And the Winner Is . . .
One man, one vote? Sometimes counting a vote matters less than how you choose to count it
The tiny nation of Yewessay (population: 121) is divided into 11 states of 11 people. Every four years the citizens vote to determine which of two candidates will become president. Each Yewessayan votes for one candidate. The candidate who receives the most votes in a state gets one electoral college vote for that state, and the candidate who receives the most electoral college votes becomes president. 1.
As in the recent United States presidential election, strange things can happen. What is the largest number of popular votes a candidate can receive and not become president?2.
What is the largest number of states a candidate can win and still not win the popular vote?3.
Suppose you can divide the 121 citizens of Yewessay into any number of states, so long as each state contains at least one person. If each state contributes one electoral college vote, what is the largest number of popular votes a candidate can receive and not become president?4.
Suppose that, again, you can divide the 121 citizens of Yewessay into any number of states, each containing at least one person. But this time the number of electoral votes for each state is the same as the state's total population. Now what is the largest number of popular votes a candidate can receive and not become president?5.
Let's go back to 11 states of 11 people each, but now there are three candidates. A candidate must garner more popular votes than any other candidate to win a state (a plurality) and more than half of the electoral college (a majority) to win the presidency. What is the smallest number of popular votes a candidate can receive and still be elected president?
The 121 citizens of Yewessay were polled about their favorite ice cream flavors. They ranked apple (A), banana (B), and chocolate (C) as follows.
|Preference order |
A > B > C
C > B > A
B > C > A
Forty-two people rank apple over banana over chocolate. If the pollsters looked just at people's top choices, they might conclude that apple was the most popular flavor, followed by chocolate and then banana.1.
Another way to determine the winner is to compare pairs of flavors. Do more people prefer A over B, or B over A? What about A versus C, and B versus C? After comparing pairs, how would you rank the three flavors? Which flavor is most popular?2.
Of course the numbers 42, 40, and 39 are extremely close, so the instability of the results may not surprise you. Can you find another trio of numbers that produce the same ranking results but that also give the popular-vote winner the widest possible margin over the popular vote second-place finisher?3.
What is the smallest possible population of Yewessay that could produce all these ranking results without any ties?4.
In another poll, participants ranked three cola brands: Pensicola, Semicola, and Texicola. The results revealed a paradoxical loop: Most people preferred Pensicola over Semicola, Semicola over Texicola, and Texicola over Pensicola. What data could have produced such strange results?
This year eight players will compete in Yewessay's Thimbledon tennis tournament. The contestants play consistently according to rank: The #1 player always beats all other players, the #2 player always beats all but the #1 player, and so on. The judges, who do not know in advance how the players are ranked, argue over scheduling.1.
Judge Knott wants a single elimination tournament: Two players play, then the winner plays one of the remaining players. The winner of that game plays one of the remaining players, and so on, until everyone has played. The winner of the last match gets the gold medal, and the loser gets the silver. Will the gold medal winner always be the best player? What is the worst rank that the silver medal winner could have?2.
Judge Dredd wants a ladder tournament. All players play one game the first day, the winners of the first day's games are paired up the second day, and so on, until two finalists play the last day. The winner of the final match is crowned champion, while the loser is runner-up. Will the gold medal winner always be the best player? What is the worst rank the runner-up could have?3.
Judge Reinhold wants to make sure the silver medal really goes to the #2-ranked player. Each day the judges choose two players to play a single game, taking into account the previous days' results. The tournament ends when the runner-up is determined with certainty. What are the smallest possible number of days and the largest possible number of days that the tournament could take?Solution
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