#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: 6

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH BY THE ANSWER

# Question 1: Describe the problem that Euler solved regarding 7 bridges.

# Answer 1: Euler's problem regarding the 7 bridges was that the city of Königsberg in Prussia 
# (now Kaliningrad, Russia) was set on both sides of the
# Pregel River, and included two large islands—Kneiphof and Lomse — which were connected to each
# other by seven bridges. Euler wanted to devise a walk through the city that would cross each of those bridges once and only once. 

# Question 2: Draw it on a piece of paper with a line segment representing
# every edge. G = [{2,3},{1,3,4},{1,2,4},{2,3}]

# Answer 2: 
#                   1 ___ 2 ___ 4
#                   |	  |	|
#                   |____ 3____ |

# Question 3: If you toss a fair coin 2000 times, what is the probability you get exactly 1000 heads? 

# Answer 3:
# (1/2)^2000 * (2000 Choose 1000) as there are two possibilites for each of the 2000 coin tosses. Then, you 
# want to choose 1000 of of the coin tosses to be heads. 

# Question 4: 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? 

# Answer 4: 
# (6000 Choose 1000)* (5000 Choose 1000) * (4000 choose 1000) * (3000 choose 1000) * (2000 choose 1000) * (1000 choose 1000)/6^6000

# Question 5: 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

# Answer 5: 
# Me (66) --> 1,2,3,4,5
# 1 --> 6,7,8
# 2 --> 9,10,11
# 3 --> 12,13,14
# 4 --> 15,16,17
# 5 --> 18,19,20
# ------------------
# 6 --> 21,22,23
# 7 --> 24,25,26
# 8 --> 27,28,29
# 9 --> 30,31,32
# 10 --> 33,34,35
# 11 --> 36,37,38
# 12 --> 39,40,41
# 13 --> 42,43,44
# 14 --> 45,46,47



# 15 --> 48,49,50
# 16 --> 51,52,53
# 17 --> 54,55,56
# 18 --> 57,58,59
# 19 --> 60,61,62
# 20 --> 63,64,65

# Graph: [{6,7,8},{9,10,11},{12,13,14},{15,16,17},{18,19,20},{1,21,22,23},{1,24,25,26},{1,27,28,29},{2,30,31,32},{2,33,34,35},{2,36,37,38},{3,39,40,41},{3,42,43,44},
 {3,45,46,47}, {4,48,49,50}, {4,51,52,53}, {4,54,55,56}, {5,57,58,59}, {5,60,61,62}, {5,63,64,65}, {6}, {6}, {6}, {7}, {7}, {7}, {8}, {8}, {8}, {9}, {9}, {9},
 {10}, {10}, {10}, {11}, {11}, {11}, {12}, {12}, {12}, {13}, {13}, {13}, {14}, {14}, {14}, {15}, {15}, {15}, {16}, {16}, {16}, {17}, {17}, {17},
 {18}, {18}, {18}, {19}, {19}, {19}, {20}, {20}, {20}, {1,2,3,4,5}]
 
# Question 6: 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
# (ii) Find the number of paths from Piscataway to Princeton
# (a) Find the actual set using Paths (and list them) 
# (b)BY USING  NuPaths
# (c) By Using GFt

# Answer 6: 

# Piscataway(1) -->  Dunellen(2), Edison(3), Highland Park(4), Middlesex(5), New Brunswick(6), South Plainfield(7), 
# Franklin Township(8), South Bound Brook(9),and Plainfield(10) 

# Franklin Township:  Bridgewater (12), Hillsborough(13), Manville(14), Millstone(15), Montgomery(16), Rocky Hill(17), South Bound Brook, 
#  Princeton(18), New Brunswick, North Brunswick (28), Piscataway and South Brunswick(29)

