You have a special eight-bladed pizza cutter. All you do is pick a point on the pizza, and the device cuts out eight straight lines from that point to the circumference of the pizza, separated by equal 45 degree angles.

You and your friend just bought a pizza and would like to have four slices of pizza each. Your friend tells you that you can make the cut using your device, using any center point you would like. After the cuts have been made, the two of you will eat alternate slices (so that nobody eats two adjacent slices).

How much of the pizza can you end up with?

I am becoming more of a believer in the 1/2 conjecture. I put the center of the cutter on the edge of the circle so as to make symmetrical pieces with 3 pieces that have zero area. The two classes of pieces with nonzero area that we are interested in comparing the areas of are

a) the big central piece and the two smaller side pieces, and

b) the two bigger side pieces. 

Using formulas I looked up, and some simple calculations of lengths, I was able to compute the total area of the first of these classes, and it came out to be exactly half the area of the pizza. Giving the pizza a radius of 1, the big central piece has area pi/4+1/sqrt(2) and the smaller side pieces each have area (1/2)(pi/4-1/sqrt(2)). 

Edited on February 5, 2005, 9:15 am

Posted by Richard on 2005-02-05 07:49:32