The diagram illustrates the first eight and the next-to-last moves of a very surprising puzzle. Take a round cake with icing only on the top. Cut out a piece making an angle of 1 radian at the center, turn it upside-down, and reinsert it into the cake. Then move 1 radian clockwise and do the same with a second piece. Continue this process in the clockwise direction. The question is: Will all the icing ever return to the top? Because 1 is not commensurable with π, the natural inclination is to say NO. But in fact the icing does return to the top after 84 moves. And no matter what angle is chosen for the pieces, the icing returns to the top in finitely many moves.