I’ve been listening to Car Talk on NPR for over a decade now and have always loved their puzzlers. This week they have a really cool geometry puzzle:
For those of you that like diagrams, here’s a summary of the puzzle. Imagine you have just baked a tray (rectangular, arbitrary dimensions) of brownies (brown rectangle below) but someone comes along and cuts out a piece, also a rectangle but smaller and could be at an angle (black rectangle below).
Question: What single cut could you make to divide the remaining brownie sheet into exactly half? Apparently, there is a simple and a complex solution.
*** UPDATE with SOLUTION ***
Today we heard back from Click and Clack with the solution to the puzzler. As I mentioned above, there was a hard and easy solution. I thought that referred to the level of math involved, but it simply referred to the complexity of cutting:
1) Hard Solution – simply slice the brownie tray in half along the “thickness” axis, resulting in identical slices with equal amounts of brownie. This was classified as hard because cutting the brownie in that manner is difficult (parallel to the baking tray)
2) Easy solution – I find this rather elegant: we all know that if you draw a diagonal line across a rectangle, the resulting areas are equal in size. If you draw two diagonal lines, you end up with a point of intersection. It turns out that any line that goes through that point will bisect the rectangle with equal areas. So, the solution is to draw those lines for the original brownie rectangle and the removed brownie rectangle, then connect the points of the two to create your cutting line. In the figure below, the black line represents the original brownie sheet and the red line represents the removed section. The green line is the single cut that splits the remaining brownie in two!