MTH110
Math
Perspectives
TEXT:
For All Practical Purposes, COMAP,
8^{th}
edition
OBJECTIVE:
The objective of this course is to introduce you
to the ways
in which mathematics impinges upon your daily lives. You will be
introduced
to quantitative concepts and skills which will enable you to interpret
and
reason with quantitative information. In addition you will be
able to see
how mathematics can be used to model many different types of realworld
situations, from presidential elections to garbage collection.
GOALS:
 learn the
mathematical principals behind the decision making processes within
groups (social choice).
 learn the
mathematics underlying various types of optimization problem
(management science).
 learn how
mathematics can aid in the allocation of resources in a equitable way
(fairness).
LAB SOLUTIONS:
ADDITIONAL MATERIAL:
EXAM REVIEWS:
HOMEWORK ASSIGNMENTS:
 Homework 1 Chapter 1, Exercises 4, 12, 16(a), 24, 36, 38, 40,
46, 50. Also, do the following problems:
 Find an Euler Circuit (starting at
A) for the graph shown in Figure 1.10, pg. 10.
 Find an Euler Path for the graph in
Figure 1.15a, pg. 14. To answer this question, label the
vertices with the letters A, B, C, etc. and then list the letters in
the order that they lie on the path.
 In the graph shown below, color in the oddvalence vertices. Draw in
dashed edges to show an optimal Eulerization of this graph
 A complete ngraph is a
graph with n vertices where each vertex is connected to every other
vertex. The figure below shows complete graphs for n = 3, 4 and 5.
Which of these graphs have Euler Circuits? What conditions on n will guarantee that a
complete ngraph will or will not have an Euler Circuit? Explain your answer.
DUE: Sept. 6
 Homework 2: Chapter 2, Exercises 2 (list the vertices for each circuit
that exists), 4a, 8, 16b (just answer the question above the 3x4 grid), 18, 26,
38 (show your work using the method of trees), 42a,b, 44, 46. For Exercises 42, 44 and 46 be sure to specify both the Hamilton circuits and the cost of those circuits. Also, do
the following problems:
 Read Exercise 24 and then answer the following question: If Jill always insists on wearing her green boots whenever she wears her green scarf, how many outfits might her friends see her in? HINT: Determine how many outfits she has when she wears a green scarf and how many outfits she has when she doesn't wear a green scarf. Then add these two values together to get your answer.
 Using the method of trees and starting at vertex A, determine the number of Hamilton Circuits in the graph below. Show your work.
DUE: Sept. 13
 Homework 3: Chapter 3,
6, 8a, 14, 16(a,b,f), 18(a,b,c), 20(a,b), 22(a,b,e). Also,
do the following problem:
 We discussed two measures of optimality for the list scheduling problem:
the length of the critical path, and the total time of all tasks divided by the number
of processors. Apply these measures to the orderrequirement digraphs in Exercise
70 in Chapter 2 (pg. 63) for a) 2 processors and b) 3 processors. For each, specify which measure is
a more accurate estimate of the true optimal time.
 Use the list processing algorithm and the orderrequirement digraph shown below to schedule the tasks
with the priority list T1, T2, ..., T10 using a) two processors, b)
three processors and c) four processors. Which, if any, of these three
schedules can you be sure is optimal?
 Determine the critical path lengths for the vertices in the orderrequirement digraph used in Problem B and create a priority list using those values. Repeat part a) of Problem B with this new list.
DUE: Sept. 25
 Homework 4: Chapter 3, 32, 38, 48, 50,
62(c,d), 70(a,b), 72, 74. Also, do the following problem:

