#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 1. Describe the problem that Euler solved regarding 7 bridges. The problem that Euler solved regarding 7 bridges was the Seven Bridges of Konigsberg problem, a historically notable problem in mathematics. The city of Konigsberg in Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands. The two islands were connected to each other and to the two mainland portions of the city by seven bridges. The problem was to devise a walk through the city that would cross each of those bridges only once. 2. Draw it on a piece of paper with a line segment representing every edge. 1----------2 | / | | / | | / | 3----------4 3. If you toss a fair coin 2000 times, what is the probability that you get exactly 1000 heads (and exactly 1000 tails)? binomial(2000,1000)*(1/2)^2000 4. If you roll a fair die 6000 times, what is the probability that each of the outcomes {1,2,3,4,5,6} occurs exactly 1000 times? binomial(6000,1000)*binomial(5000,1000)*binomial(4000,1000)*binomial(3000,1000)*binomial(2000,1000)*(1/6)^6000*(5/6)^15000 5. Pick 5 random Facebook friends. For each of them pick 3 friends. For each of the friends of the friends pick 3 friends. Label the people picked 1,2,...,n. Write the graph in our data structure. [{2, 17, 28, 29, 54, 60, 63}, {1, 39, 40, 45, 48, 51, 63}, {21, 24, 25, 36, 42, 45, 48, 52, 63}, {9, 16, 22, 34, 48, 57}, {7, 9, 13, 21, 24, 29, 44, 48, 56, 65}, {24, 27, 47, 48}, {5, 15, 35, 39, 50, 58}, {33, 40, 42, 44, 57, 63}, {4, 5, 16, 38, 47, 50, 65}, {21, 41, 45, 56, 63}, {26, 45, 50, 54, 55, 61, 63}, {37, 50, 57, 62}, {5, 22, 29, 36, 41, 42}, {17, 21, 25, 36, 37, 44, 53}, {7, 17, 19, 20, 44, 51}, {4, 9, 28, 34, 35, 55, 57}, {1, 14, 15, 19, 39, 46, 49, 50, 64, 65}, {42}, {15, 17, 22, 29, 35, 37, 61}, {15, 33, 41, 46, 50, 52}, {3, 5, 10, 14, 45, 49}, {4, 13, 19, 24, 37, 41, 42, 55, 57, 63}, {29, 46, 48, 65}, {3, 5, 6, 22, 29, 39, 41, 42, 49, 52}, {3, 14, 26, 30, 42, 48}, {11, 25, 30, 41, 46, 49, 52, 53, 58, 61, 64}, {6, 39, 53, 54, 60}, {1, 16, 36, 38, 43, 51, 52, 59}, {1, 5, 13, 19, 23, 24, 30, 50, 57, 61}, {25, 26, 29, 43, 53, 54, 64, 65}, {43, 44, 46, 65}, {46, 53, 56}, {8, 20, 43}, {4, 16, 38, 65}, {7, 16, 19, 48, 51, 65}, {3, 13, 14, 28, 41, 45, 48, 50, 60}, {12, 14, 19, 22, 50, 60, 62}, {9, 28, 34, 40, 51, 53, 55, 59, 64}, {2, 7, 17, 24, 27, 53}, {2, 8, 38, 43, 61}, {10, 13, 20, 22, 24, 26, 36, 49}, {3, 8, 13, 18, 22, 24, 25, 45, 51, 54, 64}, {28, 30, 31, 33, 40, 46}, {5, 8, 14, 15, 31}, {2, 3, 10, 11, 21, 36, 42, 49, 55, 58, 59}, {17, 20, 23, 26, 31, 32, 43, 48, 51, 54, 60}, {6, 9, 52, 65}, {2, 3, 4, 5, 6, 23, 25, 35, 36, 46, 52, 58, 64}, {17, 21, 24, 26, 41, 45, 51, 55, 63}, {7, 9, 11, 12, 17, 20, 29, 36, 37, 57, 63}, {2, 15, 28, 35, 38, 42, 46, 49, 64}, {3, 20, 24, 26, 28, 47, 48}, {14, 26, 27, 30, 32, 38, 39, 59}, {1, 11, 27, 30, 42, 46, 59, 62}, {11, 16, 22, 38, 45, 49, 60, 62}, {5, 10, 32}, {4, 8, 12, 16, 22, 29, 50, 59, 63}, {7, 26, 45, 48, 60}, {28, 38, 45, 53, 54, 57}, {1, 27, 36, 37, 46, 55, 58, 63}, {11, 19, 26, 29, 40}, {12, 37, 54, 55, 63, 65}, {1, 2, 3, 8, 10, 11, 22, 49, 50, 57, 60, 62, 65}, {17, 26, 30, 38, 42, 48, 51}, {5, 9, 17, 23, 30, 31, 34, 35, 47, 62, 63}] #Randomly generated using RandG 6. Look at the cities that border Piscataway and for each of them and again until you get to Princeton i. Construct the graph Piscataway / | / Edison / | / New Brunswick / / \ / North Brunswick --- East Brunswick / | Montgomery --- South Brunswick \ / Princeton ii. Find the number of paths from Piscataway to Princeton a. Find the actual set using paths (and list them) [Piscataway, Edison, New Brunswick, North Brunswick, South Brunswick, Princeton], [Piscataway, Edison, New Brunswick, North Brunswick, South Brunswick, Montgomery, Princeton], [Piscataway, Edison, New Brunswick, East Brunswick, North Brunswick, South Brunswick, Princeton], [Piscataway, Edison, New Brunswick, East Brunswick, North Brunswick, South Brunswick, Montgomery, Princeton] [Piscataway, Montgomery, Princeton], [Piscataway, Montgomery, South Brunswick, Princeton] b. By using NuPaths Piscataway-1, Edison-2, New Brunswick-3, North Brunswick-4, East Brunswick-5, South Brunswick-6, Montgomery-7, Princeton-8 [{2,7},{1,3},{2,4,5},{3,5,6},{3,4},{4,7,8},{1,6,8},{6,7}] seq(NuPaths([{2, 7}, {1, 3}, {2, 4, 5}, {3, 5, 6}, {3, 4}, {4, 7, 8}, {1, 6, 8}, {6, 7}], 1, 8, k), k = 1 .. 20); 0, 1, 1, 5, 9, 28, 63, 172, 422, 1109, 2814, 7318, 18817, 48819, 126210, 327373, 848285, 2200873, 5708645, 14814664 c. By using GFt GFt([{2, 7}, {1, 3}, {2, 4, 5}, {3, 5, 6}, {3, 4}, {4, 7, 8}, {1, 6, 8}, {6, 7}], 8, t)[1]; 2 / 4 3 2 \ t \t + t - 3 t - t + 1/ - ---------------------------------------------- / 3 2 \ / 4 3 2 \ \t + 3 t - 1/ \2 t + 3 t - 3 t - 2 t + 1/ seq(coeff(taylor(%, t = 0, k + 1), t, k), k = 1 .. 20); 0, 1, 1, 5, 9, 28, 63, 172, 422, 1109, 2814, 7318, 18817, 48819, 126210, 327373, 848285, 2200873, 5708645, 14814664