You are the judge, and 2 people are fighting for the ownership of a blackbox that only the owner knows the content. The traditional way is to request both people to secretly write down the content of the box on a piece of paper and give them to you so that you can verify the content. But somehow in this case, you don't have any private channel to communicate (no paper or anything to write on, they can't even whisper), all information is to be shouted out loud. What strategy can you use to decide the true owner of the box?
Note: I don't know the answer.
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8$\begingroup$ Threaten to cut the box in half: the true owner will tell you they'd rather see its contents intact yet in the other's hands, while the thief will tell you to do it! :) $\endgroup$– user62757Jul 27, 2020 at 1:08
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2$\begingroup$ This seems to me to have many answers depending on the contents of the box, and it's difficult (or even impossible) to tell whether an answer works without further details. I'm not sure this is really a puzzle - it may be a question that is unanswered, but it doesn't seem to have the capability for a definitive solution. $\endgroup$– Deusovi ♦Jul 27, 2020 at 1:57
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2$\begingroup$ @Deusovi I think the suggested answers below show that the exact content is not needed and a generalized method can be found. So I think its a valid puzzle. Some specific content could possibly make for a lot easier method, but it is not required. $\endgroup$– BmyGuestJul 27, 2020 at 8:12
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2$\begingroup$ @eru-cs Now if that box contains owner's private information/passwords/credit card, your assumption will work exactly inverted, with any logical owner asking to cut it and thief telling to give it away so they'd have another shot. $\endgroup$– val - disappointed in SEJul 27, 2020 at 13:57
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3$\begingroup$ @valsaysReinstateMonica - eru-cs is refering to this $\endgroup$– marcellothearcaneJul 27, 2020 at 15:41
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9 Answers
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One of the contestants is the rightful owner. They know what is in the box. The contents could be any one of an indefinite number of things.
Randomly choose just one of the contestants and ask them to shout out what is in the box with a full description of colour, shape, materials, inscriptions, etc. Check to see if they are right. If they get it right then they are the owner. If they get it wrong then the owner is the other person.
answered Jul 26, 2020 at 23:46
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$\begingroup$ Although I agree this would most likely work, it only really works 100% if you ask the dishonest person first (as an incorrect guess would prove certain who owns it). But if you ask the rightful owner first, then the dishonest person could just claim they would have said the same thing too and the first person merely guessed correctly. So perhaps it might be best to not be too specific with the description at first, then if person A gets it right, and person B then claims they would have said the same, you can then ask person B to give more specific info to validate their claim. $\endgroup$– musefanJul 27, 2020 at 13:01
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$\begingroup$ @musefan - I am using the strict rules as posed in the question, it says, "only the owner knows the content". Because of this we know that the dishonest person cannot claim they knew. Otherwise two people would have known the answer. $\endgroup$ Jul 27, 2020 at 13:33
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$\begingroup$ The second person might not know from the start, before the question is asked, but they will know once the first person has said it out loud. If the first person is dishonest they will guess, and then maybe they will guess correctly. Guessing it right, is not the same as knowing the right answer. $\endgroup$– musefanJul 27, 2020 at 13:43
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$\begingroup$ @musefan - But guessing correctly is a possibility in all the solutions. In my answer (and everyone else's) the dishonest person could, in theory, guess correctly. Of course the chance of being correct is vanishingly small. $\endgroup$ Jul 27, 2020 at 13:52
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1$\begingroup$ For the record, I upvoted your answer above all the others. $\endgroup$– musefanJul 27, 2020 at 13:57
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One option would be to get them to spell out the box contents one or two letters at a time (shouting alternately). So person 1 has to call out the first letter of what they claim are the contents, then person 2 calls out the first and second letters, then person 1 calls out the second and third letters, person 2 calls the third and fourth letters etc.
In this way the non-owner will have to guess every second letter and it will soon become obvious which of the two actually knows the contents.
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1$\begingroup$ I think this is the best answer, because it leaves aside the possibility of it being randomly guessed. $\endgroup$ Jul 27, 2020 at 10:36
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2$\begingroup$ @Fivesideddice: Not really, because if they randomly guess it then they will likely know how to spell it too $\endgroup$– musefanJul 27, 2020 at 12:55
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2$\begingroup$ @ Fivesideddice - ""it leaves aside the possibility of it being randomly guessed". In fact none of the answers rules out a complete random guess. This answer *helps a guess if the owner starts by giving away one or two letters first. $\endgroup$ Jul 27, 2020 at 13:56
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$\begingroup$ But without even paper to write stuff down, you're relying on the memory and state capability of the people. I know a lot of people would have trouble keeping track of what they're spelling if they're only allowed to spell two letters at a time and another person is also shouting letters in between. Not to mention the judge having to keep track of each result at the same time without even paper. Yikes. $\endgroup$ Jul 27, 2020 at 16:39
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$\begingroup$ @IanMacDonald Sure. But for every strategy you have to assume people are capable to some extend to strategize as well. Penguino´s solution isn´t that bad if you realize that it might be interrupted after very few letters already. See my example in BlueRaja´s answer´s comment. I think a combined Penguino-BlueRaja solution would be extremely safe, general and totally doable in the mind, as long as everyone is capable of remembering 2 sequences of 3,4 or maybe 5 letters plus two key-words or numbers. $\endgroup$– BmyGuestJul 27, 2020 at 18:44
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Since this is basically a cryptography question, here's a cryptography answer that works with any number of people:
Using any secure encryption algorithm, have everyone encrypt their answers and send the ciphertext over the open channel (ie. shout them out). After everyone has finished, have them send the decryption keys.
