I have a machine that can divide a square pie into 9 equal square pieces using 4 blades:
The blades can be moved, but there is only one control - which defines the width of the blades in both dimensions
We simply have to set the width, then place the centre of the pie at the centre of the blades and we can chop any size pie into 9 equal pieces:
Someone has just brought me a round pie.
I still need to locate the centre of the pie in the centre of the blades, so I'm expecting to get 3 different shapes; 1 centre, 4 "corners" and 4 "edges".
If I set the blades to 1/3rd of the diameter of the pie, the "edge" pieces and "corner" pieces will be smaller than the middle one.
But it occurs to me that I can make the blades narrower so that the middle piece is smaller and the outer pieces are bigger.
For a pie of given radius r, is it possible to find a setting that produces 9 equal area pieces? If not, what setting will give 9 pieces of pie with the least variation, i.e. the difference between area of the smallest piece and the largest piece is minimised.