Responding to Richard Strausz's of Dec 18, 2012 3:20 PM:
Delightful problem indeed! (And I am adding it to my list of useful ideas/concepts/problems for students learning math - and indeed for all those interested in 'learning to think').
I had vaguely guessed that the answer might be something like using the line connecting the centres of the two rectangles - but I must confess I looked at the answer and did not work it out or sketch it.
GSC Richard Strausz posted Dec 18, 2012 3:20 PM: > This is one of my favorites. I didn't figure it out > before seeing the solution... > > Richard > ============ > > 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.