There are three envelopes and exactly two statements are written on each of the envelopes. The statements on one of the envelopes are BOTH TRUE, the statements on the other envelope are BOTH FALSE and the remaining envelope has ONE TRUE and ONE FALSE statement. Here is what is written on the three envelopes:
First Envelope:
(a)The formula is not in here.
(b)The formula is in Envelope 2.
Second Envelope:
(a)The formula is not in Envelope 1.
(b)The formula is in Envelope 3.
Third Envelope:
(a)The formula is not in here.
(b)The formula is in Envelope 1.
Which envelope contains the formula ?
Most of the solutions presented use a case-by-case analysis by trying out assumptions and checking to see if the assumption holds up. In this solution I will solve the puzzle without using that form of analysis.
Statements 1a and 2a both claim the formula is not in Envelope 1. They are either both true or both false, therefore either Envelope 1 or Envelope 2 is the mixed true/false envelope.
Then Envelope 3 must be either the both true or both false envelope. The statements on Envelope 3 then imply that the formula is not in Envelope 2.
With this information statements 1a and 2a are equivalent to "The formula is in Envelope 3". This then implies that both statements 2a and 2b are the same. Therefore Envelope 2 cannot be the mixed envelope, which means Envelope 1 is the mixed envelope.
Either the Envelope 2 or Envelope 3 is the double true envelope, which implies the formula is in either Envelope 1 or Envelope 3. Therefore statement 1b is False.
Envelope 1 is the mixed envelope. Then statement 1a is true, which means the formula is in Envelope 3.
Finally, this then means that Envelope 2 is the double true and Envelope 3 is the double false. All of the conditions of the problem have been satisfied so the conclusion that the formula is in Envelope 3 does not introduce any paradoxes.
Solution without using case-by-case analysis
