Science & Technology

The Ham Sandwich Theorem Is a Delicious and Puzzling Mathematical Principle

If we asked you to cut a piece of bread in half with only one cut, could you do it? Sure you could. You could probably cut an entire ham sandwich in half just as easily. But if we knocked over the plate and left the bread and a single piece of ham in a big mess, could you cut them exactly in half with a single cut? There's a mathematical principle that says, yes, that's technically possible. It's called the ham sandwich theorem. What else would they call it?

I'll Take n Pancakes, Please

To understand how this is possible, let's scale it back. Ham sandwiches are far too delicious for our mathematical palates, so let's think about pancakes instead. If you have a single pancake, you can easily figure out where to cut it to get two equal halves. If you add a second pancake — no matter what shape, or where it is on the plate — you can eventually find one line that will go through both pancakes to divide them in half equally. This is known as the pancake theorem. Surprised? At this point, we hope not.

If you really want to get into the math of the pancake theorem, it technically says that in the case n = 2 (that is, if you have two objects), given two disjoint regions of the plane (regions that aren't next to each other), there's a line that simultaneously divides both regions into two pieces of equal area. Easy.

Hyperplanes and Ham

Ok, now swap out the pancakes for pieces of bread, and add a piece of ham into the mix. For one more twist to our scenario, let's remove the force of gravity. This is a magical ham sandwich with three pieces that can float in midair, and you've got a magical knife to boot that can grow as long and tall as you need it to. The ham sandwich theorem says that no matter where the pieces of that sandwich are in three-dimensional space — touching each other, across the kitchen from each other, across the universe from each other — there is a way to slice your infinitely long, infinitely tall knife through all three pieces in such a way that you get exactly equal halves of each.

In mathematical terms, the ham sandwich theorem states that given n objects floating in n-dimensional space (in our case, that's 3 objects floating in 3-dimensional space), there exists a single [n - 1]-dimensional (again, in our case, that's 3-1, or 2-dimensional) plane that simultaneously cuts all n objects into two pieces of equal volume.

So if you're ever floating out in space with a friend who's really particular about splitting food equally, you know for certain that you can cut your ham sandwich to fairly share it with them. The only problem? The ham sandwich theorem doesn't tell you where to cut it. For that, you're on your own.

The Ham Sandwich Problem

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Written by Ashley Hamer February 15, 2018