Five doors

Logic Level 2

You wake up in a very cliché fashion in a room. You hear a voice. "You're finally awake. This will be fun! As you can see, in front of you are the numbers 1-5. Underneath each number is a door, and there's a sign in front of each door. Only one of these doors will grant you a safe passage to freedom. The other four will kill you instantly. How? Doesn't matter. You won't be alive long enough to learn. Each sign provides you with some infomation regarding the other doors and signs, but 3 of them are false. If you do not wish to wait and die here, you know what you must do. Good luck!"

You move to each number and read each sign.

Sign 1 : Door 2, 4 and 5 are incorrect doors.
Sign 2 : At least one sign next to me is lying.
Sign 3 : Door 2 is the correct door.
Sign 4 : The sign in front of the correct door is true.
Sign 5 : All signs underneath an even number are true.

If only 2 of these signs are true, which door will lead you to safety?

Door 5 Door 2 Door 4 Door 3 Door 1

1 solution

Jessiah O'Neill
Jun 28, 2019

One logical approach is to determin which Signs are True or False.

Sign 5 suggests that Sign 2 and Sign 4 are telling the truth, but if this is so, 3 statements would be true. We know there are only 2 true statements, therefore Sign 5 must be false. We can also infer from Sign 5's fallacy that Sign 2 and Sign 4 can't both be true. Sign 1: Unknown, Sign 2: Opposite of Sign 4, Sign 3: Unknown, Sign 4: Opposite of Sign 2, Sign 5: False.

If Door 2 is correct, Sign 3 is true and Sign 1 is false. If Door 2 is incorrect, Sign 1 is true and Sign 3 is false. Either way, Signs 1 and 3 cannot both be true or false. They must be different, proving Sign 2 true and thus Sign 4 false. Sign 1: Opposite of Sign 3, Sign 2: True, Sign 3: Opposite of Sign 1, Sign 4: False, Sign 5: False.

Since Sign 4 is false, the sign in front of the correct door must be false. This eliminates Door 2, proving Sign 3 false and thus Sign 1 true which then eliminates doors 4 and 5. Sign 1: True, Sign 2: True, Sign 3: False, Sign 4: False, Sign 5: False.

Sign 4's fallacy also eliminates Door 1 since Sign 1 is true, leaving a single door.

With some hesitation, you open Door 3 and are greeted with a corridor. You make your way down the corridor step by step, shaking with every breath you take until you are greeted with a beautiful scene. You're free.

"Underneath an even number" could be interpreted as meaning a sign with an even number on it.

Also, if something is false, does that mean that the opposite is true?

Caleb Zylstra - 1 year, 6 months ago

My apologies. I tried to be careful and explicit with my wording. I'll see how I can edit it. As for you question about true and false, false means the logical opposite is true, but be careful how you consider this. One way to consider the opposing statement is to "negate" the statement. When saying it out loud, try adding "Not" to the start of the sentence. eg, I'm connected to a lie detector. I flip 2 coins, cover the result, and say "Both coins landed on heads". The Lie detector informs you that I'm lying. This could mean both coins landed on tails, but it could also mean I flipped 1 Head and 1 Tail. Note that there are 3 total possible outcomes. I told you 1 of them and that was false. this results in 2 possible outcomes.

If it clears things up, here are each of the signs and their logical opposite: Sign 1: Doors 2, 4 and 5 are all unsafe False Sign 1: Door 2, 4 or 5 is safe Sign 2: Sign 1 and/or Sign 3 are false False Sign 2: Sign 1 and Sign 3 are both true Sign 3: Door 2 is Safe False Sign 3: Door 2 is unsafe Sign 4: If Door X is safe, Sign X is true False Sign 4: If Door X is safe, Sign X is false Sign 5: Sign 2 and Sign 4 are both True False Sign 5: Sign 2 and/or Sign 4 are False

Note sign 5 for an example of what I tried to explain. if Sign 5 is false, it means that Sign 2 and Sign 4 are not both True. This doesn't specifically mean both are false (although this is possible), it means that at least 1 of them is false. The opposite statement implies any scenario that does not meet the stated requirement.

I hope this helps. let me know if this makes no sense.

Jessiah O'Neill - 1 year ago

Well I chose door 2. I guess I'm dying today.

Emily Peng - 1 year, 5 months ago

YAY I managed to solve it! Nice problem! (+1)

Noel Lo - 1 year ago

Glad you enjoyed it.

Jessiah O'Neill - 1 year ago

Ok there is a bunch of confusion here (at least on my part) because of the wording. I think "incorrect door" means a door leading you to death, not a door with a false sign, correct? If you labeled them "safe door" and "death door", it would be clearer. "Incorrect" and "false" are too similar. You can also just say "door" instead of "sign" for simplicity. With additional cleanup, it should read:

Door 1: Door 2, 4, and 5 are death doors. Door 2: At least one door next to me is false. Door 3: Door 2 is the safe door. Door 4: The safe door is true. Door 5: All even numbered doors are true.

I kept getting a different answer because I thought door 1 said "doors 2, 4, and 5 are false".

Martha Sapeta - 1 year ago

Five doors

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