[Math] Smullyan’s Lady or Tiger — incorrect solution

logicpuzzle

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?

Best Answer

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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