Tuesday, March 11, 2014

Sicilian Solution II: The Sicilian Solution Solution

Sonny and Fredo are finishing a Sicilian pie. Both men are hungry, and there's only one slice left.

Nobody wants any trouble here. By making a single straight-line cut, can you produce two pieces with equal areas and equal ratios of crust to cheese?

That was the puzzle in my last post. Reader jeff posted the correct solution in the comments and also sent the following diagram:

Another friend used the same method too.

It was an honor to receive an email about this puzzle from Harold Jacobs, author of Mathematics: A Human Endeavor. Harold mentioned several different ways to solve the problem, including an elegant approach based on the fact that a line bisects the area of a rectangle if and only if the line passes through the center of the rectangle. I reproduce his diagram here:

Both diagrams are reminiscent of the genre of "proofs without words" (a book of which is here).

What if the dimensions of the slice are made general?

We can apply Harold's method: To share the cheese equally, cut through the center of the cheese rectangle. To share the crust equally, share the whole equally, thus, cut through the center of the whole. These two centers determine the cut:

The equation of this line is easy to find. The center of the cheese is point (a + (b/2), a + (c/2)). The center of the whole is point ((a + b)/2, (a + b)/2). The line through these two points is immediately found using the two-point formula as y = x + (1/2)(c - b). This is the desired cut.

Just as in the specific case that we started with, the desired cut has a slope of 1.

So next time you're in a pizzeria and there's trouble brewing, just make a 45-degree cut passing through the center of the cheese rectangle!

Another way to think about it is to go for a 45-degree cut that is equidistant from the two corners.

One can view the y-intercept (1/2)(c - b) as the height of the parallelogram that the bottom player deserves to have, in view of the "excess rectangle" of dimensions (c - b) x (a + b) that sits in the top player's territory atop the (a + b) x (a + b) square:

The parallelogram is half as tall and just as wide as the dashed rectangle, so it has half the area. Similarly, the crusty part of the parallelogram (a smaller parallelogram) has half the area of the crusty part of the dashed rectangle (a smaller rectangle). And likewise the cheesy part of the parallelogram has half the area of the cheesy part of the dashed rectangle.

For the sake of amusement, I also tried to come up with the most abstruse solution possible:

Indeed, Harold told me that he liked this puzzle precisely because there are so many ways to solve it. He gave it to some mathematician friends to see what they come up with--if we hear back, I'll let you know.

Thanks for all the responses! And get ready for Sicilian Solution III: Pepperoni Protocol, coming soon to a blog near you!

UPDATE 1/8/2015 - fixed a typo or error in the solution image, thanks to a reader who pointed out the mistake.


Anonymous said...

My daughter and I believe it should say 7.5 blocks of cheese, not 6.5.

JasonZimba said...

Yes! Thanks for catching this. I published the image submitted without checking it carefully enough. It takes a little work to edit image text, and I'm on the road - but I'll correct the image at my first opportunity!

JasonZimba said...

Fixed! Thanks again.