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?

(In reply to thoughts by Charlie)

The difficulty 5 probably doesn't mean "really messy," so there likely is a trick of some kind that makes this one fall out.  However, your analysis leads to a way that is only somewhat messy: The integrals of the even-numbered segments, say, each depend on the starting angle of one of them, and the total result is to be proved constant.  Thus, if we differentiate with respect to the starting angle, we need to get 0.  Applying the rules for differentiating with respect to a variable in a limit (which appears to the 1st power + a constant) the result can be expressed in terms of the integrand evaluated at various angles.  The job then is to use identities to show that the result is identically 0.

Posted by Richard on 2005-02-06 01:14:54