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1.) Let n = (2,3,9). Then n is a digit from 1 to 9. Put down the value of n. I will be a digit between 0 and 9. From 9342n62i2, paste 79 copies of it to another, thereby forming a number with 720 digits (9 times 80). Call the number M. then answer the following: a) For which value of i is M divisible by 11? b) For which value of i is M divisible by 9? c) Do a) and b) when m is formed by 81 copies (instead of 80 copies)?

2.) Sketch the feasible region for the following set of constraints:

3y - 2x ≥ 0
y + 8x ≤ 53
y - 2x ≤ 2
x ≥ 3.

Then find the maximum and minimum values of the objective function z = 5x + 2y.

3.) A farmer grows apples on her 600 acre farm and must cope with occasional infestations of worms. If she refrains from using pesticides, she can get a premium for "organically grown" produce and her profits per acre increase by $600 if there is no infestation, but decrease by $400 if there is. If she does use pesticides and there is an infestation, her crop is saved and the resulting apple shortage (since other farms are decimated) raises her profits by $500 per acre. Otherwise, her profits remain at their usual levels. How should she divide her farm into a "pesticide-free" zone and a "pesticide-use" zone? What will be her expected increase in profits per acre with this strategy?

4.) Suppose research on three major cell phones companies revealed the following transition matrix for the probability that a person with one cell phone carrier switches to another.

                                          Will Switch to
                              Company A        Company B      Company C
Company A                   .91             .07                      .02
Now has Comp. B         .03             .87                       .10
Company C                  .14             .04                       .82

The current share of the market is [.26, .36, .38] for Companies A, B, and C respectively. Find the share of the market held by each company after
a) 1 year
b) 2 years
c) 3 years
d) What is the long-range prediction?

5.) There are five states with the populations as follows:

A) 1234
B) 3498
C) 2267
D) 5558
E) 306

There are 125 delegates to be apportioned based on population. Find the apportionment based on Hamilton's Method; on Jefferson's Method; on Webster's Method; on Adam's Method; on Huntington-Hill Method. Do any of these violate the quota criterion?

Please note, whenever you use a modified divisor, state (correct to 1 decimal point) the divisor you used and the range of divisors that would produce the same result.

6.) Consider the following set of preference lists:

Number of Voters (7)
Rank       1     1     1     1     1     1     1

First        C    D     C     B     E     D     C
Second   A     A     E     D    D     E      A
Third      E     E     D     A     A     A     E
Fourth    B    C     A     E     C     B     B
Fifth       D     B    B     C     B     C     D

find out the winning using:

a) Plurality voting
b) The Borda count
c) The Hare Method
d) The Condorcet Method
e) Sequential pair wise using the agenda A, B, C, D, E
f) Find another agenda for pair wise voting that will produce a different winner than in e)

7.) It can be shown that all configurations of the Fifteen Puzzle can be reduced, using the allowable moves to either:

            S1                                             S2
1    2        3     4             or           1    2      3         4
4     6       7     8                           5     6      7         8
9     10     11   12                         9     10    11      12
13   14     15     X                        13     15   14        X

For each of the following initial configurations, show which of S1 or S2 they can be reduced to:

           T1                                               T2
1     2     3        4                         1     12     11     10
12   13   14       5                         2     13     X       9
11     X   15      6                         3     14     15      8
10     9     8     7                         4       5       6      7

              T3                                          T4
1     5     9     13                         4     3       2       1
2     6     10     14                       5     14     13    12
3     7     11     15                       6     15     X      11
4     8     12     X                         7     8      9       10

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