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This question is inspired by this one, inspired by another Stack Exchange network question.

How do I explain zero knowledge proof to my 7 year old cousin?

My colour-blind friend has two balls, one red and one green. However, he is convinced that they are the same. In fact, the balls are identical in every way except for their color. Your friend is skeptical that they are actually distinguishable. Plus, I don't want to tell which one is red and which one is green.

How can I convince my colour-blind friend that his two balls have different colours, without telling him which one is red and which one is green?

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    $\begingroup$ This reminds me of zero knowledge proofs $\endgroup$
    – phflack
    Commented Mar 23, 2018 at 14:17
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    $\begingroup$ I thought “inspired by another StackExchange network question” was a reference to this one. $\endgroup$
    – ShreevatsaR
    Commented Mar 23, 2018 at 18:36
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    $\begingroup$ This sounds like the beginning of an excellent joke. $\endgroup$
    – Sentinel
    Commented Mar 23, 2018 at 21:12
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    $\begingroup$ Well, even if it wasn't original, it could yet be posted here if it hasn't already been. As for benefits, we get a slew of attempts to devise new proofs. $\endgroup$
    – can-ned_food
    Commented Mar 24, 2018 at 20:00
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    $\begingroup$ Voting to close as too broad????? $\endgroup$
    – NL628
    Commented Mar 24, 2018 at 20:03

16 Answers 16

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Pick either the red or the green ball and tell him it is the "special ball" (i.e. the one you will be pointing out to him each time). Ask him to switch the two balls as many times as he wants behind his back, remembering which one was the special ball. You should be able to consistently tell him which was which, since you can distinguish between the two colors.

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    $\begingroup$ I was 5 seconds too late. Five. Seconds. $\endgroup$
    – Keelhaul
    Commented Mar 23, 2018 at 13:46
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    $\begingroup$ This does not necessarily show that someone is colour-blind, only that the tester can perceive a difference that the subject cannot. For example, one could spray the ball with transparent paint that reflects infrared. The tester can then be able to distinguish two green balls to a non colour-blind person by wearing infrared night vision goggles.A similar experiment can be done with UV reflecting paint and the tester having aphakia allowing the perception of UV. $\endgroup$
    – March Ho
    Commented Mar 23, 2018 at 15:27
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    $\begingroup$ @MarchHo - Sure, but you are violating the specific terms of the question: " the balls are identical in every way except for their color" and in context "color" is confined to visible. $\endgroup$
    – WhatRoughBeast
    Commented Mar 23, 2018 at 15:53
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    $\begingroup$ This relies on the colourblind person continuing to know which ball is which: a tall ask for someone that claims they are indistinguishable. $\endgroup$
    – Ian MacDonald
    Commented Mar 23, 2018 at 18:04
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    $\begingroup$ @IanMacDonald The balls are always with the friend; he just has to keep track each time whether he's switched the balls between hands or not. (Basically all that is being tested is that if the “asker” switches hands behind his back, the “guesser” also switches their guess, and vice-versa, consistently. This would not be (consistently) possible if the balls were indistinguishable to the guesser.) $\endgroup$
    – ShreevatsaR
    Commented Mar 23, 2018 at 18:34
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You can for example photograph the balls in two separate photos and use a program Color Picker (like this one) showing your friend that the RGB color codes are different.

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    $\begingroup$ Not the answer I expected but a really good one. congratulation $\endgroup$
    – Kepotx
    Commented Mar 23, 2018 at 13:48
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    $\begingroup$ Wouldn't this reveal which ball is red and which is green? The one with the higher R channel value is the red one. If you're looking at hex codes, even a small change in lighting would result in different codes even for colors that are very similar. Unless your friend is familiar with hex codes, I wouldn't expect him to be able to distinguish two hex codes of similar shades of red from two hex codes of visually distinct colors. $\endgroup$
    – Nuclear Hoagie
    Commented Mar 23, 2018 at 14:44
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    $\begingroup$ Create a two axis graph, one axe by colour (without revealing which one is red or green), and place each pixel on it. do this for both balls. each ball will be in one particular axis, so he can see that there are different, without knowing which ball is red and which is green $\endgroup$
    – Kepotx
    Commented Mar 23, 2018 at 14:57
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    $\begingroup$ @NuclearWang You don't have to tell them which picture corresponds to which ball. However this does sound quite vulnerable to tampering then. $\endgroup$
    – Peter
    Commented Mar 23, 2018 at 14:58
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    $\begingroup$ @Octopus ok, you can calculate an average of many points for each balls to avoid this kind of problem $\endgroup$
    – Stefano Lonati
    Commented Mar 23, 2018 at 15:58
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Show him the balls through

a red or green filter. Do not mention the filter's colour.

Edit: He might agree that there is some difference that he can't see, but he's still not convinced that it's the colour. Then

bring two filters, one red, one green (exactly the same colours as the balls), but otherwise identical. He may use both on the ballls. Switch them behind you back. Then he may use them on any reference object he likes, e.g. traffic lights.

