Let's say you have a donut. You are allowed to slice it 3 times. Each slice must be a perfectly straight cut. What is the highest number of donut pieces you can end up with after 3 slices?
Assume that no crumbs are created during the slicing.
Also assume that no pieces move until after you have finished all 3 slices. That way while you are making your third slice, the pieces made from the first and second slices don't start moving and falling off.
You can create 10 pieces. Make two cuts that are perpendicular to the table as shown, tangent to the hole, creating five pieces. Then, slice the donut parallel to the table, splitting each piece into two.
Assuming we're cutting just perpendicular to the surface, the maximum is
9 Pieces. This is by first cutting at bearing 000, to the left of the centre but still going through the circular gap in the centre, Then, rotate the donut by 120 degrees and make the same cut. Repeat this once more and you should have 9 separate pieces if the distance from the centre that you used was correct. Simply scale the distance from the centre for each cut until the cut clips the hole, but also crosses with another cut before doing so. Of course, if we allow non-perpendicular cuts to the surface...