"Just Move It Over There, Dear!"
The mathematics of moving mom's sofa 

by J. Schattman
Sir John A. Macdonald Secondary School
Waterloo, Ontario, Canada 

My mother once asked me if I could please move her living room sofa into the guest bedroom down the hall and around the corner.  Before I broke my back dragging this battleship down the hallway only to discover that it wouldn't make the turn, I decided to take some measurements and work out the math first. 

Starting simple: Assume a widthless sofa! 

 

Let's start with the simpler problem by assuming the sofa has zero width.  The general case will follow easily. 

 

Idea:  Compute the shortest gap that the sofa has to pass through as it makes the turn.  That is, minimize with respect to If we're lucky, the sofa will be longer than this gap, and we won't have to move anything.

Let h be the width of the hallway and be the length of the line segment that just grazes the corner and touches both outer walls at an angle of  θ with the inner wall.  We first need to find the angle at which is minimum.  (Intuition demands that , but rigor demands we prove it formally, which we do here.) 

 

We have  

(1.1)

 

 

We solve for  

=  

 

 

As with any 3rd degree equation, this one has three solutions.  It looks like we want the real solution, .   
Our intuition was right! 

 

This means the smallest gap we have to squeeze through is the one formed when the sofa is at an angle of with the wall.  How big is this gap in terms of h? 

 

=  

 

Punch Line:   

A zero-width sofa whose length is less than will make the corner.
(We'll see how to account for the width of the sofa presently.)   

 

Let's visualize this result when h = 1.5. 

 

=  

 

Accounting for the sofa's width 

 

Now for the almost incredibly simple result:   

 

If the sofa's length plus double its width is less than , the sofa will make the turn! 

 

Here a picture is worth words. 

 

A quick rule of thumb when you don't have a calculator on you 

 

1.  Measure the width of the hallway (h) 

 

2.  Measure the length (L) and width (w) of the sofa. 

 

3.  If is comfortably less than triple the width of the hall (a liberal estimate of ), you'll make it! 

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