Name________________________________
No homework yet
Last-Diminisher Method UPDATE
Steps for solving each problem:
A) Students have an order in which they divide up the shares. 1st player always chooses
unless they are the final claimer of the land.
B) Player 1 divides up booty into fair share – claims a piece C and R is remaining
C) Next player passes or diminishes piece C. Last diminisher gets piece C.
D) 1st player left in order is the next chooser from the remaining booty and the process starts
again until all booty is given out.
1. A cake valued at $30 is divided among five players (𝑃1, 𝑃2, 𝑃3, 𝑃4, and 𝑃5 ) using the last-diminisher
method. The order in which the players play is 𝑃1 first, 𝑃2 second, and so on. Round 1: 𝑃1 makes the
first cut and makes a claim on a 𝐶 −piece. Each of the other players, in turn, has a chance to
evaluate the current 𝐶 −piece (which may or may not be the same as the previous player’s). The
following table gives the value of the 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝐶 − 𝑝𝑖𝑒𝑐𝑒 in the eyes of a player at the time it is that
player’s turn to play.
a. Determine which players are 𝑑𝑖𝑚𝑖𝑛𝑖𝑠ℎ𝑒𝑟𝑠 in round 1.
b. Determine which player gets a fair share at the end of round 1 and the value of that share to
the player.
c. (Fill in the blanks.) Round 2: ___________ makes the first cut. The value of the 𝑅 −piece to
each of the remaining players is _______ $________ (enter = or ≥ and the dollar amount).
2. A piece of land is divided among 12 players (𝑃1 , 𝑃2 , 𝑃3 , . . . 𝑃12 ) using the last-diminisher method.
The order in which the players play is 𝑃1 first, 𝑃2 second, and so on. Round 1: 𝑃1 makes the first cut;
𝑃2 passes; 𝑃3 diminishes, 𝑃4 , 𝑃5 , and 𝑃6 pass; 𝑃7 diminishes, 𝑃8 passes, 𝑃9 diminishes,
𝑃10 , 𝑃11 , and 𝑃12 pass. Round 2: After the first claim is made, 𝑃5 is the only diminisher. Round 3:
After the first claim is made, all the remaining players pass.
a. Which player gets his or her share at the end of round 1?
1
b. Which player makes the first cut to start round 2?
c. Which player gets his or her share at the end of round 2?
d. Which player gets his or her share at the end of round 3?
e. Which player makes the first cut to start round 4?
f.
What is the total number of rounds needed to play this game through to the end?
3. Angela, Boris, and Carlos are dividing the vanilla-strawberry cake shown below using the lastdiminisher method. The figure shows how each player values each half of the cake. In your answers,
assume that 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 the vanilla and strawberry parts as in "15°
strawberry - 40° vanilla” or 60° vanilla only.
Suppose the order of play is Carlos first, Boris, second, and Angela last. Round 1: Carlos claims the
entire strawberry half of the cake. Round 2: The first player to cut claims a vanilla only-zero
strawberry share.
a. Who was the last diminisher in Round 1? Describe his or her share?
b. Who was the first player to cut in round 2?
c. Who was the last diminisher in Round 2? Describe his or her share?
d. Describe the value of each player’s share (as a fraction of the total value of the cake) in the eyes of
the player receiving it. (Hint: Each of these fractions should have a different denominator.)
2
4. Jared, Karla, and Lori are planning to divide the half-vegetarian-half meatball foot-long sub sandwich
shown below using the last-diminisher method. Jared likes the meatball and vegetarian parts
equally well; Karla is a strict vegetarian and does not eat meat at all; Lori likes the vegetarian part
three times as much as the meatball part. Assume also that when the sandwich is cut, the cut is
made perpendicular to the length of the sandwich. (You can describe different shares of the
sandwich using the ruler and interval notation. For example, [0, 6] describes the vegetarian half, [6,
8] describes one-third of the meatball half, etc.).
Suppose that the order of play is Lori first, Karla second, and Jared last. Round 1: Lori starts by claiming
the C-piece [0, 𝑥].
a. Find the value of 𝑥 in Lori’s claim in round 1.
b. Who was the last diminisher in round 1? Describe his or her share.
c. Describe the final fair division of the sub.
3