Assume we have to cut 30 pipes of the
following lengths: 22, 1, 24, 26, 27, 14, 8, 27, 7, 17, 16,
11, 16, 23, 11, 21, 27, 29, 14, 3, 8, 14, 12, 16, 26,
6, 15, 30, 11, and 29. If pipes only come in lengths
of 40 determine how many pipes would be needed if we used each
of the following algorithms: next fit, first fit, worst fit, best
fit, nextfit decreasing, firstfit decreasing, worstfit decreasing
and bestfit decreasing.
For all problems no credit
will be given for a simple numerical answer (min number
of bins, min time to schedule a set of tasks, etc.); you must
show your work (for example, show how the bins are filled,
or how the tasks are scheduled).
DUE: Oct. 9
 Homework 5: Chapter 4, Exercises 4, 12, 16, 21 (just
show your algebra here to find the intersection point; you do not
need to graph the equations), 22, 24, 26, 34 (change the 90 min to
120 min), 36 (change the 600 loaves to 700), 38 (change the $50 profit
for hot sites to $150), 40 (change the $2,600,000 to $2,800,000),
44, 46.
Please note the following:
 Always show your work when determining any corner
point.
 For problems 34 through 40 be sure to list the profit
formula and constraint equations, and to label all of the
corner points of the feasible region.
 For problems 44 and 46, you need only list the profit
formula and constraint equations (you cannot graph the feasible
region nor easily apply the corner point principle for
these problems).
 Please be as neat as possible when drawing your graphs. Use a ruler when drawing lines and give yourself enough space to clearly label all points and lines.
DUE: Oct. 18
 Voting Systems: Chapter 9
Exercises 10, 12, 14, 16, 28, 36(a,b); Chapter
10, Exercise 12. Also, do the following problems (NOTE: all reference to Exercises below refer to exercises in Chapter 9):
 Determine if there are any Condorcet
Winners in Exercises 10, 12, 14 and 16.
 Read the description of the Coombs
procedure in Exercise 17, and apply it to the preference lists in
Exercises 10, 12, 14 and 16.
 Apply the Plurality Runoff method to
the preference lists in Exercises 10, 12, 14 and 16.
 Assume we use the preference lists
in Exercise 10 and are using approval voting. Who wins if
everyone votes for their top two favorite candidates? Who wins if
everyone votes for their top three favorite candidates?
 Assume we have an election with 5
candidates and 11 voters with the preference lists shown below.
Determine the winner if the following methods are used: plurality,
plurality runoff, Borda count, Hare and sequential pairwise
with the agenda AEDBC.
DUE: Oct. 25
 Weighted Voting Systems:
Chapter 11,
Exercises 4, 8, 14, and 18. Also do the following problems (be sure to show your work for all of these problems):
 Determine the ShapleyShubik indices
for the weighted voting systems in Exercise 14.
 The current # of representatives for
the New England states are as follows: Connecticut  5, Maine  2, Massachusetts  9, New Hampshire  2, Rhode Island  2, and Vermont  1. Suppose
they all got together to vote on newenglandy
type matters, and decided that the quota to pass any measure would be
18 (we assume all voters from any state always vote together). Determine the BPIs for this weighted voting
system. Which states would you say are the
biggest winners in this system? Which are
the biggest losers?
 Suppose we wanted to create a
5person basketball team using members of the class (which currently has 23 students). Determine the
number of ways we could do this if
 we pick a particular person for
each position (point guard, center, etc);
 we just pick five people who can play any of the
positions.
 Euchre is a card game played with a
deck of 24 cards divided into 4 suits (clubs, diamonds, hearts and
spades) each containing 6 cards (the A, K, Q, J, 10 and 9).
 Determine the number of possible 5
cards Euchre hands.
 Determine the number of possible
full houses in a 5card hand.
 Determine the number of possible
flushes in a 5card hand.
 Use the techniques discussed in
class to determine the Banzhaf Power Indices for the following weighted
voting system: [9: 3, 2, 2, 2, 2, 2, 2].
Repeat with a quota of 10 and 11. For
which of these quotas does the voter with weight 3 have the most power?
 Determine the Banzhaf Power Indices
for the following weighted voting system: [16:5, 5, 5, 4, 4, 4, 4]. Repeat with a quota of 17.
DUE: Nov. 15
 Fair Division : Chapter 13,
Exercises 2, 4, 8, 12, 16, 18, 26 and 28. Also, do the following problems:
 Solve problem 6 in the Skills Check section (pg. 426)
completely (i.e., show the steps and the final results for a
complete division of the items listed).
 Solve problem 8 in the Skills Check section (pg. 426)
completely (i.e., show the steps and the final results for a
complete division of the items listed).
 Four coworkers  Tessa, Alex, Abby and Sarah  have
decided to use the Knaster Inheritance method to decide who gets two
highly prized items at
the office: an electric foot massager and a mousepad with a picture of
Elvis on it. Tessa values the foot massager at
$24 and the mousepad at $10; Alex values them at $36 and $12;
Abby values them at $24 and $18; and Sarah values them
at $28 and $8. Determine who gets which item, and how much money each
coworker either makes or spends.
 Read the description of the lonechooser method in Exercise 29,
then
apply it to the cake shown in Exercise 27, where
Player 1 is Bob, Player 2 is Carol and Player 3 is Ted, and Bob divides
the cake initially. Determine if any of the
three are envious after the cake has been divided up.
DUE: Nov. 29
 Apportionment: Chapter 14, Exercises 6, 8, 12, 16, 18, 20
and 22. Also, do the following problems:
 Five people have pooled their financial resources to start a small company. The amount of money put in by each person is $56,000, $41,000, $25,000, $15,000 and $4,000. In order to decide on company policies they decide to create a weighted voting system by dividing 100 votes between them based on the amount that they originally contributed. Determine how the 100 votes should be divided using the Hamilton, Jefferson and Webster methods.

A country has four states, A, B, C and D. Its house of
representatives has 100 members, apportioned by the Hamilton
method. A new census is taken, and the house is
reapportioned. Here are the data:
State

Old
Census

New
Census

A

6390

6395

B

5890

5890

C

2920

3015

D

1389 
1389

Totals

16589 
16689

Apportion the house using both the old and new census. Explain
how this is an example of the population paradox described on pages 440441 in the textbook.
 The ThreeBears Corporation has four stock holders: Mama Bear, who owns 971 shares;
Papa Bear, who owns 807 shares; Baby Bear, who owns 510 shares; and Goldilocks, who
owns 315 shares. At the end of year they have 50 units of porridge as dividends to
be distributed to the four stock holders. Use the Webster method to come up with
a fair apportionment of the porridge. Repeat this procedure for 51 units and 52 units.
(Can you figure out why the Three Bears scenario was used for this problem?)
DUE: Dec. 6