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 also think it's exactly half regardless of the cut.

First, consider any point on the pizza. Cut the pizza such that one of the cuts goes through the center. Now, you have two equal halves. It is easy to see that this is divided half and half, as the symmetric piece (regardless of where you start) must be eaten by the other person since nobody can eat adjacent slices.

If the cut doesn't go through the center the problem becomes unclear to me but I'm still confident it's half. For example, you will have one chord that gives you a small semicircle-type piece of pizza and a large semicircle-type piece of pizza. Trial and error shows that unbalanced portions in a particular area always  lead to equal and opposite unbalances in neighboring areas.

No proof yet.

Posted by Eric on 2005-02-04 20:59:00