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    $\begingroup$ Feels like the only answer that doesn't rely on heuristics. $\endgroup$
    – PuzzlingFerret
    Commented Mar 23, 2018 at 15:36
  • $\begingroup$ A practical problem with this is that most red/green colorblind people can distingush some shades of red and green, but not others. While the balls are likely close together, it might be harder to get filters that are indistinguishable. $\endgroup$
    – trlkly
    Commented Mar 24, 2018 at 0:17
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    $\begingroup$ This doesn't work. Suppose I have three polarising pieces of glass, like the lenses from sunglasses. I place two of them side by side, in different orientations, then look at them through the third. Depending on the orientation of the third piece of glass, the first two pieces may appear different - probably one dark and one light. I'd be incorrect to conclude that they're different colours; but this is precisely what your colourblind friend will observe with the balls and the filter. $\endgroup$
    – Dawood ibn Kareem
    Commented Mar 24, 2018 at 11:46
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    $\begingroup$ @DawoodibnKareem the balls are identical in every way except color. I guess that means they reflect light in the same way. If you're nitpicky, you could still say the balls have the same coating but are rotated differently. $\endgroup$
    – Eric Duminil
    Commented Mar 24, 2018 at 16:23
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    $\begingroup$ @DawoodibnKareem Your objection is irrelevant if it is established that the orientation of the filters does not change. $\endgroup$
    – Acccumulation
    Commented Mar 25, 2018 at 22:53
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Use physics:

take a (strong) red or green lamp and let the light shine equally on his balls. After some time, one of the balls will be warmer. Repeat as necessary with an opposite colo(u)r lamp.

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    $\begingroup$ as one ball would absorb more light, wouldn't one ball glow more than the other? isnt't that enough to make a difference, even without waiting the ball to be warm? $\endgroup$
    – Kepotx
    Commented Mar 23, 2018 at 15:57
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    $\begingroup$ @Kepotx of course, but such an (or very similar) answer has been already given... $\endgroup$
    – Radovan Garabík
    Commented Mar 23, 2018 at 15:59
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    $\begingroup$ > let the light shine equally on his balls. $\endgroup$
    – htmlcoderexe
    Commented Mar 24, 2018 at 3:33
  • $\begingroup$ You could just look at the balls. If the only illumination is green light, the green ball will look bright, and the red one dark. $\endgroup$
    – Acccumulation
    Commented Mar 25, 2018 at 22:48
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Simply

Show him a ball as the one you have to find and ask him to rearrange them while you don't look. Then show him the same ball (easy, they are the same color!). It could be luck (1/2 probability), so repeat until your friend is convinced.

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    $\begingroup$ yes, friend could perhaps make one ball in a conceiled way to help him keep track. $\endgroup$
    – Jasen
    Commented Mar 25, 2018 at 8:32
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Give your friend a red light and a green light. The red ball will appear dark under the green light, while the green ball will appear dark under the red light. He can see they are different without knowing which is which

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    $\begingroup$ Welcome to Puzzling.SE! Could you edit your answer to include spoiler tags, so as not to spoil the solution for anyone who wants to have a go at the puzzle themselves? Thanks! $\endgroup$
    – F1Krazy
    Commented Mar 23, 2018 at 17:08
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    $\begingroup$ OK, its all done! $\endgroup$
    – Norm
    Commented Mar 23, 2018 at 17:14
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    $\begingroup$ @F1Krazy Actually it would be more helpful in this case to remove the tags so we don't have to click every answer to see what has already been suggested $\endgroup$
    – Sentinel
    Commented Mar 23, 2018 at 22:52
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You could walk up to random people somewhere in public, letting your friend pick who to approach, and

ask them if the balls are the same color.

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Doesn't the procedure described here work?

Place the balls before your friend and tell them the green one is 'A' and the red one 'B'. Ask him to randomly show one of them after swapping the balls an even or odd number of times. From the colour, you'll be able to say 'A' or 'B'; repeating this $n$ times should convince him you're able to tell them apart with probability $1 - (1/2)^n$.

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Find an old pair of 3D glasses and have him look at the balls while wearing them.

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  • $\begingroup$ Must they be old glasses? $\endgroup$
    – Dr. Shmuel
    Commented Nov 10, 2018 at 23:56
  • $\begingroup$ @Dr.Shmuel Old 3D glasses use lenses of different colors (usually red and blue, I believe). Newer ones use polarizing filters, so they don’t interfere with the color of the picture — but that makes them useless for this purpose. $\endgroup$
    – Peregrine Rook
    Commented Nov 16, 2019 at 20:33
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If your colour blind friend doesn't have any knowledge in physics, so that he can conclude which ball is red/green, this approach might work: In a dark room, let a (very) powerful spotlight with a continuous light spectrum illuminate the ball, mark the ball position (so that it's the same for both balls), place a slit (1-3mm) in the reflected light so that you get a 1-3mm beam only. Let the beam enter a prism with very high index of refraction. The photons in the green light got a higher frequency (more energy), and will deviate more from the original direction. If you alternate the balls fast enough your friend will see the difference in the angles of refraction. Or just mark one direction before changing balls.

