by Jonathan Kujawa
If you have a sibling you are familiar with the problem of dividing up something desirable between selfish people. For some things, like ice cream or money, your only preference is to get as much as you can. If you divide it equally, then at least nobody will be envious of anyone else. My parents used the trick of letting one of us make the divisions, but in the knowledge that the other kids would get first pick. NASA can only dream of the atomic scale cuts made by me and my siblings [1]!
But what happens if the item in question isn't all the same? If it's a cake, a corner piece with roses made of frosting might be more desirable than a piece from the center. In an inheritance, a taxidermy collection and jewelry are hardly the same. Worse, each person might have very different preferences! My brother has a huge sweet tooth. He loves frosting while I'd definitely steer clear of corner pieces. One person's mounted deer head is another person's diamond earrings.
You might guess math has useful things to say about dividing things. But we'll soon see that there are more than a few surprises, too. The first question you might ask is if it is even possible to always divide something into two equal pieces with a single straight cut. It's not too hard to see if you have a single uniform object (say a plain cake), then no matter its shape, a single, well-chosen slice with a Samurai sword will split it into two equal sized pieces.
But what if your cake is a Baked Alaska? Surely you can't make a single cut which equally splits the cake and the ice cream and the chocolate drizzled on top? The Ham Sandwich Theorem is a remarkable result which says that no matter how elaborately intertwined three objects are, it is always possible to make a single cut which separates each of the three into two equal sized pieces!