Moving Sofas

 It seems like when I move, I’m always ending up in places where I have to haul a sofa around some corner.  You’ve done it.  You and someone else are trying to move it around a corner and inevitably, you start and stop, back up, swear, adjust it and after a while you’ve either done it or you give up.  Well in 1966 mathematician Leo Moser wondered “What is the shape of the largest area in the plane that can be moved around a right-angled corner in a two dimensional hallway of width 1?” Eventually it became called “The Moving Sofa” problem.

When we think of a sofa, we think of something long and straight almost like a rectangle.  A normal sofa usually seats three people and generally can be between 70 and 96 inches long but the average sofa is 84 inches long.  It is extremely rigid and doesn’t curve around a corner easily.  

When this problem was proposed, the idea was to find a shape that allowed the sofa to be as large as possible and still go around corners, so people had to decide the size of the “sofa” and its shape.  If the sofa is a square with a length of one and the hall is also one unit wide, so it slides around the corner easily without any struggle.  It hits the wall and then changes direction without actually rotating.  This has a size of 1 unit.

On the other hand, if the shape is a solid semi-circle, it is able to rotate around the corner so the flat edge is along the interior wall and the curved is along the outside wall.  As it goes around the corner, the curved outside is moving the sofa but it only has an area of around 1.57.

For a semi-circle with a cut out center, it is a bit different.  This version can be longer because the cut out part allows it to rotate around the corner more easily because it has that scoop which provides some leeway as the sofa works its way around the corner.  There is more wiggle room so to speak which makes it easier to move it around a corner. This one has the largest area at about 2.2 times the square one.  This one is composed of 18 curves glued together.

The last shape explored is an unusual one for most sofas.  It looks like the base part of a guitar with two indentations, one on each side.  The rounded edges on both sides make it easier for it to move through the corners and the indentations make it so it has space to move around the corner.  Although this shape slides through the corners better, it only has an area of 1.6.

This site has some wonderful animations of each situation.  I just know that when moving a sofa, one usually ends up swearing and getting frustrated.  Let me know what you think, I’d love to hear.  Have a great day.