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We could deal with this by many solutions, one amongst is:

Bring an arbitray bottle/can of vingar of any odoriferous liquid, pour some of it onto one previously determined ball, tell your friend to taste it, now tell him to swap these balls as many times he want, then tell him wether each one taste like vinegar or not remotedly, your friend will get baffled when you always point on the different one without even get enough closer to smell it.

Another approach if the previous one is kinda deja-vu:

Let's bring a dog, ok forget it, a dog can smell from distance, a cat, hmm even cats are attracted to round things no matter they are, but well let's give it a chance, dip a ball of predefined color in fish soup, bring it few inches next to this cat, the cat will taste it and enjoy it, take it off from it then retry the same process, put it forward a cat's scope of view tens to few hundred feet, until enough far from it, the cat will arrive following the conjunctive view-taste phenomenon that has been insinuatingly impressed into its subconsciousness, always under our observation, we carry on the process but without sinking the ball in any soup, the cat intuitively comes into the ball, if ever it hasn't an appeal to play with it, it goes away disappointedly. Now if we call another cat, iterating the same process with it, until the point the ball is not merged with fish soup, instead of revealing the ball of same colour to the cat's field of view, we use the ball of different colour, the cat wouldn't be expected to apporach the ball, why? it's because the second ball doesn't bear a common conjunctive factor of view with the other ball, the cat isn't turned on. From this behavior, the achromatic person can understand just from events sensed from his surrounding, the imperative existence of another factor apart the ones he enjoys, which has influence on living beings and their reactions. so called colour.

If purposely we can break some of "zero knowledge proof" 's rules, we could approach the question in another way. This is contrarily not supposed to be a valid and conforming proof due to "conspiration" factor especially considering the "skeptical" nature of the person in question.

Calling out a random person occasionally passing by, blind-folding him, then we hand him the balls at many reprises with telling him to drop "the one he thinks special". The subject would, at some extent, drop the wrong ball. Always under the witnessing eye of the colour-blind person, we tell the one in experience to drop the special one without wearing the band, he would never miss. This can be same way expressed with that person, telling him to pick a special ball, then asking him to drop that ball without seeing it, at some round he would definitely say "Oh I missed the right one!". What makes him think he misses? Why haven't he had mistaken the special ball when the band was removed from his eye? what kind of this insensible factor which is acting upon that person's choices?

Such kind of questions and methods are used in real life by psychologists and social experts to map the patient to his abnormalities that he isn't aware of, without forcing him unquestionably to get convinced without self asking why. (this is a little off topic margin so my excuse)

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    $\begingroup$ Are you intentionally going for a pretentious, confusing, poorly worded, and overly wordy phrasing? $\endgroup$
    – Acccumulation
    Commented Mar 25, 2018 at 23:02
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Ask your friend to look at the reflections in the balls' glossy surfaces. They will appear differently.

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    $\begingroup$ could you further explain your answer? I don't think the surfaces will have differences because of different color, or at least not with human eyes. $\endgroup$
    – Alex
    Commented Mar 23, 2018 at 22:23
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    $\begingroup$ A green ball will reflect red light differently than a red ball therefore affecting the grayscale luminosity. It's like looking through a red or green filter, but the ball is the filter. $\endgroup$
    – Sentinel
    Commented Mar 23, 2018 at 22:43
  • $\begingroup$ I think it won't work for really glossy surfaces - think of the surface as having a thin layer of transparent lacquer, where the reflected light bounces off the surface of that layer without encountering red/green material. You should probably be looking at the diffuse reflection instead, after placing them very near other red/green objects. $\endgroup$
    – Jaap Scherphuis
    Commented Mar 24, 2018 at 6:38
  • $\begingroup$ @Jaap Scherphuis That's just incorrect. You have no cause to assume a reflective lacquer. $\endgroup$
    – Sentinel
    Commented Mar 24, 2018 at 7:07
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Add a small lump (like of hardening glue or tape or a chewing gum) to one ball and make sure to know which it is. Then tell him to shuffle the balls behind his back and then take them out to the front again but only make sure the lumpy part stays out of sight (i.e the lump will be between his palm and the ball so you can't see it). Then, you would be able to distinguish by color alone which ball has the lump and tell him that. Do it several times until he believes you.

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Use a different physical effect:

Place a thin black screen with a small hole in such a way that the ball touches the screen at the hole, illuminate the hole from the other side of the screen by white light. Use a high dispersion lens (a lens with different focus lengths for red and green light) and show that light from the red and green balls is focused at different points.

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You could post a YouTube video with a photo of the balls, asking if they are the same colour or not, then show him the comment section.

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BASICALLY what I would do is get him to Believe a third entity (a detector or a person or anything that can point out the difference).If he has full trust on this third entity then third entity can concur your results then ,he will believe that the two balls will have different colors.Else you can add on more entities to concur your results............
OR

Since hes is color blind take a picture of the green ball Now tell him the inverted color of green is red Show the image of the inverted green ball to him He will say it is the exact same as the red ball Then how can the invert of one colour look the same as the original? this will be the question in his mind Then he will understand that the only possibility is that they were initially different

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