Cutting the Holey Cake

Dec 08, 2012

RAY: My neighbor Joan decides to bake some brownies for her two little grandsons. She got one of those rectangular pans. Because these kids are really competitive, she knows she has to divide what she bakes right in half -- which is pretty easy to do if you've got a rectangular pan.

So, she bakes the brownies, takes them out and puts them on the cooling rack. Before she cuts it in half, however, her husband comes along and cuts a rectangle out of the middle, at random.

Imagine, now, you've got a rectangular cake, and he cuts a rectangle out of the middle. He wasn't even nice enough to cut it out of the corner! The cuts aren't even parallel to the sides of the original cake.

It's not touching an edge, but it could be, and it's not necessarily parallel to any of the sides.

She says, "What's this all about? How am I going to cut it in half now?"

He says, "Well, I guess you'll just have to bake more brownies, and I'll eat this."

She was bemoaning this to one of her girlfriends on the phone, and the girlfriend says, "I have a remedy for your dilemma."

Joan asks, "Does it involve rat poison?"

The girlfriend says, "No, I have a way that you can cut this cake in half and make sure that each of your grandchildren gets the same amount of brownie. In fact, I have two ways to do it. There's a hard way and an easy way."

Joan says, "Give me both ways."

So, here's the question. How can you, with one cut of the knife, cut the brownies in half?


RAY: Here's the answer. The hard way is to take your knife, and holding it parallel to the cooling rack that the brownie is sitting on, slice the brownies that way. One kid's going to get the top half of the brownie and the other kid is going to get the bottom half. I don't like that solution because the top half and the bottom half are not truly equal.

And here's the other solution. If you take a rectangle, how would you find the center of a rectangle? You would draw diagonals and you would find the center.

Any line drawn through the center of a rectangle, that's not a diagonal, also cuts the rectangle in half. So if you were to draw a line through the center of the big rectangle and it went through the center of the hole, then you would cut the brownie into exactly two, equal pieces.

So what you do is find the center of the little rectangle that her husband cut out, by making two diagonals, then you draw two diagonals for the big piece of the brownie.

It makes no difference where the other rectangle is, if you connect the two centers with a straight line, and continue right through to the edges. You will have cut the brownie in half. Do we have a winner?

TOM: We do! The winner is Jim Loftus, from Cambridge, MA. Congratulations, Jim!

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