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There is a guy who can determine if a diamond is genuine or fake by just looking at it. His name is Richard.

One day, he is summoned by the leader of a gang of thieves. It turns out that thieves have stolen a lot of diamonds from a jewelery store. However, the problem is, some of the diamonds are fake. If the thieves try to cash out a fake diamond, they will be caught.

Naturally, none of the thieves want to go to a jewelery store and borrow a jeweler’s loupe, because this will cause them to be discovered.

The leader, Nick, wants to learn which diamonds are fake ones. Richard agrees to tell them, but only if they agree to share 20%.

Both parties do not trust each other. Richard does not want to tell thieves which ones are real, because then the thieves would take them and run away. The thieves think Richard might lie to them and cause them to go to jail.

Thus, they set some rules:

  1. Richard has to prove that he can distinguish any diamond by only looking at them.
  2. Richard should not tell if any of the diamonds are fake or real.
  3. Thieves know that out of 500 diamonds, only 100 of them are genuine. Therefore, they will let Richard take any 20 and leave. If he takes some fake ones, then he won’t be able to cash them out.

How can Richard prove that he is able to do what he says, without showing them any real diamond among all the fake ones?

Note that all the diamonds look, weigh and feel identical.

Hint:

Richard takes one fake diamond and one genuine diamond from the stack, and hands them to the leader.

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    $\begingroup$ Is Richard also afraid to get robbed once he has the 20 diamonds, or can we assume that any diamond he ends up with during the sorting process is safe with him? $\endgroup$ – Karsten Köpnick Jan 12 '18 at 16:17
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    $\begingroup$ I'm confused as to why Richard can't tell them if the diamonds are real or fake. Is that the whole point of his presence? At the end I assume the thieves will want to know which diamonds are real, otherwise they've just given away 20% of the loot for nothing, $\endgroup$ – PopularIsn'tRight Jan 12 '18 at 16:30
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    $\begingroup$ @Bachrach44 Richard will prove first that he can distinguish, and only then he will separate the stack, and take 20. Thus, the thieves will be sure that he didn't just take a blind guess to take some diamonds. $\endgroup$ – padawan Jan 12 '18 at 16:49
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    $\begingroup$ Can I just hire a crew to shoot them and steal all of the diamonds? $\endgroup$ – Richard Jan 12 '18 at 20:11
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    $\begingroup$ I am intrigued by the comment "Note that all the diamonds are identical." It makes me wonder what, precisely, it means for a diamond to be either "genuine" or "fake", in the world of this puzzle. $\endgroup$ – Trevor Powell Jan 13 '18 at 1:20
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Assuming that, other than real/fakeness, the diamonds are all indistinguishable from each other:

1. Richard separates the diamonds into, say, 5 piles of 100, each one containing a different number of real diamonds, and writes the number of real diamonds in each pile down secretly. 2. Richard leaves the room. The leader randomly rearranges the order of the piles (without mixing them), remembering the order. 3. Richard comes back in and re-counts the piles, and as a proof of his prowess, announces the rearrangement order (which he can figure out by comparing the counts to the ones he has written down).

With this approach,

There are 120 possible rearrangements, so a correct guess would be unlikely (and the procedure can be repeated if the leader wants a higher degree of assurance). Since none of the piles are all-real or all-fake, and since Richard doesn't tell the leader how many real diamonds are in each pile, the leader doesn't get any actionable information about individual diamonds.

Incidentally,

Rather than letting Richard take "any 20", once Richard has proven his prowess, he should rearrange the diamonds into a new 5 piles, one of them containing all the real diamonds. The leader then separates 20 diamonds at random from each of the piles, Richard takes the real subset, and the thieves take the rest. That makes it impossible (or, at least, unprofitable) for Richard to double-cross the thieves, and makes it so that Richard getting paid and the thieves getting their information happens in the same moment.

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    $\begingroup$ This demonstrates that Richard is able to distinguish some consistent characteristic of the diamonds. It doesn't prove that it has anything to do with fake versus genuine diamonds. (E.g., suppose the diamonds, both real and fake, come in different sizes. Then he can declare the biggest 100 of the diamonds "genuine" and distribute them into piles accordingly, and provided he can distinguish larger from smaller diamonds he will pass the test described here.) $\endgroup$ – Gareth McCaughan Jan 12 '18 at 17:41
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    $\begingroup$ @GarethMcCaughan The original post says the diamonds are otherwise identical. $\endgroup$ – Sriotchilism O'Zaic Jan 12 '18 at 17:41
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    $\begingroup$ Though ... if Richard can distinguish real from fake, clearly the diamonds aren't indistinguishable, and it's kinda hard to imagine a situation in which the only possibilities are (1) he can 100% reliably distinguish real from fake by eye and (2) he can't identify any differences at all between the diamonds... $\endgroup$ – Gareth McCaughan Jan 12 '18 at 17:45
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    $\begingroup$ Richard is afraid of the thieves running away with 100 diamonds after they've learned which ones they are, not of being stabbed in media skedaddilis. With this method, both sides get what they want at the same moment, so there's no opportunity to cheat. $\endgroup$ – Sneftel Jan 13 '18 at 14:53
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    $\begingroup$ Strangely this not only answered the question, but only this answer made me understand what the actual puzzle was... $\endgroup$ – BmyGuest Jan 13 '18 at 17:42
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From the hint it sounds like what you were looking for was this:

Richard takes one fake diamond and one real diamond from the stack, then hands them to the leader. He tells the leader this, but doesn't tell him which is real and which is fake. He asks the leader to hold one in each hand.

