Self-study
Whenever she considers voting in an election to select one candidate for a position and there is at least one issue important to her, Kay uses the following principle in choosing which course of action to take: it is acceptable for me to vote for a candidate whose opinions differ from mine on at least one issue important to me whenever I disagree with each of the other candidates on even more such issues; it is otherwise unacceptable to vote for that candidate. ██ ███ ████████ ███████ █████████ ███ █████ ██████████ ███ ████████ ███████ ███ ███████ █████ ██ ████ ███ █████ █████████ ██ ████ ███ ████ ██████ ██████ ███ ███████ ██ ████ ██████
Kay's Voting Principle
Kay's principle applies when she's voting in an election and there's at least one issue important to her. It has two parts:
Acceptable: If Kay disagrees with a candidate on at least one important issue, it's acceptable to vote for that candidate as long as she disagrees with each of the other candidates on an even greater number of important issues.
Unacceptable: The word "otherwise" here means "in any situation where the acceptable condition isn't met." So if Kay disagrees with a candidate on at least one important issue, but there's at least one other candidate she disagrees with on the same number or fewer important issues, it's unacceptable to vote for that candidate.
In simpler terms, Kay's principle is all about counting disagreements. It's acceptable to vote for a candidate only if she has a higher number of disagreements with each other candidate.
Here's an example. Suppose Kay has 5 important issues and three candidates are running:
Candidate A
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2 disagreements
Candidate B
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5 disagreements
Candidate C
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4 disagreements
Is it acceptable to vote for Candidate A? Yes, because Kay has more disagreements with each other candidate (5 and 4 are both greater than 2). Is it acceptable to vote for Candidate B? No, because Kay has only 2 disagreements with A and 4 disagreements with C, both of which are fewer than her 5 disagreements with B. Is it acceptable to vote for Candidate C? No, because Kay has only 2 disagreements with A, which is fewer than her 4 disagreements with C.
Anticipation
The question asks what "must be true" about Kay's course of action in any election. So the details about Legrand, Medina, and Norton aren't relevant here. We need to focus only on Kay's general principle and find an answer that is guaranteed by it.
22.
If the statements in the ███████ ███ █████ █████ ███ ██ ███ █████████ ████ ████ ██ ████ █████ █████ ██████ ██ ██████ ██ ███ ████████ ██ ██████ ███ █████████ ███ █ █████████
a
If there are ██ ██████ █████████ ██ ████ ██ ██ ████████████ ███ ███ ██ ████ ███ ███ █████████ ██ ███ █████████
Kay's principle applies when there is at least one issue important to her. If there are no important issues, the principle is silent. We can't conclude that it's unacceptable for her to vote, because the principle simply doesn't address that situation.
b
If she agrees ████ ████ ██ ███ ██████████ ██ ████ ██ ███ ██████ █████████ ██ ████ ██ ██ ████████████ ███ ███ ██ ████ ███ ███ █████████ ██ ███ █████████
Agreeing with each candidate on most important issues doesn't mean she can't vote for any of them. Imagine Kay has 10 important issues. She agrees with Candidate A on 8 issues and disagrees on 2. She agrees with Candidate B on 7 issues and disagrees on 3. She agrees with Candidate C on 6 issues and disagrees on 4. She disagrees with each candidate on at least one issue, but she disagrees with B and C on more issues than she disagrees with A. So it's acceptable to vote for A, even though she agrees with all three candidates on over half of the issues.
c
If she agrees ████ █ ██████████ █████████ ██ ████ ███ █████ █████████ ██ ████ ██ ██ ████████████ ███ ███ ██ ████ ███ ████ ██████████
Agreeing with a candidate on only one important issue doesn't make it unacceptable to vote for that candidate. What matters is how the disagreement count compares to the other candidates. For example, suppose there are 10 important issues and Kay agrees with Candidate A on only 1 (disagrees on 9). That sounds bad, but if she disagrees with every other candidate on all 10 issues, then she disagrees with each other candidate on more issues than she disagrees with Candidate A. So it would be acceptable to vote for Candidate A.
d
If she disagrees ████ ████ ██ ███ ██████████ ██ ███████ █████ ██████ █████████ ██ ████ ██ ██ ████████████ ███ ███ ██ ████ ███ ███ █████████ ██ ███ █████████
If Kay disagrees with each candidate on exactly 3 important issues, then no candidate can satisfy the principle's requirement. For any given candidate, Kay would need to disagree with each other candidate on more than 3 issues. But every other candidate also has exactly 3 disagreements, which is not more than 3. Since no candidate passes the test, it's unacceptable for Kay to vote for any candidate. This must be true.
More generally, whenever Kay has the same number of disagreements with every candidate, she's stuck. The principle requires that she disagree with every other candidate on a greater number of issues. When everyone is tied, that's impossible.
e
If there are ████ ██████ █████████ ██ ███ ██ █████ ███ █████████ ████ █ ██████████ █████████ ████ █████ ███ ████ ██████ ██ █████ ███ ██████ ████ ████ ██████████ ██ ██ ████████████ ███ ███ ██ ████ ███ ████ ██████████
Kay's principle compares her disagreements across candidates, not her agreements vs. disagreements within a single candidate. For example, suppose there are 10 important issues and Kay disagrees with Candidate A on 6 of them (and agrees on 4). If she disagrees with every other candidate on 8 or more issues, it's still acceptable to vote for Candidate A, because she has more disagreements with each other candidate.