While all logic puzzles defy intuition to some degree (hence their being “puzzles”), the Green Eyes one is among the few where the solution remains difficult to grasp even after it is known (at least for me). When I stumbled back across it recently, I spent a good hour thinking the answer must be wrong, and even now my mind flickers back and forth like I’m looking at an optical illusion.
The problem is posed as follows (borrowing from the TED-Ed video on the topic):
There are 100 prisoners on an island, each with green eyes. The prisoners can all see each other (i.e., can see 99 pairs of green eyes), but they have no way of observing their own eye color (no reflections, can’t remove their own eyeball, etc.) and cannot communicate with each other. The only way a prisoner can get off the island is by telling the warden their own eye color, which they have an opportunity to do each night; however, if they get the answer wrong, they are executed. The prisoners are all perfect logicians, so they won’t go to the warden unless they’re certain. Additionally, each prisoner knows that every other prisoner is a perfect logician, and that all prisoners know this fact, and that all prisoners know that all prisoners know this fact (some foreshadowing…).
You’ve requested that the warden free the prisoners, and while that’s off the table, he’s agreed to allow you to make a single statement to the whole group, under the condition that you can’t tell them anything they don’t already know. What can you say to the prisoners to help them get off the island?
One possible answer is that you tell the prisoners, “at least 99 of you have green eyes”. Each prisoner will then know that if they have non-green eyes (let’s say blue eyes), the other prisoners would see 98 pairs of green eyes and 1 pair of blue eyes, and would leave that night (as they’ll know they represent the 99th pair of green eyes). When no one leaves, each prisoner will then know that they have green eyes, and will leave on the second night. This solution seems straightforward enough; each prisoner knows that at least 99 prisoners have green eyes, but can’t know whether their fellow prisoners also know that.
However, if you’re worried about that being too obvious to the warden, you can instead say something much more innocuous. All that’s required for the prisoners to be able to leave (albeit after some time) is that you tell them, “at least one of you has green eyes”! This seems absurd on the surface, since all of them obviously know that at least one of them has green eyes (since they can each see 99 pairs), and they know that everyone else knows that (since, even if they themself have blue eyes, everyone else will see 98 pairs of green eyes), and they know that everyone else knows that everyone else knows that (since, even if they themself have blue eyes, everyone else will see 98 pairs of green eyes, and will know that each other prisoner must see at least 97 pairs of green eyes), and they know that everyone else knows that everyone else knows that everyone else knows that (since, even if they themself have blue eyes, everyone else will see 98 pairs of green eyes, and will know that each other prisoner must see at least 97 pairs of green eyes, which means they’ll know that everyone else must see at least 96 pairs of green eyes)… well, maybe it’s not so absurd.
It turns out that, while each of the prisoners knows privately that at least 99 of them have green eyes, they don’t know that each other prisoner knows that each other prisoner knows (x99) that even a single prisoner has green eyes. Once you tell them that at least one prisoner has green eyes, they’re able to start gaining information each night (similar to the information gained in the scenario where they’re told that 99 prisoners have green eyes), and they will all leave together on the 100th night. This was the part I found (and continue to find) confusing, as it seems like there must be a way for each prisoner to leverage their knowledge of 99 prisoners with green eyes into common knowledge of at least a single green eyed prisoner.
It’s a bit easier to wrap one’s head around the situation when dealing with a smaller number of prisoners. For example, with two or three prisoners, the same logic applies, and the gap in knowledge seems more reasonable. With two prisoners, each prisoner sees a single green-eyed prisoner, and has no idea what the other prisoner sees; with three, each sees two pairs of green eyes, but knows that if they have blue eyes the others will see only a single pair and will not be sure whether that green-eyed prisoner sees any green eyes.
For higher numbers, the logic scaffolds up in the same way, but it gets very difficult to hold the mental state of a prisoner in one’s head.
In our daily life, it’s often important for us to be able to recognize the mental states of others (particularly when they differ from our own), and occasionally this involves thinking about what others think that others think, but it’s rare to ever go beyond that depth. By challenging us to go 97 levels deeper, the Green Eyes puzzle forces us outside the realm of intuition and into the domain of logic, where we might be able to arrive at an answer but it certainly doesn’t feel right.
If it goes beyond the calculation puzzle, it is concerned with the problem of deference or how one can rely upon other’s view or opinion, and this is always risky