From here,

Richard asks the leader to either keep the diamonds in the same hands or swap them while Richard keeps his back turned. Richard then can turn around and tell him if he swapped them or not. They can repeat this any number of times until the leader and thieves are satisfied.

With this, he's proved the minimum of being able to distinguish between real and fake diamonds.

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    $\begingroup$ this sounds like what OP was looking for, I'm not sure why he accepted one that completely ignores the hint. $\endgroup$ – user3453281 Jan 12 '18 at 18:49
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    $\begingroup$ Because it's not a complete answer, it misses the part explaining how to share real diamonds. And even if the accepted answer does not follow the same reasoning as the OP, it is still a valid answer. $\endgroup$ – Alix Eisenhardt Jan 12 '18 at 19:11
  • $\begingroup$ Agreed. The accepted answer is far more complete. $\endgroup$ – itriedacrab Jan 12 '18 at 19:28
  • $\begingroup$ This is the answer that I had in mind, but the other one was earlier, and also more extended. Now it would be really unfair if I take the accept back. $\endgroup$ – padawan Jan 15 '18 at 8:50
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The following answer assumes Richard is greedy, and that the thieves know it.

Richard makes 5 heaps of 100 diamonds, then tells the thieves one of those has only genuine diamonds. He then asks the leader to select at random 20 diamonds from each pile. Richard then pick one of these pack of 20 diamonds and leave. The thieves now know which heap of 80 remaining diamonds is the correct one, assuming Richard didn't want to risk having even a single fake diamond.

Of course, if Richard is willing to sacrifice a bit of his pay to trick the thieves he could, for example, insert 10 fake diamonds in the "correct" heap. The odds are he would still get more or less 18 genuine diamonds, while the thieves would most probably still get some fake (and get caught !)

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    $\begingroup$ This in no way proves that Richard can distinguish real from fake. I could make 5 heaps of 100 diamonds selected randomly and do the same thing. Only 20% of the 20 diamonds I take with me will be real, but 4 free diamonds is pretty good for a guy who has no particular skills. The robbers may know Richard is greedy, but they don't know if he has the skills he claims. $\endgroup$ – Nuclear Wang Jan 12 '18 at 20:57
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    $\begingroup$ @NuclearWang Fake Richard goes to jail too in that case or needs to find a real Richard. $\endgroup$ – user19641 Jan 12 '18 at 21:29
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Richard makes two piles of 100. One pile is all real, one pile is all fake. Richard tells them that he will leave the room and they will secretly place the 200 diamonds in a long line, so that from left to right each diamond can be numbered from 1 to 200, and to write down in the leaders secret journal which diamond belongs to which pile. Richard can simply come back and redo the exact same piles.

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Richard can say how many diamonds are real. The number of real diamonds are known to the robbers, and the probability of Richard guessing the correct number of real diamonds randomly is 1/501. Actually, 1/479 if you assume that knowing that he will be paid 20 diamonds lets Richard know that greater than 20 real diamonds exist.

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The answers given so far prove that Richard can identify some trait associated with at least stones, but not that there is a trait which could distinguish 100 of them from the rest. I would suggest that much stronger approach would be for Richard to divide the stones into five piles, and then have the crooks put 20 from all piles except #1 into bag #1, put 20 stones from all piles except #2 into bag #2, etc. up through bag #5. The result will be 80 stones in each bag and 20 in each pile.

With Richard out o the room, have the thieves place the stones in bag #1 and pile #1 into line #1, keeping track of which stones came from the bag and which from the pile. Do likewise with bag #2 and pile #2, creating line #2, etc. Four of the piles will contain 20 original fake stones, 60 other fake stones, and 20 real stones. The other will have 20 original real stones, 20 other real stones, and 60 fake stones.

To prove his abilities, Richard identifies 20 stones in each pile that have come from other piles. For four of the piles, that will mean identifying the 20 real stones. For the remaining pile, it will mean identifying 20 fake ones (chosen arbitrarily from 60). Assuming Richard can keep a poker face, nothing will distinguish the whether he identifying 20 real stones among 80 fakes, or 20 fakes among 60 more fakes and 20 real stones.

While some other methods would allow Richard to fake his way through even if 90% of stones were indistinguishable to him, this approach would require him to recognize about 100 of them as being different from the other 400.

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Let's assume the characteristic of fake and real diamonds is their conductance.. this will give Richard (who for the sake of argument I'm giving the superpower of being a human ohmmeter, and even though diamond is an insulator, no insulation is perfect. (Diamond is better than glass at insulating electricity, and worse at insulating heat than glass)) the ability too pinch a diamond and know for certain that he's either holding a fake or a real one, he can then randomly sort the diamonds in any of the previously mentioned methods and the thieves can do whatever they please too the piles and Richard will know very quickly that the orders have changed and how.

As an aside, +1 for the guy who's username IS Richard and opted too hire a crew too shoot the thieves and steal all the diamonds! That gave me a good laugh!

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  • $\begingroup$ But then the fake ones would be revealed during the re-sorting process $\endgroup$ – padawan Jan 14 '18 at 20:51

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