Before the keys are transferred, no one will know what anyone else said. However, after the keys have been transferred, everyone will know what everyone said.
answered Jul 27, 2020 at 10:51
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$\begingroup$ The question specifies, "no paper or anything to write on". What purely mental encryption system are you proposing? $\endgroup$ Jul 27, 2020 at 16:52
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$\begingroup$ @chasly-reinstateMonica It doesn´t have to be complex actually, because no paper or anything is allowed. It just needs to be good enough to trick the brain. I guess a very simple ROT-X would already do the trick. Because at the time there is even a chance for the thief to make out the thing based on the answers of the owner (say 3 or even only 2 letters), he has already gone down the wrong path and can not "correct" his answer anymore. See example. For me, this is actually the best answer (better than mine.) $\endgroup$– BmyGuestJul 27, 2020 at 18:30
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1$\begingroup$ Example: You KNOW I use a rot-x cipher. I start D. You go next... You without knowing better go 'F'. Now I go 'N'. Do you know what is in the box and what letter you should choose next? If you choose anything but 'P' (D-N distance same as F-P distance) you are screwed with my next round. Which will be 'J', btw. You could only copycat (with same distance) again, but that makes it apparent that you are the thief. If the thief has to go first, it´s even worse for him. Also: Do you know what´s in the box now? :c) $\endgroup$– BmyGuestJul 27, 2020 at 18:35
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$\begingroup$ So this is essentially making Penguin's answer a bit more secure. How much more secure depends on how much complexity you allow for cryptographic encoding. Rot-X was super-simple. Anything with a key-word would make it even harder - by a lot - and could likely still be done by everyone in their head. $\endgroup$– BmyGuestJul 27, 2020 at 18:40
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$\begingroup$ @chasly-reinstateMonica: My understanding is that "no paper" meant "no private communication channel", not "computations must be simple" $\endgroup$ Jul 27, 2020 at 19:43
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The question could be solved with a zero knowledge proof. Since the example where there were private communication channels allows you, the judge, to look in the box, I'm going to assume the same is true in the no-channels case. In that case, you need to look in the box and then come up with a series of questions that 1) don't give away any information about the contents and 2) someone that doesn't know the contents would have to guess at. You need to be able to ask many questions like this as there's a chance the wrong person could guess the correct answer for a few (and/or to cover the chance that the right person misunderstands or misremembers).
Given that, it will depend heavily on the contents of the box.
Edit: While this should be theoretically sound, I think it might not be plausible to come up with enough questions to really make it work, especially for certain possibilities for the contents, at least without resorting to operations that'd be difficult for an unaided human to perform.
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1$\begingroup$ rot13(Gb or snve, vs jr'er hfvat pelcgbtencuvp gbbyf whfg nobhg nal nflzzrgevp pelcgbtencul jbhyq qb gur gevpx (rira whfg n choyvp-xrl rapelcgvba fpurzr). Ohg sbe fhpu n fbyhgvba gb or ernyyl vagrerfgvat (naq "snve-cynl") sbe fhpu n chmmyr V guvax jr fubhyq vzcbfr gur fbyhgvba or cresrpgyl (be ng yrnfg hapbaqvgvbanyyl) frpher, juvpu vfa'g gur pnfr sbe MXCf. CF: Fbeel sbe gur pelcgbtencuvp yvatb, ohg lbh oebhtug hc Mreb-Xabjyrqtr Cebbsf :) ) $\endgroup$– user62757Jul 27, 2020 at 1:05
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$\begingroup$ No, that's fair. I may be imposing unnecessary additional restrictions with my solution. rot13(V guvax n MXC vf n snve nccebnpu gubhtu. Vs fbzrbar gung xabjf gur pbagragf pbhyq nafjre n dhrfgvba jvgu zhpu terngre npphenpl guna fbzrbar whfg thrffvat, naq lbh pbhyq nfx nf znal dhrfgvbaf nf arrqrq, naq ab bar yrnearq nalguvat arj sebz lbh nfxvat, gura V guvax gung jbhyq dhnyvsl nf n fbyhgvba.) I am having trouble translating that into something that's actually doable by humans in practice though.(Also, I don't exactly see how to apply unconditional security here.) $\endgroup$ Jul 27, 2020 at 1:11
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$\begingroup$ Regarding security: I was actually mostly think of correctness, sorry... So basically we need something "humanly feasible" for which we can drop the error probability arbitrarily low $\endgroup$– user62757Jul 27, 2020 at 1:25
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$\begingroup$ Ah, gotcha. Yeah, that's the part I'm struggling with. I think I can do something with the color cube, but even then, you'd need to be able to ask multiple questions, so that alone wouldn't be enough. $\endgroup$ Jul 27, 2020 at 1:57
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$\begingroup$ No-one's solution can rule out a correct random guess. Of course the probabilities are vanishingly small in most of them. $\endgroup$ Jul 27, 2020 at 13:59
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Let both shout out (alternately) a long list of content items - true or false. Once both have given their list, either specifies the "index" of the shouted content item that is the true item. The longer the list becomes, the more secure the method becomes. A "copy-cat" person could be prevented by having each person go another time, if he repeats an item of the other person´s list - and then the first person go multiple times at the end to "catch up" with number of items. That way, the true owner can always "add unique" (correct) content which can finally only be indexed by him.
