0. We imagine two segments that are parallel and w meters apart such that the inner one touches the inner corner and ask where the outer segment touches the outer walls. It would seem that in effect the position of the corner gets shifted by w so that the outer segment touches this relocated corner. This would allow us to ignore the inner segment and work with just the outer one, thus simplifying our calculations. Is it really possible to solve this more general problem by simply imagine that the corner is somewhere else and then call on the results of the simpler problem?

Unfortunately, this is not the case, since the new position of the corner depends on the angle of the segment.

However, the idea of ignoring the inner segment and keep the outer segment at the proper position using a connecting perpendicular bar of length w is sound, since it is essentially the only reasonable way of somehow describing its position. Thus we can use this to find some structures that we can handle, namely we will try triangles.

By similarity, the four triangles in this picture share the angle a and thus we can calculate the length of the segment by the following formula:

We need to find the minimum of this function on the interval (0,π/2). We cannot somehow determine it just by analyzing the expression for l, since the formula mixes sine (which is increasing there) and cosine (which is decreasing there). Thus we will try the standard approach via critical points.

This derivative is equal to zero for a satisfying

Asin³(a) − Bcos³(a) = w(sin²(a) − cos²(a)).

That is an equation that cannot be solved analytically, so we are out of luck. Not just in general, but even if we have concrete values for A, B, and w, we are in most cases unable to solve this equation. If we really need to know this a, we have to find it numerically using a suitable software (or directly as it for the approximation of minimum of l).

Are there any simpler cases that we could solve? Obviously we do know how to solve this problem for w = 0, since then we have the case that we solve in Exercises. Another simple case is when A = B. The equation then simplifies to

1 + sin(a)cos(a) = (w/A)(sin(a) + cos(a)),

but that is of no help, since it is still impossible to solve directly. However, when we look at the original equation, we can guess that with A = B we get equality when sin(a) = cos(a). Actually, the angle π/4 is then something that we can get without doing all the complicated work, it is a matter of common sense. If A = B, then the situation is symmetric and therefore also the critical moment happens when the segment is in a symmetric position, which gives the following situation.

Thus we can use the Pythagoras rule and get