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Unformatted text preview: Math 141 Second Midterm Exam Thursday December 14 2006 Name..______._________.__ _________________ III
II
II I (15 points.) Find the cheapest route in the graph below starting at vertex A, ﬁnishining at
vertex A, and traversing each edge at lesat once. The cost of a route is computed by summing
the numbers along the edges used. II (35 points.) The following table shows the mileage between four cities: Springﬁeld (8); Urbana
(U); Eﬁingham (E); and Indinanapolis (I) E I S U E — 147 92 79
I 147  190 119
S 92 190 — 88 U 79 119 88 — (i) Represent this information by drawing a weighted complete graph on four vertices. (ii) One of the Hamiltonian Circuits in this graph is ESI U E. Find the others. Warning: ES I U E
is the same Circuit as S] U ES. (Problem I] continued.) (iii) Find the cost of each the distinct Hamiltonian circuits. Which Hamiltonian circuit gives
the minimum cost? ‘  (iv) Which circuit is obtained from the nearest neighbor algorithm starting at E? at I ? at S?
at U? ‘ ' (v) Which circuit is obtained from the sorted—edges algorithm? (vi) Find the minimal spanning tree. III (25 points.) Henry and Lisa play the following game. Henry chooses a row and Lisa chooses
a column from the matrix 3 2 1 4‘ (Each is ingnorant of the other’s choice.) Then Lisa gives Henry the entry in that row and
column. (i) Find the optimal mixed strategy for Henry. (ii) Find the Optimal mixed strategy for Lisa. (iii) What is the value of the game, i.e. how much should Henry pay Lisa before each play of
the game so that in the long run either is equally likely to be ahead? IV (25 points.) Our text explained four methods of apportionment: Hamilton’s method, Jeffer
son’s method, Webster’s method, and HillHuntington’s method. (i) Which (if any) of these methods can lead to the Alabama paradox?
(ii) Which (if any) of these methods always satisﬁes the quota condition? .(iii) Three states A, B, and C have populations 729, 337, 534, respectively. Apportion a house
of size 16 among them according to each of the four methods. Put your answers in the table.
Below the table I have done enough arithmetic so that you don’t need a calculator to solve the
problem. ' State Population Hamilton Jefferson Webster Hill—Huntington 729/100 = 7.290 337/100 = 3.370 534/100 = 5.340
729/971 = 7.508 337/971 = 3.471 534/971 = 5.499
729/9745 = 7.481 337/9745 = 3.458 534/9745 = 5.480
729/910 = 8.011 337/910 = 3.703 534/910 = 5.868 \/3 =\/l§ 23.464 1/4 =1/2'0 24.472 \/5 =\/§6 =5.477 1/6 =\/4‘2 26.481. \/7 =x/55‘ 27.483 \/8 = J73 =8.485 ...
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 Fall '07
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