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 A student requests a recommendation letter from a professor. The professor gives three sealed envelopes. Each envelope contains either a good recommendation letter or a bad recommendation letter.

  1. Make a list of all the possible scenarios.
  2. Suppose now the professor tells the student that exactly one envelope contains a good recommendation letter and the other two contain bad recommendation letters. In the list of scenarios you prepared above, mark the ones that are still possible.
  3. On envelope $1$ is written the clue "This contains a bad recommendation letter".On envelope $2$ is written the clue "This contains a bad recommendation letter". On envelope $3$ is written the clue "Envelope $2$ contains a good recommendation letter". Suppose now the professor gives the additional information that exactly one of these three clues are true and the other two are false. Can the student find out the contents of the envelopes without breaking their seals? 
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a.  $2^3=8$ possible scenarios can be there.

  $Envelope_1$ $Envelope_2$ $Envelope_3$
1. $Good\ recommendation\ letter $ $Good\ recommendation\ letter $ $Good\ recommendation\ letter $
2. $Good\ recommendation\ letter $ $Good\ recommendation\ letter $ $Bad\ recommendation\ letter $
3. $Good\ recommendation\ letter $ $Bad\ recommendation\ letter $ $Good\ recommendation\ letter $
4. $Good\ recommendation\ letter $ $Bad\ recommendation\ letter $ $Bad\ recommendation\ letter $
5. $Bad\ recommendation\ letter $ $Good\ recommendation\ letter $ $Good\ recommendation\ letter $
6. $Bad\ recommendation\ letter $ $Good\ recommendation\ letter $ $Bad\ recommendation\ letter $
7. $Bad\ recommendation\ letter $ $Bad\ recommendation\ letter $ $Good\ recommendation\ letter $
8. $Bad\ recommendation\ letter $ $Bad\ recommendation\ letter $ $Bad\ recommendation\ letter $

 

b.

  $Envelope_1$ $Envelope_2$ $Envelope_3$
1. $Good\ recommendation\ letter $ $Good\ recommendation\ letter $ $Good\ recommendation\ letter $
2. $Good\ recommendation\ letter $ $Good\ recommendation\ letter $ $Bad\ recommendation\ letter $
3. $Good\ recommendation\ letter $ $Bad\ recommendation\ letter $ $Good\ recommendation\ letter $
4. $Good\ recommendation\ letter $ $Bad\ recommendation\ letter $ $Bad\ recommendation\ letter $
5. $Bad\ recommendation\ letter $ $Good\ recommendation\ letter $ $Good\ recommendation\ letter $
6. $Bad\ recommendation\ letter $ $Good\ recommendation\ letter $ $Bad\ recommendation\ letter $
7. $Bad\ recommendation\ letter $ $Bad\ recommendation\ letter $ $Good\ recommendation\ letter $
8. $Bad\ recommendation\ letter $ $Bad\ recommendation\ letter $ $Bad\ recommendation\ letter $

 

c.

Case 1. Suppose clue written on  envelope $1$ is true.

So this means envelope $1$ contains bad recommendation letter

Clue written on envelope $2$ is false $implies$ Envelope $2$ contains a good recommendation letter is TRUE

Clue written on envelope $3$ is false $implies$ Envelope $2$ contains a bad recommendation letter is TRUE

and thus clues written on envelope $2$ and envelope $3$ are contradicting each other

so this case is NOT POSSIBLE

 

Case 2. Suppose clue written on  envelope $2$ is true.

So this means that envelope $2$ contains a bad recommendation letter.

Clue written on envelope $3$ is false $implies$ Envelope $2$ contains a bad recommendation letter is TRUE

Clue written on envelope $1$ is false $implies$ Envelope $1$ contains a good recommendation letter is TRUE

Since any one of the envelope can is containing good recomendation letter so $3$ will definitely contain false recommendation letter.

Hence envelope $1$ contains the good recommendation letter.

 

Case 3. Suppose clue written on  envelope $3$ is true.

So this means that envelope $2$ contains a bad recommendation letter.

Clue written on envelope $2$ is false $implies$ Envelope $2$ contains a good recommendation letter is TRUE

and thus clues written on envelope $2$ and envelope $3$ are contradicting each other

so this case is NOT POSSIBLE

 

Hence $CASE\ 2$ is tthe only valid case.

$\therefore$ The student can find out the contents of the envelope without breaking the seals.

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