Tom, Paul and Joe have just learnt the ranks that they reached in a competition. None of them knows the ranks of the other two, but they all know that Tom and Paul ended up in consecutive positions. As they were curious to know each other's ranks, this little conversation took place:

Paul declares: I don't know Tom's rank.

Tom: Me neither. I also don't know your rank.

Paul: I already knew this.

Joe: Now I know that I am the last among the three of us.

Tom : Me too. Before I just knew that I'm not the first among the three of us.

What's everyone's rank?


  • The ranks are strictly postive integers.
  • No rank occurs twice.


Every ones knows the maximum rank that could be obtained.

  • $\begingroup$ When did Tom know he was not first? $\endgroup$
    – kaine
    Feb 13 '15 at 19:43
  • $\begingroup$ follow the discussion order ... evey declaration implies something . $\endgroup$
    – Abr001am
    Feb 13 '15 at 19:56
  • $\begingroup$ I did; the wording is ambiguous. "I knew"? When did you know? The start? Before you made your statement? Before Paul made the statement that told you that you are last? Those are the 3 options. Is determining which part of the puzzle? Punctuation on that line would help. $\endgroup$
    – kaine
    Feb 13 '15 at 20:04
  • $\begingroup$ please i did rephrase my problem , the ranking intended in these declarations is between tom ,paul and joe , but not the global grades ! $\endgroup$
    – Abr001am
    Feb 13 '15 at 20:33
  • $\begingroup$ the edit on the last statement was delusive ... tom knew just after joe s statement that he s not the fist because if joe didnt speak paul might be behind tom . and staying speechless is not indicative situation btw. $\endgroup$
    – Abr001am
    Feb 14 '15 at 16:09

Note, my numbers are for if a higher rank is better..

Joe's rank is 4, Tom's rank is 5, and Paul's rank is 6

Paul: "I don't know Tom's ranking."

Paul knows that he and Tom have consecutive positions, so if his degree was either 1 or N (the max), he would know Tom's ranking.

Tom: "I don't know yours, either."

Tom now knows that Paul's ranking is neither at the top nor at the bottom, but since he still doesn't know Paul's ranking that means his ranking is not 2 or N-1.

Paul: "I already knew that."

This means Paul already knew Tom wasn't at 2 or N-1, so Paul's ranking must be greater than 3 and less than N-2. Note: I'm assuming here that Paul would have mentioned if they knew Tom's ranking at this point. If that's not the case, then Paul could have 2 or N-1 and know (although he hasn't said it) Tom's rank.

Joe: "Now I know that I am last among the three of us."

Right before Paul's second statement, Joe would only know that Paul's ranking must be at least 2, with Tom's possibly being 3. After Paul's second statement, Tom could still be at 3 with Paul being at 4. Joe's ranking must be 4 because he now knows he is last - up until Paul's second statement there was the possibility that Tom and Paul could be below Joe, but since Paul cannot be lower than 4, and Joe has 4, he knows Paul and Tom must be above him.

Tom : Me too I just knew that I'm not the first between the three of us.

Up until Joe's statement, Paul could have been in rank 4. Now that Tom knows he is below Paul, he must have rank 5, otherwise he would not have been able to tell. That leaves Paul to be rank 6.

  • $\begingroup$ i think your last statement is mistaken ..... paul isnt 6th graded ! it is right but inversed hahaha $\endgroup$
    – Abr001am
    Feb 13 '15 at 20:43
  • 1
    $\begingroup$ Thanks, I still don't like some of the wording in the question but this is the first answer without a logical flaw. $\endgroup$
    – kaine
    Feb 13 '15 at 20:45
  • $\begingroup$ kaine may because english is not my native tongue . the first declaration means paul isnt 1 or n. second means T is not 2 or n-1. third means paul paul is between 4 and n-3. when J heared this (order has importance) he d just declared that he is the last means n-3 because the ranking(T,P)or(P,T)=(n-3-k,n-3-k-1) is impossible the last locality remained which made tom declare he s the second is n-4 because he was surethat noone is located between him and joe . so this make paul in front = 4 and n=9 $\endgroup$
    – Abr001am
    Feb 13 '15 at 20:58

J,T and P's grade are respectively 3, 4 and 5

let's presume that the max grade is 100, but it makes no difference

Paul doesn't know Tom's grade. since they are consecutive integers, Paul cannot have the grade 1 or 100, because then he'd know Tom's grade

even with that additional information, Tom doesn't know Paul's grade. so Tom cannot have 2 or 98.

Paul already knew that Tom wouldn't know his grade. if Paul was 98, there would be a chance for Tom to be 99. so Paul cannot be 2 or 98 either.

now Joe says that he knows for sure he is last. he must be have a grade of 3

since now Tom knows that he is not first, it means Paul has a higher grade than him. the only way Tom can be sure of that is if there is no possible grade in between himself and Joe

  • $\begingroup$ joe stated that he is the last !!! how could he be the third ? $\endgroup$
    – Abr001am
    Feb 13 '15 at 20:06
  • $\begingroup$ you are right, I said it wrong. he is not third, he has a grade of 3. big difference there. $\endgroup$ Feb 13 '15 at 20:08
  • $\begingroup$ i dont see where was your adjustment on your post be more enlightening ? $\endgroup$
    – Abr001am
    Feb 13 '15 at 20:13

My answer:

Paul claims that he doesn't know Tom's ranking. Because Paul also knows what the best ranking is, he cannot have a perfect score, else he would know what Tom's ranking was, knowing that they are consecutive. However, he may still be first if the other two scored less than him.
Tom says the same as Paul, indicating that his ranking is also not a perfect score.
Paul says that he knows this fact already, proving that Paul's ranking must be at least 3.
Joe states that he knows he is last. That places him last.
Tom states that he knows he is not first. We know he isn't last, either. That places him in the middle.
Paul must then be first.
If Paul must be at least 3 and comes in first, then we can say his rank is 3; Tom in the middle is rank 4; and Joe being last is rank 5.

  • $\begingroup$ i need numbers and i v already noted that $\endgroup$
    – Abr001am
    Feb 13 '15 at 20:03

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