Chapter 3, Problem 40E

Exercises 39 and 40 refer to the following: Arthur, Brian, and Carl are dividing the cake shown in Fig. $3-27$ using the lone-chooser method. Arthur loves chocolate cake and orange cake equally but hates strawberry cake and vanilla cake. Brian loves chocolate cake and strawberry cake equally but hates orange cake and vanilla cake. Carl loves chocolate cake and vanilla cake equally but hates orange cake and strawberry cake. In your answers, assume all cuts are normal “cake cuts” from the center to the edge of the cake. You can describe each piece of cake by giving the angles of its parts, as in “ $15°$ strawberry $-40°$ chocolate” or “ $60°$ orange only.”

Figure $3-27$

Suppose that Carl and Arthur are the dividers and Brian is the chooser. In the first division, Carl makes the cut shown in Fig. $3-29$ and Arthur picks the share he likes better. In the second division, Arthur subdivides the share he chose into three pieces and Carl subdivides the other share into three pieces.

Figure $3-29$

a. Describe which share ( $s_1$ or $s_2$) Arthur picks and how he might subdivide it.

b. Describe how Carl might subdivide the other share.

c. Based on the subdivisions in (a) and (b), describe a possible final fair division of the cake.

d. For the final fair division you described in (c), find the value of each share (as a percentage of the total value of the cake) in the eyes of the player receiving it.