Question 13.56

26. Suppose Players 1, 2, and 3 view a cake as in Exercise 25. Illustrate the last-diminisher method (still restricting attention to vertical cuts and, in addition, assuming that the piece potentially being diminished is a piece off the left side of the cake) by following steps (a) through (f) below.

  1. Draw a picture showing the third of the cake (6 squares) that Player 1 will slice off the cake.
  2. Determine whether Player 2 will pass or further diminish this piece. If he or she would further diminish it, make a new drawing.
  3. Determine whether Player 3 will pass or further diminish this piece. If he or she would further diminish it, make a new drawing.
  4. Determine who receives the piece cut off the cake and what size or value he or she thinks it is. (Actually, we knew what size the person receiving this first piece would think it was, assuming that he or she followed the prescribed strategy. How did we know this?)
  5. Finish the last-diminisher method using divide- and-choose on what remains, with the lowest-numbered player who remains doing the dividing.
  6. Redo step (e) with the other player doing the dividing.