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In a contest there are 10 questions asked, which have to be marked as true (T) or false (F).
You get 1 point for a correct answer and 0 points for a wrong answer.
Joe, Jack, William, and Averell are answering the questions as follows:

name 1 2 3 4 5 6 7 8 9 10
Joe T T T F T T F F T T
Jack F T F T T F F F F F
William T F F T F F T F T F
Averell T T F T T F F T T F

After test evaluation, Joe gets 6 points, Jack gets 8 points and William gets 7 points.
How many points does Averell get?

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  • $\begingroup$ Please mention the source of this puzzle. $\endgroup$ Nov 16, 2021 at 16:58

5 Answers 5

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Another possible reasoning:

We only look at the first three people. They have given $30 - 6 - 8 - 7 = 9$ wrong answers in total.
However for every question except question 8, the three of them don't have identical answers, hence there is at least one wrong answer.
These already add up to $9$ wrong answers, which means that exactly one answer is wrong for each question, except that for question 8 there is no wrong answer.
Thus the correct answer to each question is simply the "majority" of the three answers.

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Combining the reasoning of AxiomaticSystem and WhatsUp, for what I feel (after upvoting theirs) is a slicker answer:

  • Each question has one outlier answering differently from the rest: Joe for 4 questions, Jack for 2 questions, William for 3 questions, Averell for 1 question.
  • So the total number of points awarded to Jack, Joe, and William is either 1 or 2 for each of nine questions, either 0 or 3 for one question. Their overall total is between 9 and 21 (inclusive).
  • The upper bound is achieved, so the majority answer is always right, and each person's score is exactly ten minus the number of questions where they were an outlier.

  • So Averell's score is 9. Congratulations, Averell is top of the class!

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  • $\begingroup$ Upvoting you back for arriving at something that looks like it could be the intended solution. $\endgroup$ Nov 11, 2021 at 20:09
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Averell's score is

9

Reasoning

Joe and Jack differ in six of their answers. If Jack had five or six of those correct then Joe can at most score five. Hence Jack must only have four of those answers correct and for the questions on which they match - 2, 5, 7, 8 - they must both have the correct answers.
William answered differently on three of those - 2, 5, 7 so these must be his the incorrect ones and the rest for William are correct.
Hence the correct answers are: t t f t t f f f t f
This makes Averell's score 9

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The answers are

TTFTTFFFTF, and Avarell got 9 points.

First, notice that

for every question, exactly one person answered differently from the rest. Depending on whether the outlier was correct or not, either 1 or 3 points are added to the sum of all four scores, and after 10 questions the sum is even. Therefore, Avarell's score is odd.

In addition,

Avarell's answers only differ from Jack's in three places: 1, 8, and 9, so her score cannot be less than 5. What remains is to determine: How many of these three questions can be true?

We can divide the questions neatly into three sets: 1/8/9, where Avarell differs from Jack, 3/4/6/10, where Joe differs from everyone, and 2/5/7, where William differs from everyone.
Clearly all three of 1/8/9 cannot be true, because Jack could only have up to 7 points. If two are true, then those are Jack's misses, and the rest are correct. Joe misses his entire set, and William misses his; in order for these scores to all match up, they can't miss any more and the two trues must be 1 and 9. If we do that, everything is valid, and Avarell has nine points!
If only one is true, then Joe and William miss at least one of Jack's set. But Jack has to miss one from each of Joe and William's sets to match their scores, and he only has one miss left; a contradiction. The case for zero true is similar, so the solution we already found is the only one!

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I just grouped answers into clusters and brute forced on Jack's answers as he had only 2 wrong answers. my solution

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