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This book contains section about a King who gives his prisoners a chance to break free and win a bride in a process -- by choosing one of two presented doors. The catch -- there's either a Lady or a tiger behind a door and prisoner needs to reason, using signs on the doors and additional hints from the King.

One of puzzles, "Day 1 Trial 2", have, in my opinion, incorrect solution, contradicting conditions author gave for first day's trials, specifically:

Two rooms which can contain:

  1. One a lady, the other a tiger
  2. Ladies in both rooms
  3. Tigers in both rooms

For 2nd trial, doors have following signs on them:

  1. "At least one of these rooms contains a Lady"
  2. "A tiger is in the other room"

and King states that signs either both true or both false.

Now, my solution:

  1. Consider both false -- there's no contradiction: "None of rooms contains a Lady" and "A tiger is in this room", as conditions state both rooms can contain tigers;
  2. Consider both true -- again, no contradiction, and Lady must be in second room.

So, I guess, feeling-lucky prisoner can choose 2nd room and not-so-brave-one can skip the chance and go back to prison cell (although, author doesn't seem to mention such a possibility).

The thing is, the book gives different solution, which, I think, is based on 'there's one Lady and one tiger' precondition.

Can you please verify if my reasoning correct?

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    The negation of "A tiger is in the other room" is "a lady is in the other room". – 2017-02-21
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    @lulu .. or more precisely "there is no tiger in the other room", which by the overall rules of the game is equivalent to "a lady is in the other room" – 2017-02-21
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    @HagenvonEitzen Yes, that's right. I am assuming the rules of the game. – 2017-02-21

2 Answers 2

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The proper negation of "a tiger is in the other room" is "there isn't a tiger in the other room", not "there is a tiger in this room" as you would have it.

Now, the rules of the game assure us that every room must contain either a tiger or a lady (but not both). Thus the statement "there isn't a tiger in the other room" implies "there is a lady in the other room". Were that true, it would imply that the first sign was true (as we'd know that there was at least one lady present). Thus, assuming the second statement is false implies that the first statement is true and (since we know that either both are true or both are false) we can conclude that both must be true.

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Since lulu has already answered this question, I'd like to point out that working logic puzzles by assuming statements to be true or false is a best a tricky proposition, and often such problems are designed exactly to trap people who attempt them in this way. (A common trick is to make one of the statements equivalent to "this statement is true". The person who assumes it is true will find that it is true. The person who assumes it is false will also find that it is false.)

A far safer approach when possible is to list the possibilities and check which possibilities match the conditions for a solution. In this case, we are given that there are only 4 possibilities. So it takes very little effort to examine the truth values of the signs in each case: $$\begin{array}{cc|cc}\text{Room 1}&\text{Room 2}&\text{Sign 1}&\text{Sign 2}\\\hline\text{Lady}&\text{Lady}& T & F\\\text{Lady}&\text{Tiger}& T & F\\\text{Tiger}&\text{Lady}& T & T\\\text{Tiger}&\text{Tiger}& F & T\end{array}$$

As can be seen, only the case of Tiger in Room 1, Lady in Room 2 makes the statements both true, and no combination makes them both false.