This is somewhat a refinement of @zovits suggestion above.
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1$\begingroup$ OP specifies, "no paper or anything to write on" - This going to put a strain on everyone's memory! $\endgroup$ Jul 27, 2020 at 16:56
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$\begingroup$ @chasly-reinstateMonica Fair point, but to be honest: Provided there is one true thing in the box, it's possibly enough to have a list of 3 to 5 items each, with a strategy for the "true" owner to start with one or two dummies. If the thief plays copycat, the judge can call it quit after very few items and the true owner can "sneak in" the unique item at the end without the thief being able to copy (because he has no slots to fill anymore.) The list only gets long if the thief does not play copycat, as the owner then needs to guess when it is safe to say the true item. $\endgroup$– BmyGuestJul 27, 2020 at 18:18
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$\begingroup$ ...as the judge knows the item, he will realize when the true item has been named by only one person (not copied by the other) and call it quit at that time as well. $\endgroup$– BmyGuestJul 27, 2020 at 18:19
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$\begingroup$ But as mentioned in a comment at BlueRaja´s answer, I think that one is superior to mine when combined with Penguino´s approach of piecewise giving the answer. $\endgroup$– BmyGuestJul 27, 2020 at 18:39
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Have each claimant shout a list of a dozen random items plus the name of the real content mixed in somewhere. Note these items, then open the box and see which one had the real item included in their list.
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$\begingroup$ The more guesses the contestants make, the more chance has the baddy to guess correctly. This actually makes things worse than allowing a single "Guess". $\endgroup$ Jul 27, 2020 at 17:12
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$\begingroup$ @chasly-reinstateMonica Not quite. Only the thief has a (good!) motivation to name things his opponent has already named. So if a lot of items are named, he needs to duplicate a lot of guesses, making it apparent that he is the thief. The judge, on the other hand, will know the true content and realize the first mention of a true item if it isn´t "copied" immediately by the other person. $\endgroup$– BmyGuestJul 27, 2020 at 18:22
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$\begingroup$ @BmyGuest - "The judge, on the other hand, will know the true content" - Nope. As the OP says, "only the owner knows the content" The judge can only check afterwards, once the shouting is over. $\endgroup$ Jul 27, 2020 at 22:10
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Ask very abstract and generic questions about contents with only true-false answers individually to each contestant. For instance: "Does that round object have red colour", "Is it only one green object in the box?", "Does gold object not exist?", etc. True owner should have answer rate near 100% (suppose he may not remember details exactly) while impostor should give correct rate only near 50%. Even if box is already empty we can fabricate some questions about non-existent objects. (They should be also mixed in anyway)
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Have each claimant take turns shouting out an item they know is not in the box but could logically be in the box.
No shout by an individual can be a duplicate of another shout by either.
The first to guess accurately is not the owner.
Keep having guesses thrown out by the claimants until the probability of guessing by the 'non-owner' which item(s) they got right is so low in probability it approaches useless.
Should both shout out a correct answer you know neither is the owner.(assuming the actual owner has not forgotten the contents or made a 'shout' error)
If both never shout out a correct answer given a significant number of attempts in relation to the complexity of items in the box we know both already know the contents...and ownership cannot be determined without more stringent means.
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$\begingroup$ You can remove your own answer by clicking the "delete" button at the bottom. But please don't vandalise it by removing its content and leaving a non-answer in place. $\endgroup$ Jul 27, 2020 at 17:47
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Ask one individual, and then the other. Whoever gives the better description is the owner, or if the second person simply repeats the first, it is the first person that's the owner. The dishonest person knows that they can give no better description than the true owner, so their best strategy is just to repeat the true owner's words if they go second. If they don't simply repeat, they can only give a worse description than the person who actually knows what they are describing. If the dishonest person goes first, they have an infinitesimally small chance of guessing correctly in a manner that cannot be refined upon by the true owner.
This is essentially the same as @chasly's answer, but avoids the problem of a dishonest person going first being very vague. With just a single person answering, there's no hard cutoff of what constitutes "sufficient detail" to prove ownership, since it's not immediately clear if one would accept the answer of "money, or "\$100 cash", or "$100 cash in the following denominations", or any increasing level of detail. Getting both answers and comparing their accuracy avoids this issue.
