Tuesday, 21 February 2012

Women Or Tigers? — Solution to Final Challenge!

Did you find out? Actually, it wasn’t that difficult after all …once you got on the right path. Here’s what I have come up with — tell me if you have the same, or another answer, by commenting on the post.

If the prisoner wants to know whether Room VIII is empty or not, the answer he gets must be crucial to solve the whole puzzle. So let’s take that room as our point of departure.

We know already that Room VIII cannot contain a woman, as its sign says it hosts a tiger, and the sign on the woman’s door is correct. Leaves us with two possibilities: Room VIII is either empty or hosts a tiger.

It will not help us much to assume it is empty. Then its sign can be either true or false. So let’s assume there is actually a tiger in Room VIII. This means the sign is not correct. The sign gives two statements, with an ‘and’ in between. If one or both statements are false, the sign is false, as it should be if there is indeed a tiger behind the door. The first statement ‘this room hosts a tiger’ is true (so we assume for now) — ergo, the second ‘Room IX is empty’ must be false for the sign to be false.

So Room IX is not empty. It contains either a woman or a tiger. Again, its sign gives two statements linked by an ‘and’. The first statement says the room contains a tiger. If there is a woman in it, the sign must tell the truth, which it then can’t as it talks about that tiger. So… it does contain a tiger. And, as with Room VIII, the second statement must be false in order for the sign to be false. The sign on Room VI therefore tells the truth (because the sign on Room IX, lying, says that it doesn’t). Let’s keep in mind (write down) that Room VI either hosts a woman or is empty.

The sign on Room VI says that the sign on Room III is false. This one gives two statements, this time linked by an ‘or’, meaning that both statements must be incorrect in order for the sign as a whole to be false. To take the second statement first (you’ll see in a second that in this case, that’s easier, in order to keep some sort of overview): ‘sign VII is lying’ …so sign VII must be true, and sign VII says the woman is not in Room I. Now that is good to know, let’s write it down too.

…and continue with the first of the two statements on Room III, which is also false (see the paragraph just above): ‘sign V tells the truth’. So it does not. Sign V is yet another two statements linked by an ‘or’. They are both false, according to the logic explained just before. For one, this means that Room II is indeed empty, as its sign states. OK. It also means that sign IV is lying.

Sign IV says that sign I is incorrect. So it is correct …that ‘the woman is in a room with an uneven number’. We know from two paragraphs further up that she is not in Room I. We also know that she is not in Room III as its sign is false. Same goes for Room V (go up just one paragraph from here), and for Room IX (go up three paragraphs from here).

Leaves us with one room carrying an uneven number: Room VII. A quick check: The sign on this room must tell the truth if the woman is indeed behind the door. It says that the woman is not in Room I. Indeed, as she cannot be in two rooms.

One doubt remains though! The premise for this string of logic was that Room VIII is not empty. But what if it is, what if that’s the answer the king gave the prisoner? Can he use that answer to follow a different logic and reach the solution in a different way, or even reach a different solution?

I would like to have your answers to this! Comment below if you have it!


Trent Osland said...

After spending several hours working on challenge seven, I finally gave up and looked at the answer. And I agree, the solution is obvious, if you discount the word "either." You see, by saying "either A or B" you indicate that only one is true. This completely changes the amount of information that can be deduced from signs three and five. When you say that either sign 2 is false or sign 4 is true, that implies that only one of them is true. Therefore, if both those statements are true, then the overall either/or statement is false. If the word "either" had simply been omitted, then your solution would have worked.

As it is, the prisoner's best bet is room six, assuming the king told him that room 8 is not empty. In this case, if the woman were in room 1, room six would be empty. The woman cannot possibly be in rooms 2,8, or 9, so those can be ignored. Likewise, room 8 cannot be empty if the woman is in room 3 or 5. If the woman were in room 4, room six would also be empty. Similarly, room six would be empty if the woman were in room seven, and obviously if the woman were in room six, room six would have the woman.

That solution assumes that saying "either/or" means only one of the possibilities can be true, similar to saying: "Either you will win the race or I will" meaning that if the statement was false, either both statements are true, or both are false. Technically it does not actually answer the question asked (which room had the woman) but it does give a safe option for the prisoner to choose.

Joseph Ward said...

Sorry, I just came across this, which is why I'm commenting on an post that's so old. I do have a complete solution.

The key piece of information in the problem is that after the king told him whether room 8 was empty or not, the prisoner was able to solve the problem - it says he knew where the woman was hiding. Thus, if a possible response creates more than one solution, this cannot have been the response the king gave, since the prisoner would not have known for certain where the woman was if that answer was given.

If the king told the prisoner that room 8 was empty, there are at least two possible solutions that I can list. One is that the woman is in room 1 and all other rooms are empty. Room 1 says the woman is in an odd-numbered room, which is true since she is in room one, and all other rooms are empty and can be true or false as required. Another option is that the woman is in room 7 and all other rooms are empty. Again, room 7 states that the woman is not in room 1, which is true, and the other rooms are all empty, so no contradiction results.

Since two solutions are possible under this scenario, the prisoner was not told that room 8 was empty. Thus, he was told that room 8 was not empty, and your solution takes over from here. The woman is in room 7.

Thank you for the problems! I enjoyed solving them!