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An archaeologist who has just unearthed a long-sought-after pair of ancient treasure chests.

One chest is plated with silver and the other is plated with gold.

According to the legend one of the two chests is filled with a great treasure whereas the other chest houses a death cursing spell in the form of a dragon.

Faced with a dilemma archaeologist then noticed that there are inscriptions on the chest. Based on these inscriptions which chest should he open?

$$\mathrm{Gold\, Chest}- \mathrm{One\, of\, these\, inscriptions\, is\, true}$$

$$\mathrm{Silver\, Chest- \mathrm{This\, chest\, contains\, the\, dragon}}$$

Note: Thanks to his understanding of the culture from which these chests came, the archaeologist knows that the gold chest's inscription could also be translated as "One and only one of these inscriptions is true".

I already know the answer, what I am seeking here is a nice explanation.

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  • 3
    $\begingroup$ @generalcrispy Unless the question is purely: "Why is <this> the answer"? $\endgroup$ – d'alar'cop Nov 25 '14 at 16:40
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    $\begingroup$ Correct, but everyone who posts a riddle here ostensibly knows the answer. @Enigmo didn't ask what you did. $\endgroup$ – generalcrispy Nov 25 '14 at 16:41
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    $\begingroup$ Should the gold chest say "ONLY one of these..."? $\endgroup$ – Ste Nov 25 '14 at 16:57
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    $\begingroup$ The ancients should've put a curse in both chests. The treasure could just be the removable plating. :) $\endgroup$ – Rob Watts Nov 25 '14 at 16:59
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    $\begingroup$ Is the dragon good or bad? $\endgroup$ – user2357112 Nov 25 '14 at 18:12

10 Answers 10

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He should open the

Silver Chest

Explanation of the logic (assumes the inscriptions are intended to be helpful):

Assume the inscription on the gold chest is true. Because that inscription is true, we already have the one true inscription, so the other inscription must be false. Thus the silver chest does not contain the dragon, so it must have the treasure.

Now assume the inscription on the gold chest is false. That means either both inscriptions are true, or both are false. Since the gold chest's inscription is false, the silver chest's inscription must also be false, so it must have the treasure.

So regardless of whether the gold chest's inscription is true or false, the silver chest's inscription must be false and therefore must be the one with the treasure in it.

I'm also assuming here that the dragon referred to by the silver chest's inscription is the same thing as the death curse.

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    $\begingroup$ All that we learn is that the silver chest does not contain a dragon. We are not given anything in the puzzle linking a dragon to either a treasure or the curse. Interpreting the inscriptions in this way does not provide any useful data about the actual contents of each chest. $\endgroup$ – eclipz905 Nov 25 '14 at 18:17
  • $\begingroup$ @eclipz905 I forgot to mention that I'm assuming that "dragon" means the death curse $\endgroup$ – Rob Watts Nov 25 '14 at 19:46
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    $\begingroup$ I figured that, but the puzzle does not make it clear. Someone else would be equally justified in assuming "dragon" is a code name for the treasure. $\endgroup$ – eclipz905 Nov 25 '14 at 19:51
  • $\begingroup$ @DoubleDouble I agree that it is ambiguous, but Enigmo originally included which chest was correct as a part of the question. So the dragon is not the treasure. $\endgroup$ – Rob Watts Nov 25 '14 at 21:08
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    $\begingroup$ @Cthulhu the gold chest says that only one of the inscriptions is true $\endgroup$ – Rob Watts Nov 26 '14 at 17:53
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This is a more formal solution

We have a small system of formulae to define the problem:

If a chest $c_i$ is $true$ then it contains the treasure. If a chest $c_i$ is $false$ then it contains the dragon. We have 2 propositional variables, $c_1$ and $c_2$.

The first formal assertion we can define is:

$c_1 \oplus c_2$

We express the inscriptions as further assertions about each other, themselves and the propositional variables. $f_1$ expresses the inscription on the Gold chest and $f_2$ that of the Silver chest. Conjointly they are $F$.

$f_1 = f_1 \oplus f_2$

$f_2 = (c_2 = false)$

We must deduce for which $i$, $c_i$ is true.

We do this by observing contradictions in $F$.

If $c_2 = false$, then that makes $f_2 = true$. Now, if $f_1$ is $false$ then $f_1$ is true. If $f_1$ is true then $f_1$ is false, this is a contradiction. Thus, $c_2 \not= false$.

