Elections

There were three candidates in the elections - Amber, Pearl, and the last minute candidate Crystal. 100 people were eligible to cast a vote. The voters were asked a series of questions at the end of the vote.

* if it had been just Amber and Pearl, which would you have voted for?
* is Crystal your least favorite candidate?
* which of the two candidates you didn't vote for do you like better?
* is there anyone you would never vote for?

61 people preferred Pearl over Amber. 18 said Crystal is their least favorite candidate, but she took some comfort in learning that 50% of the people eligible to vote would have voted for her, had they been required to change their vote, giving her a win in this category by a margin of two votes. Amber was brought almost to tears when she learned that 59 of the 100 voters considered her their least favorite candidate.

The election itself was a simple one. The 100 people each wrote a name on a piece of paper and the one who got the most votes was elected. Is the above information sufficient to tell who won and by getting how many votes?

Originally posted by talzamir
There were three candidates in the elections - Amber, Pearl, and the last minute candidate Crystal. 100 people were eligible to cast a vote. The voters were asked a series of questions at the end of the vote.

* if it had been just Amber and Pearl, which would you have voted for?
* is Crystal your least favorite candidate?
* which of the two candidates ...[text shortened]... was elected. Is the above information sufficient to tell who won and by getting how many votes?

I'm surprised this hasn't been solved by now.

Yes, I think it is sufficient, with a couple of reasonable inferences. I say Amber wins with 39 votes, Crystal gets 32, and Pearl 29.

Show your work, please.

Originally posted by apathist
Show your work, please.

OK, although the voters are only required to vote for their favorite candidate, we can infer that they had in mind an order of preference. This is seen from the Q&As after the vote. So let's see what would have happened if they had been asked to rank the 3 candidates in order of preference.

From the questions and answers:
1) A would be placed last by 59 voters, C last by 18 voters, and therefore P would be ranked last by the other 23.

2) 50 people said they would have voted for C, if they had been required to change their vote. This implies that 50 voters would have ranked her second, if they had been required to rank all 3 candidates. Also, the "margin of two votes" means that either A or P would be ranked second by 48 voters.

3) As we already know that A is placed last by 59, there cannot be another 48 people who put her second. So P is ranked second by 48 and therefore A is placed second by only 2 voters.

4) Now we know the first preferences must have been A39, C32, P29

Nicely done.

As background for this.. my son's class is starting studying a new foreign language next year in addition to our native tongue and English. Everyone was given a questionnaire asking to put in preference whether they want to take Russian, German, Swedish, French, or nothing at all. As he's interested in languages in general and learns them easily, he listed four choices. A bit later we heard that there had been enough interest for a Swedish class, so it is starting next year - for everyone who had picked it as first or second choice, so despite interest to study anything and everything, my son got excluded from the class. Disappointed, I put the feeling up as a math puzzle.

Good puzzle!.

I understand your disappointment. It was a good idea for the school to take a vote to determine which language to offer; but why use it to exclude someone who wants to do the course, even if it wasn't his first or second choice?? You might be interested to read this, if you haven't already seen it.:

http://www.sss.ias.edu/files/papers/papereleven.pdf

Thank you. A good read.