# Dunellen: Middlesex, Piscataway, Green Brook Township (11), Plainfield

# Edison: East Brunswick (19), Highland Park, New Brunswick, Piscataway, Sayreville (20), South Plainfield, Woodbridge Township (21), Clark (22), 
# Plainfield and Scotch Plains (23)

# Highland Park: Edison, New Brunswick, Piscataway 

# Middlesex: Dunellen, Piscataway, Bound Brook (24), Bridgewater, Green Brook Township, South Bound Brook

# New Brunswick: Piscataway, Highland Park, Edison, North Brunswick, East Brunswick, Franklin Township

# South Plainfield: Piscataway, Edison, Plainfield, Scotch Plains

# South Bound Brook:Bound Brook, Bridgewater, Franklin Township,  Middlesex, Piscataway

# Plainfield: Scotch Plains, Fanwood (25), South Plainfield, Piscataway, Dunellen, Edison, Green Brook Township, North Plainfield (26), Watchung (27)

# (i) G = [{2,3,4,5,6,7,8,9,10}, {5,1,11,10}, {19,4,6,1,20,7,21,22,10,23}, {3,6,1}, {2,1,24,12,11,9}, {1,4,3,28,19,8}, {1,3,10,23}, {12,13,14,15,16,17,9,18,6,28,1,29}, {24,12,8,5,1}, 
 {23,25,7,1,2,3,11,26,27}, {2,5,10},  {8,5,9}, {8}, {8}, {8}, {8}, {8}, {8}, {3,6}, {3}, {3}, {3}, {3,7}, {5,9}, {10}, {10}, {10}, {8}, {8}]

# (ii) 

# Paths(G,1,18,1) = {}

# Paths(G,1,18,2) = {[1, 8, 18]}

# Paths(G,1,18,3) = {[1, 6, 8, 18], [1, 9, 8, 18]}

# Paths(G,1,18,4) = {[1, 2, 1, 8, 18], [1, 3, 1, 8, 18], [1, 3, 6, 8, 18], [1, 4, 1, 8, 18], [1, 4, 6, 8, 18], [1, 5, 1, 8, 18],
# [1, 5, 9, 8, 18], [1, 5, 12, 8, 18], [1, 6, 1, 8, 18], [1, 6, 28, 8, 18], [1, 7, 1, 8, 18], [1, 8, 1, 8, 18], [1, 8, 6, 8, 18], 
# [1, 8, 9, 8, 18], [1, 8, 12, 8, 18], [1, 8, 13, 8, 18], [1, 8, 14, 8, 18], [1, 8, 15, 8, 18], [1, 8, 16, 8, 18], [1, 8, 17, 8, 18],
# [1, 8, 18, 8, 18], [1, 8, 28, 8, 18], [1, 8, 29, 8, 18], [1, 9, 1, 8, 18], [1, 9, 12, 8, 18], [1, 10, 1, 8, 18]} 

# (iii) NuPaths(G,1,18,1) = 0, NuPaths(G,1,18,2) = 1, NuPaths(G,1,18,3) = 2, NuPaths(G,1,18,4) = 26

# (iv) GFt([{2, 3, 4, 5, 6, 7, 8, 9, 10}, {1, 5, 10, 11}, {1, 4, 6, 7, 10, 19, 20, 21, 22, 23}, {1, 3, 6}, {1, 2, 9, 11, 12, 24}, {1, 3, 4, 8, 19, 28}, {1, 3, 10, 23},
# {1, 6, 9, 12, 13, 14, 15, 16, 17, 18, 28, 29}, {1, 5, 8, 12, 24}, {1, 2, 3, 7, 11, 23, 25, 26, 27}, {2, 5, 10}, {5, 8, 9}, {8}, {8}, {8}, {8}, {8}, 
# {8}, {3, 6}, {3}, {3}, {3}, {3, 7}, {5, 9}, {10}, {10}, {10}, {8}, {8}], 1, t)[1] 
# coeff(taylor(%, t = 0, 19), t, 18) = 2263101840843