## The Lady or the Tiger?

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Mike Rosoft
Posts: 63
Joined: Mon Jun 15, 2009 9:56 pm UTC
Location: Prague, Czech Republic

### The Lady or the Tiger?

Inspired by a series of puzzles by Raymond Smullyan (the puzzles themselves, of course, inspired by a short story by Frank Stockton).

[original puzzle by Raymond Smullyan]
A prisoner has to choose between two rooms, each containing either a lady or a tiger. If he chooses a lady, he gets to marry her. If he chooses a tiger ... well, he'd better not. Except for that this king decided not to leave the prisoner's fate on chance: he put signs on the doors, so that he can logically deduce which room to choose.

The king tells the prisoner that if the room on the left contains a lady, then the sign on its door is true; if it contains a tiger, then it's false. The opposite is true for the door on the right: if it contains a lady, then the sign is false; if a tiger, then it's true.

The prisoner enters the arena, and sees that there are no signs on the doors. "The signs were just made", explains the king, "but we didn't have time to install them. But do not worry; you should be able to solve the puzzle even without knowing which sign belongs on which door."

Signs:
1) This room contains a tiger.
2) Both rooms contain a tiger.

[my puzzle]
It turns out that it's a needless complication that the signs weren't put up on the doors; it's immediately obvious that the first sign cannot belong on the left door. Can you find a different setup where the prisoner can solve the puzzle without it being obvious (or even possible to determine) which sign belongs on which door? (I have found two such pairs of signs.)

Gwydion
Posts: 336
Joined: Sat Jun 02, 2007 7:31 pm UTC
Location: Chicago, IL

### Re: The Lady or the Tiger?

Sorta trivial solutions:
Spoiler:
Making two identical signs: "both rooms contain a lady", "neither room contains a lady", "both rooms contain a tiger", or "neither room contains a tiger". When placed on both doors, none of these 4 possible signs can be true, so a lady is always in the room on the right and a tiger is always on the left.

SDK
Posts: 703
Joined: Thu May 22, 2014 7:40 pm UTC

### Re: The Lady or the Tiger?

Spoiler:
Pretty easy, I guess, since the lady simply can't be on the left since neither of those signs are true when on the left door. Therefore, she's on the right (where the first sign belongs, not that that matters).
The biggest number (63 quintillion googols in debt)

Mike Rosoft
Posts: 63
Joined: Mon Jun 15, 2009 9:56 pm UTC
Location: Prague, Czech Republic

### Re: The Lady or the Tiger?

Gwydion wrote:Sorta trivial solutions

Spoiler:
I guess it's within the spirit of a Smullyanesque puzzle to come up with the solution that matches the puzzle's exact words. If the signs are identical, then of course there's no way to find out which one belongs to which door. You could also have two signs with different texts: "Both rooms contain a lady" and "Neither room contains a tiger". (Who cares that the statements are equivalent.)

But now for a genuine solution, where it really matters that the signs weren't installed on the doors.

PeteP
What the peck?
Posts: 1451
Joined: Tue Aug 23, 2011 4:51 pm UTC

### Re: The Lady or the Tiger?

Spoiler:
"stone is soft" "birds are mammals" two lies => only one possible result and technically fulfills the requirement that they can't be assigned to a door.

Btw am I right in assuming that there is always one tiger and one lady (because otherwise the choice hardly matters) if so either both shields lie or both are true.

Mike Rosoft
Posts: 63
Joined: Mon Jun 15, 2009 9:56 pm UTC
Location: Prague, Czech Republic

### Re: The Lady or the Tiger?

PeteP wrote:Adding another different trivial solution:
[snip]

Btw am I right in assuming that there is always one tiger and one lady (because otherwise the choice hardly matters) if so either both shields lie or both are true.

Good, but as I have said, I am looking for non-trivial solutions where the fact that the signs are not installed on the doors really matters.

Spoiler:
Curiously, in the case of one of my solutions it is not necessarily true that either both signs are true or both signs are false.

PeteP
What the peck?
Posts: 1451
Joined: Tue Aug 23, 2011 4:51 pm UTC

### Re: The Lady or the Tiger?

Mike Rosoft wrote:
Spoiler:
Curiously, in the case of one of my solutions it is not necessarily true that either both signs are true or both signs are false.

