I'm confused by issues about α and β and the meaning of "such operations". Are we supposed to be given arbitrary β < α < 180 degrees and then alternate them? Or is the question whether β < α < 180 can be chosen so that alternation returns the pie to its original state after finitely many operations using α and β alternately? (Assuming a self=healing pie.) Or ??? --Dan << A round pie is cut by a special cutter that cuts off a fixed sector of the angle measure α, turns this sector upside down, and then inserts back; after that the whole pie is rotated through an angle of β. Given β < α < 180 degrees, prove that after a finite number of such operations (the beginning of the first and the second operations are shown on Fig. 67) every point of the pie will return to its initial place.