Discrete Math
Fair Division Review Worksheet
Name___________
1. Anne, Beth, and Jay are heirs to an estate that includes a computer, a used car,
and a stereo. Each heir has submitted bids for the items in the estate as
summarized in the following table. Find the final settlement for each heir after
applying the estate division algorithm you learned in this unit.
Computer
Car
Stereo
Anne
$1,800
$2,600
$1,000
Beth
$1,500
$2,400
$800
Anne
Jay
$1,650
$2,000
$1,200
Beth
Jay
Total of bids and cash
Fair share
Items received
Value of items received
Initial Cash received
Share of remaining cash
Final settlement
2. Arnold, Betty, and Charlie are dividing a cake in the following way.

Arnold divides the cake into what he considers six equal pieces.

The pieces are then chosen in this order: Betty, Charlie, Betty, Charlie,
Arnold, Arnold.
Who is guaranteed a fair share by his or her own assessment?
3. Describe how the following cake division algorithms work:
a. cut and choose method for dividing cake between two people.
b. lone divider method for dividing cake among three people.
c. last diminsher method for dividing cake among four or more people.
4. States A, B, and C have populations of 647, 247, and 106, respectively. There are
100 seats to be apportioned among them.
a. What is the ideal ratio?
b. Apportion the 100 seats among the three states by the Hamilton method.
c. Apportion the 100 seats among the three states by the Jefferson method.
d. Apportion the 100 seats among the three states by the Webster method.
e. Apportion the 100 seats among the three states by the Hill-Huntington
method.
f. Suppose the populations of the states change to 650, 255, and 105,
respectively. Reapportion the 100 seats by the Hamilton method.
g. Explain why the results in part (f) constitute a paradox.
5. Compare and contrast the four apportionment methods with respect to the quota
condition, the Alabama paradox, the population paradox, the new states paradox,
and favoritism towards particular states.