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?

The general equation (general enough for our purposes) of a circle in polar coordinates is r^2 - 2*r*cos(theta) + 1 = a^2 where the circle is of radius a, with the center located at theta = 0 at a distance 1 from the origin.  This gives r = 2*cos(theta) + sqrt(4*(cos(theta))^2 + 4*(a^2-1)) -- the positive sign indicating the center is within the enclosure of the circle.  Half of this is cos(theta) + sqrt((cos(theta))^2 + a^2 - 1).  Does anyone care to integrate this? And then prove that no matter where integration begins, in increments of pi/4 radians, that odd intervals of integration add up to the same as even segments.

Wolfram's The Integrator will do this for various constant values of a^2 - 1, and one can see the general pattern, but it involves elliptic integrals.

Posted by Charlie on 2005-02-05 02:37:54