Why Captain Picard Sticks to the Prime Directive: the Red-Eyed Monks

6 min readAug 6, 2022

A monk looking down — Picture credits: demian1975 @ 123RF.com

I like puzzles a lot. There is one, though, that earned my utmost appreciation, be it for the beauty of the reasoning involved, for the apparent paradoxes it raises, or for the fundamental concepts that it gathers from several disciplines. This puzzle would fit perfectly well in a discrete mathematics lecture, but also in an introductory game theory lecture. Yes, this is a hint. Or not?

Ready? Fasten your seat belts, here we go.

Mountains and sky — Picture credits: sam74100 @ 123RF.com

The pristine setting

Once upon a time, on a remote part of our planet, somewhere in the middle of the Pacific ocean, there was a beautiful island. No other place on Earth could match the landscapes it offers: Its lakes were bluer than blue, its snowy mountains were whiter than white, its luxurious forests greener than green — and don’t get me started on the fair weather.

This distant island, which did not appear on any map, was home to a tiny tad more than a thousand monks whose ancestors chose to enjoy a quiet and peaceful retreat from the world, for the greatest enjoyment of the indigenous birds.

These monks were abiding by a set of very simple rules. Indeed, 76 of them had red eyes, while the others had black eyes. Red eyes were considered a curse. Yet, there were no mirrors on the island, and the water of the ocean was so blue that they could not distinguish red from black in their own reflections. Also, they were not allowed to mutually mention their eyes in conversation. Hence, all that each monk directly knew was the others’ eye colors. Maybe it is for the best, because, for some obscure reason, should any monk know at any point that they have red eyes, they would have to leave the island at the next full moon.

The monks have lived with no incident and nobody leaving for more than 42 years.

The disruptive statement

One day, an anthropologist (we will not call him Jean-Luc) discovered the island. He stayed with the monks for a bit more than a year, learning about their ways, and carefully avoiding any interference in their eye scheme: you don’t want to mess up with the Prime Directive, do you?

However, upon leaving, he felt playful and, in front of the entire island population assembled before him, solemnly declared:

“At least one of you has red eyes.”

and left with an amused smile, considering that there is nothing here that they didn’t know before he came.

The mind-blowing questions

Why does nothing happen at all?
Why does something happen 76 full moons later?
Both 1 and 2 can’t be true simultaneously. Which one of these is wrong, and why?

Scroll down for the solution.

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Scroll more.

This line left empty intentionally.

And here you go.

The absent reasoning

Question 1 almost answers itself: of course, nothing happens! The statement that the anthropologist made does not convey any new information to anybody: black-eyed monks have seen 76 of their peers with red eyes for 42 years and thus knew that at least one monk is red-eyed. Red-eyed monks have seen 75 of their peers with red eyes for 42 years and thus knew that at least one monk is red-eyed.

Yes. Obvious. Trivial. Too trivial?

The crazy recursion

Now, let us investigate question 2.

[R1] Imagine for a moment that only one monk had red eyes — and hence saw all other monks black-eyed. If such had been the case, then the statement of the anthropologist would be new information to him: he would imply that he is the one with red eyes, and would leave the island on the next full moon.

[R2] Now, imagine that two monks instead had red eyes. Bob is one of them. He sees one other red-eyed monk, whom we shall call Alice because Ron Rivest would be very happy about this. Bob reasons as follows: “If I have black eyes, then Alice is the only one with red eyes, and she will logically leave the island at the next full moon because of the above reasoning.” Then, seeing that nothing happens at the first full moon after the visit of the anthropologist, Bob concludes that he has red eyes too and, just like Alice, leaves the island on the second full moon.

The same reasoning can be performed with increasing values in the number of red-eyed monks, finally leading to the conclusion that, since there are 76 red-eyed monks, [R76] seeing that nothing happens at the 75th full moon, they will then all know that they have red eyes, and will leave the island at the 76th full moon.

The not to innocent statement

Well, this is awkward: we have just carried out two reasonings that lead to exactly opposite conclusions. One of them has to be wrong.

It turns out that the wrong reasoning is the first one. Indeed, the argument that the monks learn no new information through the anthropologist statement is fallacious.

And the reason for this can be expressed with just two words:

Common Knowledge

Common knowledge is a concept widely used in game theory to express that everybody knows that everybody knows that everybody knows, etc, with any number of levels, a fact.

Imagine that there are only two red-eyed monks, Bob and Alice. Before the anthropologist comes, both Alice and Bob know that there is a least one red-eyed monk. However, Bob does not know whether the other red-eyed monk, Alice, knows as well. Indeed, if Bob had black eyes, they Alice would not see anybody else than her with red eyes.

But when the anthropologist makes his statement, Bob sees that all monks hear this statement, including Alice, including any other red-eyed monks if there are more. When the anthropologist makes his statement in front of everybody, it becomes common knowledge: Bob knows Alice knows. And they know they know they know, etc, this is the new piece of information.

So, if we now go back to having 76 red-eyed monks, there are 76 levels of “knows that he knows:” after the anthropologist comes, it becomes true that

All monks know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that they know that at least one has red eyes.

And this new piece of information triggers the recursion that unfolds in the next 76 full moons.

So, granted, 76 levels of “know that” are tough to handle for our human minds. Yet, in the ideal world of game theory where agents are assumed to be perfectly rational and logical, the reasoning is flawless and strikingly beautiful.