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Alice would like to determine the relative salaries of three coworkers using two facts. First, she knows that if Pauline is not the highest paid of the three, then Paul is. Second, she knows that if Paul is not the lowest paid, then Peter is paid the most. Is it possible to determine the relative salaries of Pauline, Peter, and Paul from what Alice knows? If so, who is paid the most and who the least?

Explain your reasoning.

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  • $\begingroup$ Alice is a man?? :-o $\endgroup$ May 3, 2015 at 18:45
  • $\begingroup$ @randal'thor I know that my english is not the best ;-) I learning it now $\endgroup$ May 3, 2015 at 18:47
  • $\begingroup$ @randal'thor do you have some answer this time? $\endgroup$ May 3, 2015 at 18:48
  • $\begingroup$ Yes! See below :-) $\endgroup$ May 3, 2015 at 18:51
  • $\begingroup$ But what happened to Mary???? $\endgroup$ May 3, 2015 at 18:57

3 Answers 3

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The only valid order is

Pauline > Peter > Paul

We have 2 sentences:

A) Pauline is paid the most OR Paul is paid the most.
B) Paul is paid the lowest OR Peter is the richest

If we suppose Paul being the richest, then Peter can't be the richest, forcing Paul to be the least paid. Contradiction!
On the contrary, if we suppose Pauline being paid the most, Peter can't still be the richest, so Paul's wage is the lowest, putting Peter in the middle of the wageboard.

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  • $\begingroup$ You win this time to @randal'thor (puzzling.stackexchange.com/q/13272/12112) ;-) $\endgroup$ May 3, 2015 at 18:27
  • $\begingroup$ Ahah he wasn't fast enough this time! When I temporarily lost the internet connection while writing my answer I was thinking "Oh no, I'll be the second again!" $\endgroup$
    – leoll2
    May 3, 2015 at 18:28
  • $\begingroup$ jajaja, very funny :-O $\endgroup$ May 3, 2015 at 18:29
  • $\begingroup$ I was away from the computer for a while, but now I've composed a nice answer :-) No question about who was first this time though! $\endgroup$ May 3, 2015 at 18:53
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From the second statement "if Paul is not the lowest paid, then Peter is paid the most." any order without Paul at the bottom must have Pauline at the bottom, Peter first and Paul second.

But the first statement "if Pauline is not the highest paid of the three, then Paul is" would contradict that situation so therefore Paul must be the lowest paid.

To fulfil the second statement and ensure Paul is the lowest paid Pauline must be paid the most.

Therefore the order from highest paid to lowest is:

Pauline, Peter, Paul.

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The other answers are great, but I'd just like to present it in a more mathematical-logic style.


We know:

  • S1: if Pauline is not the highest paid, then Paul is the highest paid;
  • S2: if Paul is not the lowest paid, then Peter is the highest paid.

We deduce as follows.

ASSUME Pauline is not the highest paid. By S1, Paul is the highest paid and therefore not the lowest paid. By S2, Peter is the highest paid. Contradiction, so Pauline is the highest paid.

ASSUME Paul is not the lowest paid. By S2, Peter is the highest paid. But we already know Pauline is the highest paid. Contradiction, so Paul is the lowest paid.

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  • $\begingroup$ there is not problem with questions like my firsts two question? I know that they are easy to solve, but I think that they are good to some people that loves this kind of question $\endgroup$ May 3, 2015 at 18:57
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    $\begingroup$ @AdrianCidAlmaguer I've enjoyed both your questions so far. The first one was on the edge of being a maths problem rather than a maths puzzle (see here), but this one was great. As long as you don't make them too easy though... :-) $\endgroup$ May 3, 2015 at 19:15

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