Math Forum Discussions

GS Chandy

Posts: 7,866
From: Hyderabad, Mumbai/Bangalore, India
Registered: 9/29/05

Re: Car Talk Geometry Puzzler (and solution)
Posted: Dec 18, 2012 11:26 PM

Plain Text

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.