#ATTENDANCE QUIZ FOR LECTURE 13 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p13
#with an attachment called
#p13FirstLast.txt
#(e.g. p13DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Oct. 20, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 5

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH BY THE ANSWER
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1. Describe the problem that Euler solved regarding the  bridge.
A1. The citizens of Koenisburg made a game out of seven bridges. They could go around the cities 
crossing each of the bridges only once. They could not prove that it was impossible and so they 
brought it to Euler. He later worked on this and published a solution  called 'Solutio problematis
and geometriam situs pertinentis'. 

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2. Draw it on a piece of paper with a line segment representing every edge.
A2. 
attached seperately.
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3. If you roll a fair die 6000 times what is the probability that each of the possible outcomes 
{1, 2, 3, 4, 5, 6} occurs exactly 1000 times. 
A3.
add(x[rand(1 .. 6)()], i = 1 .. 6000);

970 x[4] + 1044 x[3] + 965 x[5] + 984 x[1] + 1026 x[2] + 1011 x[6]

From above we can see that the probability of choosing 1..6 have probabilities close to 1000 but 
not exactly. 

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4. Pick 5 random facebook friends. For each of them pick 3 friends. For each of the friends of 
friends pick 3 friends. Label the people picked 1,2,...,n. Write the graph in our data structure.
A4. 
G := RandG(11, 1);
G := [{2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, 

  {1, 3, 4, 5, 6, 7, 8, 9, 10, 11}, 

  {1, 2, 4, 5, 6, 7, 8, 9, 10, 11}, 

  {1, 2, 3, 5, 6, 7, 8, 9, 10, 11}, 

  {1, 2, 3, 4, 6, 7, 8, 9, 10, 11}, 

  {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}, 

  {1, 2, 3, 4, 5, 6, 8, 9, 10, 11}, 

  {1, 2, 3, 4, 5, 6, 7, 9, 10, 11}, 

  {1, 2, 3, 4, 5, 6, 7, 8, 10, 11}, 

  {1, 2, 3, 4, 5, 6, 7, 8, 9, 11}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

  ]


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5. Look at the cities that border PISCATAWAY and for each of them those that border them and again 
until you get to Princeton. 
(i) Construct the graph
{2,3,4,5},{1,3,4,5},{1,2,4,5},{1,2,3,5},{1,2,3,4}

(ii) Find the number of paths from Piscataway to Princeton
12

(a) Find the actual set using Paths (and list them)
H := [{2, 3, 4, 5}, {1, 3, 4, 5}, {1, 2, 4, 5}, {1, 2, 3, 5}, {1, 2, 3, 4}];
 H := [{2, 3, 4, 5}, {1, 3, 4, 5}, {1, 2, 4, 5}, {1, 2, 3, 5}, 

   {1, 2, 3, 4}]


Paths(H, 1, 1, 3);
   {[1, 2, 3, 1], [1, 2, 4, 1], [1, 2, 5, 1], [1, 3, 2, 1], 

     [1, 3, 4, 1], [1, 3, 5, 1], [1, 4, 2, 1], [1, 4, 3, 1], 

     [1, 4, 5, 1], [1, 5, 2, 1], [1, 5, 3, 1], [1, 5, 4, 1]}

(b) By using NuPaths
NuPaths(H, 1, 1, 3);
                               12


(c) By using GFt

GFt(H, 1, t);
 [     3 t - 1                 t                    t          
 [-----------------, - -----------------, - -----------------, 
 [(t + 1) (4 t - 1)    (t + 1) (4 t - 1)    (t + 1) (4 t - 1)  

             t                    t        ]
   - -----------------, - -----------------]
     (t + 1) (4 t - 1)    (t + 1) (4 t - 1)]


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