If $c_2 = true$, then that makes $f_2 = false$. Now, $f_1 \leftrightarrow f_1$. No problem.

So,

$c_2 = true$. The Silver chest contains the treasure.

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    $\begingroup$ Nice symbolism, but I think Rob's answer is more understandable. +1 anyway :-) $\endgroup$ – Rand al'Thor Nov 25 '14 at 16:56
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    $\begingroup$ @randal'thor Yes, Rob's is probably more suitable here. But it's good to have this here as a record anyway :p $\endgroup$ – d'alar'cop Nov 25 '14 at 16:57
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Most of the existing answers assume that you meant "Only one of these inscriptions is true", rather than "One of these inscriptions is true", so I will go ahead and assume that you meant what you said instead, for completeness, or just in case.

In that case:

He should pick the Silver chest, but he only has a 2/3 chance of this being the correct one.

Let's see why:

We know we only have 4 possibilities:

1. Gold is true, Silver is true: Definitely possible if we assume you didn't mean "Only one...". In this possibility, both statements are true and therefore Gold contains the treasure.

2. Gold is false, Silver is false: This is also possible since rendering the "one of these is true" statement false allows (and requires) for both to be false. The treasure chest would be Silver

3. Gold is true, Silver is false: Since "this contains death" is false, the Silver chest contains the treasure.

4. Gold is false, Silver is true: This is a contradiction and should be discarded. "One of these is true" being false (as a statement) would mean that neither inscription is true. Since by definition we are looking for a possibility where Silver is true, this possibility contradicts itself.

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    $\begingroup$ Excellent answer, instead of harping on "dragon" nailing the actual imprecise expression. $\endgroup$ – No. 7892142 Nov 26 '14 at 9:04
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Assuming this is a bit of a trick question...

Neither. There is not enough information to safely pick a chest. According to legend, one chest has treasure while the other has a death cursing spell. The fact that the silver chest must be lying and thus does not contain a dragon has no bearing on which chest has the curse and which has the treasure. A chest that lacks a dragon can instead contain a curse or a treasure. This assumes the question is correctly worded.

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I'm sure I've seen something similar in one of Ray Smullyan's books...

If the position is exactly as stated, and we take the legend as true (no "I don't believe in curses" type solutions), the archaeologist should still not open either casket. If he does, he is making a big assumption: that the chest inscriptions have valid truth values.

The standard solution is essentially that the dragon is in the gold chest, because if he's in the silver one then the silver inscription is true, and the gold inscription can neither be true nor false. Without additional information, though, it's perfectly possible for the gold chest to have a meaningless inscription - just as there's nothing physically stopping me carving "This statement is false" on to the chest. Compare this with the canonical knight/knave/joker type puzzle - there we know that we only ever deal with people who make true or false statements.

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  • $\begingroup$ I agree. For it to work, the puzzle would need the appearance of some "guru" or "magician" that guarantees that the statements have valid truth values. $\endgroup$ – Daniel Daranas Nov 26 '14 at 13:06
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    $\begingroup$ Found it - in "What Is The Name Of This Book?". The scene is set by several series of puzzles where the chest-makers are known to be knights/knaves (who inscribe the same way as they speak). Late in the book along comes one like this where you have no such context. $\endgroup$ – Julia Hayward Nov 26 '14 at 13:18
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Hint:

Consider four possibilities: (1) Both inscriptions are true, (2) both are false, (3) only the gold chest inscription is true, and (4) only the gold chest inscription is false. There are no other possibilities.

My answer:

Let's look at four possibilities:

1. Both inscriptions are true. This is a contradiction because the Gold inscription says "One of these inscriptions is true", so we can rule this out (see note below).

2. Both inscriptions are false. This means that the Silver inscription "This chest contains the dragon" is false and therefore, by elimination, the silver chest has the treasure.

3. The gold chest's inscription is true. If we assume the gold chest's inscription is true and the gold chest's inscription says that one of the inscriptions is true (see note below), then the silver chest's inscription is necessarily false and thus the silver chest has the treasure.

4. The gold chest's inscription is false. If we assume the gold chest's inscription is false, then either both inscriptions are false or both inscriptions are true. I've shown in #1 above that both inscriptions being true leads to a contradiction, so both inscriptions must be false. Therefore the silver chest has the treasure.

These possibilities either are contradictory or point to the silver chest, so the silver chest is the answer.

Note, I'm assuming "One of these inscriptions is true" means "only one" is true.

