Thursday, March 13, 2014

Sicilian Solution III: Pepperoni Protocol

This time it's artisanal.

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 cheese to pepperoni?

(I won't be posting the answer to this one - feel free to post in the comments.)

This puzzle and the last one are part of an interesting genre called bisection problems. I looked around online, but I couldn't find a good collection of problems to link to, just various items scattered here and there. But some classic problems in this area include bisecting a triangle; bisecting the area and perimeter of a shape simultaneously; bisecting two shapes simultaneously, and so on. By googling around, I found this book which looks interesting. If others have any gems in this area, feel free to link in the comments.

Hope you enjoyed the Sicilian Trilogy! Thanks as always for all the comments and emails.


jeff said...

Form a triangle with vertices at the center of each pepperoni. Any of the 3 bisections of that triangle also will bisect the rectangle and will have exactly 1.5 pieces of pepperoni in each bisection. Thus equal areas and equal pepperoni:cheese ratios.

I can draw it freehand and see the validity of this method but i can't do it using my clunky old software.

P.S. Did you include the "classic problems" sentence as a clue to this solution?

jeff said...

I no longer trust that solution because i reread Sicilian Solution II's discussion. Stay tuned for a complete redaction and rewrite.

jeff said...

(i emailes you attempts 2 & 3, which i eventually discarded as incorrect)

Solution attempt #4
(generalizable for any 3-pepperoni slice?)
a) The rectangle must be bisected, so one point on the ideal cut line must be the center of the rectangle.
b) There must be 1.5 pepperonis per bisection of the rectangle.
c) Find the center of each circle; their points will be on the same line or they can be viewed as the vertices of a triangle.
d) The center of the line or triangle defines the second point on the ideal cut line.

JasonZimba said...

I will post some answers and further thoughts soon!