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# Rigging a Vote

There are five candidates participating in an unusual election. There are four rounds of voting, and only two candidates participate in each round, with the winner moving on and the loser dropping out. This means first candidate 1 will run against candidate 2, then the winner will run against candidate 3, and so on, with the winner of the last round being elected.

Polling shows that Stephen Hawking is not very popular. There are five equal-sized groups in the voting population, and polling shows they prefer the candidates in the following order:

• Group 1: Einstein > Newton > Maxwell > Hawking > Curie
• Group 2: Newton > Curie > Maxwell > Einstein > Hawking
• Group 3: Curie > Maxwell > Einstein > Newton > Hawking
• Group 4: Einstein > Newton > Hawking > Curie > Maxwell
• Group 5: Newton > Maxwell > Einstein > Hawking > Curie

But Hawking has a secret weapon – he gets to choose the order the candidates participate in each round.

Is there a way for Hawking to organize the rounds such that he wins?

#### Solution

Yes, Hawking can win, despite being one of the three least popular candidates in every group.

First of all, whichever candidate is left to the last round has the best chance of winning, since they only need to beat one person. So Hawking must put himself last.

Then Hawking has to determine the candidate he has the best chance of winning against, and figure out a way to get them to win the second-to-last round. It turns out that Hawking would only win against Curie in a head-to-head (he polls ahead of her in 3 of the 5 groups), so Curie is in the second-to-last round.

Then the same logic applies to getting Curie to win the second-to-last round: determine the candidate she has the best chance of winning against, and figure out a way to get them to win the third-to-last (second) round. This would be Maxwell (Curie polls ahead of Maxwell in 3 of the 5 groups).

Maxwell, in turn, wins against Einstein head-to-head, and Einstein wins against Newton head-to-head. This means for Hawking to win, he should arrange the following rounds:

• Newton vs. Einstein [Einstein wins 3/5 groups]
• Einstein vs. Maxwell [Maxwell wins 3/5]
• Maxwell vs. Curie [Curie wins 3/5]
• Curie vs. Hawking [Hawking wins 3/5]