The 3 Doors Of Doom

Logic Level 1

You are in a room with 4 doors and a guard. The guard says, "Welcome to the end of your quest! Behind one of these four doors lies the treasure you have been searching for. However, behind the other 3 doors lies certain death. Each door has a sign on it, but you are not sure which signs are truthful or which are lies. However, you are certain that the door that contains the treasure is truthful. Choose your door wisely and remember, "if you pick wrong, death will be brought upon you."

Here are what the four signs read:

Door 1: Exactly one of doors 3 and 4 is truthful.
Door 2: Both odd-numbered doors are untruthful.
Door 3: Neither of the odd-numbered doors contains the treasure.
Door 4: One of the even-numbered doors contains the treasure.

Door 1 Door 2 Door 3 Door 4

6 solutions

Nik Olson
Aug 29, 2016

Relevant wiki: Truth-Tellers and Liars

"However, you are certain that the door that contains the treasure is truthful."

This is a VERY important statement when solving the problem

Let's say Door 1 has the treasure:

Then one of Doors 3 and 4 are truthful
Door 4 must be a lie because it would disprove the fact that Door 1 contains the treasure
Door 3 must be true because one of Door 3 and 4 are truthful
Door 3 being true disproves Door 1 containing the treasure making this all impossible

Door 1 does NOT have the treasure

Let's say Door 2 has the treasure:

Then both odd numbered Doors are untruthful
Then Door 3 is untruthful meaning one of the odd numbered doors contains the treasure
This disproves Door 2 containing the treasure making this impossible

Door 2 does NOT have the treasure

Let's say Door 3 has the treasure:

Then neither one of the odd numbered doors contains the treasure...

Door 3 does NOT have the treasure

Let's say Door 4 has the treasure:

Then one of the even numbered doors has the treasure
Door 4 is even numbered so it is possible that Door 4 has the treasure

Since no other door can have the treasure Door 4 MUST have the treasure

If door 4 has the treasure, that would make statement one true because either of door 3 and 4 one of them has the treasure. Therefore statement one would have the treasure because it is truthful. However both can't have the treasure so how door 4 have the treasure when only it can be the true statement when door 1 supports it?

fred jones - 4 years, 9 months ago

The statements on doors 3 and 4 are equivalent, so they are either both true or both false. The statement on door 1 states that exactly one, and not both, (nor neither), of statements 3 and 4 are true. But this is impossible because of their equivalence, and hence the statement on door 1 is necessarily untrue.

Brian Charlesworth - 4 years, 9 months ago

But door 1 statement is true. Just because it doesnt have the treasure does not mean door 1 is stating a lie

Greg Grapsas - 4 years, 9 months ago

The statements on doors 3 and 4 are equivalent, so they must be either both true or both false. The statement on door 1 states that exactly one of these equivalent statements is true and the other false, which is impossible, allowing us to conclude that the statement on door 1 is necessarily false.

Brian Charlesworth - 4 years, 9 months ago

@Brian Charlesworth Since statement 4 is true then every other statement is wrong. If 1 is wrong then it means both 3 and 4 are either true or both false. Now if both 3 and 4 are true then it makes 2 doors true(only one door can be true) and if both 3 and 4 are wrong then treasure is in one of the odd numbered doors. How's that possible?

Ramtin Yousefgorji - 4 years, 5 months ago

@Ramtin Yousefgorji – It is not stated that it must be the case that only one statement is true, but only that one door leads to treasure and the other three to death. So it is possible that more than one statement is true even though the door with one of those true statements could lead to death. This is the case here, where the only logically consistent conclusion is that statements 1 and 2 are lies and statements 3 and 4 are both true, with the treasure then necessarily lying behind door 4.

Brian Charlesworth - 4 years, 5 months ago

@Brian Charlesworth – @Brian Charlesworth

It makes sense now. thanks

Ramtin Yousefgorji - 4 years, 5 months ago

Used the same logic as you and got the same answer 👍🏼

angela thomas - 4 years, 5 months ago

Hey,I have the same solution. U used the method of Trial and Error.😇😇😇😇😇😇😇😇

Xylene resha terminio - 4 years ago

Same logic here too

Devansh Sharma - 4 years ago

Aura Odum
Sep 7, 2016

Let's start by noting Door 2 as paradoxical.

Why is Door 2 paradoxical? It claims both odd numbered doors are untruthful. The treasure can only be behind a truthful door. Door 3 claims neither of the odd numbered doors contain the treasure. If the odd numbered doors are untruthful, then they do not contain the treasure. This makes Door 3's claim truthful. But Door 3 is odd-numbered. Therefore Door 2 is wrong.

Knowing that Door 2 is wrong makes the rest easy. Door 3 and Door 4 have a synergy.

Door 4 states that one of the even numbered doors contain the treasure. Door 3 states that neither of the odd numbered doors contain the treasure.

If no odd numbered doors contain the treasure, than an even numbered door must. If Door 3 is true, then Door 4 must also be true. If Door 3 is false, then Door 4 must also be false.

However, Door 1 insists they must be different, only one of them is true. Therefore, Door 1 is wrong.

The only doors left, Door 3 and Door 4 must both be truthful. And as they claim, the treasure lies behind an even numbered door. The only even numbered door remaining in fact: Door 4.

Well, you have the shortcut on how to solve problems like these. Were you studying in a chapter called Paradox?

Xylene resha terminio - 3 years, 8 months ago

Shreyash Mhalagi
Sep 5, 2016

Door 4 is correct

David Ganezer
Mar 29, 2017

Doors 3&4 agree, so their both lying that means that it is an even number & 4 is correct

Tom Verhoeff
Nov 12, 2016

Observe that the statements on doors 3 and 4 are logically equivalent (since there is exactly one treasure). Thus, the statement on door 1 is false (no treasure there).

Since the treasure is behind a truthful door, the statement on door 2 implies the statement on door 4 (hence, on door 3). The statement on door 2 also implies that the statement on door 3 is false. These two implied results are contradictory. Hence, the statement on door 2 is false (no treasure there).

Since the statement on door 2 is false, and on door 1 as well, the statement on door 3 (and hence, door 4) is true. Hence, the treasure is behind an even numbered door, not being door 2 (because its statement is false).

Thus, the treasure is behind door 4.

Saya Suka
Feb 15, 2021

Door 3 & Door 4 are complementary of each other, so if the treasure really does exist behind ONE of the doors, then both doors have the same truth values. As such, Door 1 must be lying.
Since Door 3 is either honestly denying to contain the treasure (in its truth) or ineligible to actually contain one (in its deception), anyway it ended up being, we can never find it behind Door 3. Thus, with no odd numbered doors left possible, then both Door 3 & Door 4 are telling the truth while Door 2 turned out to be lying by Door 3 truthfulness.
Between the 2 doors left, it must be behind the even Door 4.

The 3 Doors Of Doom

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