# 3 chest logic puzzle, is there something you can say that is guaranteed to give you treasure without the risk of poisoning?

You are locked in a room, with 3 chests: left, middle and right. One of the chests contain treasure, the other two contain a poisonous gas. You don’t know what is inside each chest.

The left chest will open if a true statement is said. The middle chest will open if a paradoxically true and false statement is said (eg this statement is false) and the chest on the right will open if a false statement is said. If you say a paradoxically true and false statement, the left and right chests will remain closed and only the middle will open.

Is there something that you can say that’ll be guaranteed to open the chest with the treasure? This is a harder version of the two chest riddle, but that was too easy so I made a harder version with three chests.

• If I say "The left chest will open". Which chest will open if any? Aug 27, 2022 at 12:40
• @FlorianF I think this is undecidable at the time this statement is made. If the chests have a memory, Left will open if it hears a true statement, but it will do so anyway. You could try "the chest on YOUR left will open/won't open" as part of your statement. (This site is addictive!) Aug 27, 2022 at 16:14
• @FlorianF, both chests will open. Aug 29, 2022 at 1:56

I will try with the following statement.

"The treasure is in the left chest or [it is in the middle chest and this sentence is false]."

If the treasure is
- in the left chest: the sentence is true regardless of the second part (after "or").
- in the right chest: both parts of the sentence are false.
- in the middle chest: it claims that the sentence is false. Paradox.

• Some brackets in the statement might help Aug 27, 2022 at 23:41
• As a programmer I follow that "and" has higher precedence than "or". But brackets don't hurt, I guess. Aug 27, 2022 at 23:47

I don't have a full solution. But here is a method that I believe will increase the chances. Maybe something to build on.

L=Left, M=Middle, R=Right

The statement

"If I say chest(L) contains the treasure the chest containing the treasure will open"

Now

a) If it is contained in chest(L), then chest(L) will open and we get the treasure. Job done. b) If it's contained in chest(R) then it can't open. If it opens then that would mean the statement the chest containing the treasure would open was true and not false even though it opened. Contradiction. So it can't open. Now since this is a "not true-not false"-statement (paradoxal), what about chest(M)? Chest(M) only opens when something is both/neither true and false but if it would open and the treasure isn't in there that would mean the statement was false and not true-false. Contradiction. So it won't open. Like-wise in situation c) if the treasure is contained in chest(M), it won't open but for the opposite reason - if it did open it would mean the statement was true and not true-false. Contradiction. So it won't open.

In summary:

After the statement we get two scenarios: 1. it's in chest(L) and we get the treasure. Or 2. none of the three chests will open and its in either chest(M) or chest(R).

• a) or it does not open because/therefore the statement is false. Also possible. Aug 27, 2022 at 15:04
• if the treasure is in the middle chest then the right chest opens and you die. Aug 27, 2022 at 23:19
• @Jasen the way "if, then" is defined in logic, if the "if" part is false, then the whole statement is vacuously true. Ergo, R won't open no matter what. Sep 22, 2022 at 11:39

"(The chest 2 steps on your left and your neighbour both contain gas) or (The chest 2 steps on your right or your neighbour contains the treasure) or (if both your neighbours contain the same thing then this statement is false)".

Part 1: L evaluates it to true iif it holds the treasure. For M and R it is undecidable (they don't have another chest 2 steps on their left).

Part 2: R evaluates it to false iif it holds the treasure. For L and M it is undecidable.

Part 3: L and R only have one neighbour, undecidable for them. iif the initial condition holds (both L and R contains gas) M evaluates the statement and finds a paradox.

(iif means "if and only if")

• Using undecidability to not open any chest, interesting approach. Aug 27, 2022 at 14:56
• By "you" you mean the chest that is evaluating the statement, right?
– MJD
Aug 27, 2022 at 19:41
• I have come across iff as an abbreviation for “if and only if”, but not iif Aug 27, 2022 at 23:59
• @justhalf Well, now we know what the fourth chest is gonna do Aug 28, 2022 at 13:58