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.
