# The salary of three of Alice's coworkers

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?

• Alice is a man?? :-o – Rand al'Thor May 3 '15 at 18:45
• @randal'thor I know that my english is not the best ;-) I learning it now – Adrian Cid Almaguer May 3 '15 at 18:47
• @randal'thor do you have some answer this time? – Adrian Cid Almaguer May 3 '15 at 18:48
• Yes! See below :-) – Rand al'Thor May 3 '15 at 18:51
• But what happened to Mary???? – Ian MacDonald May 3 '15 at 18:57

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.

• You win this time to @randal'thor (puzzling.stackexchange.com/q/13272/12112) ;-) – Adrian Cid Almaguer May 3 '15 at 18:27
• 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!" – leoll2 May 3 '15 at 18:28
• jajaja, very funny :-O – Adrian Cid Almaguer May 3 '15 at 18:29
• 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! – Rand al'Thor May 3 '15 at 18:53

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.

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.

• 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 – Adrian Cid Almaguer May 3 '15 at 18:57
• @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... :-) – Rand al'Thor May 3 '15 at 19:15