Spoiler:
Hmm mind providing it? Since the truth value is determined by the solution that shouldn't be possible when it is possible to find a definitive solution.

Edit: here is a solution where you don't know to which door a sign belongs but it is super easy to solve so not particularly interesting:
Spoiler:
"The other door would say the lady is behind the left door" "The other door would say the tiger is behind the right door"
So the prisoner has to realize that with "The other door would say X" statements the X is true if both are true or false.

Mike Rosoft
Posts: 63
Joined: Mon Jun 15, 2009 9:56 pm UTC
Location: Prague, Czech Republic

### Re: The Lady or the Tiger?

PeteP wrote:Hmm mind providing it? Since the truth value is determined by the solution that shouldn't be possible when it is possible to find a definitive solution.

You need to understand that it is not given that one of the rooms contains a lady and the other a tiger; at least, the king promised no such thing. Who knows - perhaps he has angered the king, who sentenced him to death by putting a tiger behind each door. Or maybe the king has decided to pardon him by putting a lady in each room.

As far as the prisoner is concerned, the only thing he needs to do is finding a door behind which certainly is a lady - if there is in fact such a door!

Spoiler:
As I had already said: I have found multiple solutions. In the first the prisoner can determine that behind one of the doors is a lady and behind the other a tiger (without knowing which sign belongs where). In the second he can fully determine the occupants of the doors given one placement of the signs, so can he given the other - but the solutions are not the same. In the third he can only determine the occupants of both rooms given one placement of the signs, but not given the other. (Of course, in both the previous cases there is still a room which is guaranteed to contain a lady.) Finally, I have found a solution using only one sign, from which it can be proven that behind the left door is a lady; it can't be determined what is behind the other door, it can't be determined if the sign should be on the left or right door, nor can be any combination thereof excluded. Naturally, there is also a dual sign which proves that the lady is behind the right door.

Edit: here is a solution where you don't know to which door a sign belongs but it is super easy to solve so not particularly interesting:
Spoiler:
"The other door would say the lady is behind the left door" "The other door would say the tiger is behind the right door"
So the prisoner has to realize that with "The other door would say X" statements the X is true if both are true or false.

Spoiler:
You can't really say "The other sign would say X"; the prisoner already knows what is written on the signs. You mean something like:

1) The sign on the other door is true, if and only if a lady is behind the left door.
2) The sign on the other door is true, if and only if a tiger is behind the right door.

This solution in fact works! It is not possible for both the rooms to contain tigers; otherwise, the signs would have been equivalent to:

1) The sign on the other door is false.
2) The sign on the other door is true.

And vice versa, if both rooms contain ladies. Mr. Smullyan would sure warn us that it is indeed possible for a room to have a paradoxical statement on it, but in this case the king specifically said that each sign is either true or false. (And since a lady is behind the first door and a tiger behind the second one, both signs are in fact true.)

Mike Rosoft
Posts: 63
Joined: Mon Jun 15, 2009 9:56 pm UTC
Location: Prague, Czech Republic

### Re: The Lady or the Tiger?

Spoiler:
That said, the above solution is not exactly what I was looking for. Here the reasoning is identical, regardless of if the first sign should be on the left or on the right door. In my solutions, it matters where the sign is installed: the arguments are different, even though the conclusion (or a part of the conclusion) is the same.

Mike Rosoft
Posts: 63
Joined: Mon Jun 15, 2009 9:56 pm UTC
Location: Prague, Czech Republic

### Re: The Lady or the Tiger?

Here is a pair of signs from which you can deduce that a lady is in one of the rooms and a tiger in the other one, but can't prove which sign belongs where.
Spoiler:
1) If this room contains a lady, then the other room contains a tiger.
2) The room on your right contains a tiger.

From this pair you can deduce that one of the room contains a lady. Who or what is behind the other door depends on which sign goes to the left door, and which one goes to the right door.
Spoiler:
1) This sign belongs to the door on your left.
2) Behind the other door is a tiger.

Can you find more examples?