Question 2: Pocket Politics (any resemblance to recent events is purely coincidental) Long time ago, in a country far far away, a Totally Racist Unqualified Malicious President ruled the land with an iron fist. His rivals had to do everything in their power to free the people from his terrible regime and make the country great again. Many people offered to take the TRUMP down, but six wise and brave men and women stood out from the rest: 1. Joe "Busy Hands" Bye-then (1) 2. Bernie "Crazy Eyes" Slanders (B) 3. Elizabeth "Pocahontas" Warden (E) 4. Tulsi "Go Land Crabs!" Globbard (T) 5. Mike "Mini Me Broomfield (M) 6. Pete "Father-of-Chickens Boot-a-Judge (P) a. It's everyone's favorite part, the sword fight! This is a tournament style competition. The first attempt is a round-robin tournament, where every candidate fights every other candidate exactly once. Every fight ends when one candidate is on the ground for more than five seconds. There is no draw. How many fights are there in total? (that's an easy one). b. After the fights end you realize that no candidate won all of his/her fights. Use the pigeon hole principle to show that there must be at least two candidates with the same number of wins (be rigorous in your proof). c. Since no clear ranking could be established at the previous fights, the candidates are now paired up tournament style and each pair fights it out until someone gives up. How many ways do you have to divide the six candidates into three pairs? (Hint: How many ways can you select one pair? How many ways does it leave you to select the second pair? Then the third? Don't forget to eliminate redundancy due to order). d. Finally, three candidates are left - "Busy Hands", "Crazy Eyes" and "Pocahontas". It's time for the audience to vote. Each citizen casts his/her ballot by ranking the three candidates in preferred order: 1 being the favorite candidate and 3 being the least favorite. The votes are as follows (in millions): 30 25 | Voters (millions) 1st choice 2nd choice 3rd choice 45 Busy Hands Crazy Eyes Pocahontas Crazy Eyes Pocahontas Busy Hands Pocahontas Crazy Eyes Busy Hands The votes are assigned using three different methods: 1. "Borda method". Each candidate gets three points whenever s/he is voted first, two points when s/he is voted second and one point when voted third. 2. "Plurality method": The candidate with the most first places wins. 3. "Majority method": The candidate with the absolute majority wins. Who wins in each case (if at all)? Question 2: Pocket Politics (any resemblance to recent events is purely coincidental) Long time ago, in a country far far away, a Totally Racist Unqualified Malicious President ruled the land with an iron fist. His rivals had to do everything in their power to free the people from his terrible regime and make the country great again. Many people offered to take the TRUMP down, but six wise and brave men and women stood out from the rest: 1. Joe "Busy Hands" Bye-then (1) 2. Bernie "Crazy Eyes" Slanders (B) 3. Elizabeth "Pocahontas" Warden (E) 4. Tulsi "Go Land Crabs!" Globbard (T) 5. Mike "Mini Me Broomfield (M) 6. Pete "Father-of-Chickens Boot-a-Judge (P) a. It's everyone's favorite part, the sword fight! This is a tournament style competition. The first attempt is a round-robin tournament, where every candidate fights every other candidate exactly once. Every fight ends when one candidate is on the ground for more than five seconds. There is no draw. How many fights are there in total? (that's an easy one). b. After the fights end you realize that no candidate won all of his/her fights. Use the pigeon hole principle to show that there must be at least two candidates with the same number of wins (be rigorous in your proof). c. Since no clear ranking could be established at the previous fights, the candidates are now paired up tournament style and each pair fights it out until someone gives up. How many ways do you have to divide the six candidates into three pairs? (Hint: How many ways can you select one pair? How many ways does it leave you to select the second pair? Then the third? Don't forget to eliminate redundancy due to order). d. Finally, three candidates are left - "Busy Hands", "Crazy Eyes" and "Pocahontas". It's time for the audience to vote. Each citizen casts his/her ballot by ranking the three candidates in preferred order: 1 being the favorite candidate and 3 being the least favorite. The votes are as follows (in millions): 30 25 | Voters (millions) 1st choice 2nd choice 3rd choice 45 Busy Hands Crazy Eyes Pocahontas Crazy Eyes Pocahontas Busy Hands Pocahontas Crazy Eyes Busy Hands The votes are assigned using three different methods: 1. "Borda method". Each candidate gets three points whenever s/he is voted first, two points when s/he is voted second and one point when voted third. 2. "Plurality method": The candidate with the most first places wins. 3. "Majority method": The candidate with the absolute majority wins. Who wins in each case (if at all)