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    $\begingroup$ What reason do you have to believe that "the dragon" refers to the curse? $\endgroup$ – eclipz905 Nov 25 '14 at 21:03
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    $\begingroup$ It seems pretty clear to me that the author made a mistake with this. I'm assuming "curse" is supposed to be "dragon". The logic puzzle would be underdetermined without that assumption. $\endgroup$ – user2023861 Nov 25 '14 at 21:10
  • $\begingroup$ "There are no other possibilities" is demonstrably false - it is possible for the inscription on a container to be a logical paradox as I have just confirmed by writing "this statement is false" onto a shoebox with a Sharpie. $\endgroup$ – Joe Lee-Moyet Nov 26 '14 at 14:31
  • $\begingroup$ @yjo I suppose that someone could put the treasure in the gold chest just to mess with us. Or they both have dragons. Or the dragon is actually a brand new car. Or whatever else you can dream up. Given the context of the question, there are only four possibilities. $\endgroup$ – user2023861 Nov 26 '14 at 14:54
  • $\begingroup$ @user2023861 I disagree: the question gives evidence (albeit as 'legend') that one chest contains treasure and the other a dragon. On the other hand there is nothing to support the idea that the inscriptions are not paradoxical or self-contradictory. Furthermore it's not possible for both of the inscriptions to be true so we're already starting from a point of distrusting the inscriptions. $\endgroup$ – Joe Lee-Moyet Nov 26 '14 at 15:11
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Three possibilities.

  1. The archaeologist can CT scan both the chest. The one in which is finds treasure he can open it.
  2. He can tap on both the treasure chest or do something that creates noise. If there is a dragon it will react
  3. Ask someone else to open both the chest on his behalf
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It's most likely the...

Gold Chest

And here's why:

Let's pretend for a while that I'm the person who made the treasure chests and was responsible for storing the treasure in them. (As well as making sure nobody made off with them.)

A chest allows me to move treasure from point A to point B and provides a reasonable protection from external weather conditions, it does not however, provide a decent protection from thieves, raiders and other varieties of hoodlums.

So what do I do?

I make another chest, this time plated with silver and rig it with a deathtrap inside.

Ok, what next? If I make a scary story about a curse residing in these chests, that should put off at least a good deal of people wanting to get their clutches on this treasure. But what about the people who do not believe in mystical curses?

An idea springs into the mind: "Their rational way of thought will be their undoing." I will engrave inscriptions upon both of the chests portraying a conundrum. But what would initially seem to be a logical puzzle is actually a deception leading to their demise. A solution of which will not lead to the chest containing the treasure but instead to the chest with the deathtrap.

For I have been given the task of protecting these chests and I am aware that the gold chest contains the treasure, and all staff who need access to the treasure know that the gold chest is where the treasure lies.

For I have no reason to hand over the treasure to just about anyone who can solve a puzzle just as how the fair people of the future will not be able to solve a logical puzzle to access an account of a complete stranger.

Some might argue that:

- What if thieves anticipate such a cunning way and decide to pick the gold chest?

To which I answer:

Kudos to you, sir thief, you have bested my guards as you have bested my wit and are worthy of the treasure inside.

For if I switch the chests around, my initial ploy becomes void and the measures taken described above would become ineffective.

I think this fits quite nicely with the tag.

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I fully agree with Julia Hayward's answer. Apparent laws of logic cannot prevent anyone from putting a treasure into the gold chest, a dragon into the silver one, and then make arbitrary inscriptions on them. Valid conclusions about physical world require some preliminary assumptions that themselves are not purely logical, and here we are supposed to derive one from propositions of unknown logical value, based only on their logical structure. This is not possible.

In Smullyan's book cited by Julia, there is also a statement which "proves" that unicorns exist. I don't remember it exactly, but I believe it was, "This statement is false, and unicorns do not exist," which seems to imply, on a purely logical basis, that unicorns exist indeed. (In fact, anything could be "proved" this way.) Self-referencing propositions have been a known problem in logic for over a century, leading to contradictions and antinomies. That's why contemporary mathematics became strictly formalized in a way that does not allow them as valid statements.

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I hate to be 'that guy' but since dragons and death curses aren't real, he can safely open either chest as long as he takes elementary precautions against explosives, booby-traps and poison.

And yes, I fully appreciate that this answer sucks. Sorry.

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  • $\begingroup$ You should change your profile pic to 'that guy' :-p $\endgroup$ – Rand al'Thor Nov 26 '14 at 13:08

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