Riddles and Limited Rationality

I have always loved riddles, and my favorite is the riddle of the hats (well, one particular hat riddle among many). Here’s a short version.

Three prisoners, two sighted and one blind, are shown five hats: three red and two white. One hat is placed on each of them in a way so that they can’t see the color of their own hat (or the unused ones). They are told that if they correctly guess the color of their hat they can go free, if they guess wrong, they will be killed. If they choose not to guess they go back to their cell. The first prisoner (sighted) looks at the hats on the other two and refuses to guess. Then the second prisoner (also sighted) looks at the other two and also refuses to guess. Finally, the blind man correctly declares the color of his hat, and is set free. How does he know?

Ok, I am going to provide a solution to the riddle below, so if you want to think about it for a while go ahead and do that now. Know that the blind prisoner makes a logical inference (not a guess).

With thee prisoners and two colors, there are eight combinations:

  1. WWW
  2. RWW
  3. WRW
  4. RRW
  5. WWR
  6. WRR
  7. RWR
  8. RRR

We can immediately eliminate the first possibility because there aren’t three white hats.

We know that if either of the first two prisoners saw two white hats, they would immediately know their hat was red. So after the first prisoner refuses, we can eliminate the second possibility. After the second prisoner refuses, we can also eliminate the third possibility.

So it looks like the answer for the third hat is probably red (after all, 4 out of 5 remaining choices have an R for the third hat). But it’s still a guess. If we can’t eliminate possibility #4, the blind prisoner is taking a big risk.


Ok, so everything before this is pretty simple, and the whole fun of the riddle is determining whether we can really eliminate the RRW possibility. And it turns out we can! To see how, let’s show this from the second prisoner’s perspective:

  1. WWW = WXW (not possible)
  2. RWW = RXW (elimated based on P1’s choice)
  3. WRW =WXW
  4. RRW = RXW
  5. WWR = WXR
  6. WRR = WXR
  7. RWR = RXR
  8. RRR = RXR

Note that there are two WXRs and two RXRs (one for each possibility of X). However, there is only one WXW, and only one RXW. We were easily able to see that if the second prisoner saw WXW he would choose red, because there are only two white hats. However, we now realize that if the second prisoner sees RXW, there is also only one choice (again, red) because the other RXW (i.e., RWW) was eliminated by the first prisoner’s choice!

So if the second prisoner had seen either WXW or RXW he would have been able to guess his hat color. Since he didn’t guess, the blind person eliminates the both 3 and 4. Then only 5–8 remain, all of which end with R.


Ok, so that was the fun part. Now I am going to connect this to my new favorite book that I have never actually read, The Secret of Our Success. Basically, people aren’t that smart, including me. This riddle took me three damn days to solve the first time I heard it!!

If we properly programmed all the rules into a logic machine, the solution would come instantaneously without a whole lot of computing power. But the solution did not appear immediately to me (and probably not to you). Our brains do some pretty amazing stuff without even thinking about it, like recognizing faces, or maintaining our balance when walking on a boat. But when it comes to logic, we are very, very limited.

It’s like we have this veil in our brains that that only allows us to see one tiny logical step at a time. The second prisoner’s choice requires us to contemplate several things simultaneously, so we can’t really do it without great difficulty. The fact that this problem took me so long (and that it stumps many, many people entirely) says to me that it is a pretty good indicator of the bounds or our rationality.

In other words, once you know the solution and understand that it doesn’t take a huge number of complicated steps, it becomes clear that our rationality is almost embarrassingly limited. Spock would be disgusted.

So what’s the connection to The Secret of Our Success? The premise of that book is that the human ability to develop and transmit culture, not our rationality, is really what allowed us to rise above the animals. Almost every real problem is way more complex than our human brain is capable of comprehending. Over the years we have become reasonably good at making up toy problems and models that simplify real life to the point where it is tractable to our paltry reasoning capacity. That sometimes gives us this sense like we can actually understand the universe. But it’s a trick! Things only seem rationally comprehensible because whenever they are complicated we simplify them until they fit within our brains!

This isn’t to say that rationality isn’t important. I love trying to apply reason to things, and I do think we can create card castles of rationality that bear just enough resemblance to the real world so as not to be totally useless. And the tiny little bit of usefulness afforded by our rational brains can sometimes accumulate into some pretty amazing accomplishments. But don’t get too cocky! Our rationality only accomplishes great things when we stand on the shoulders of giants in a way so that our abstractions and simplifications aren’t totally useless.

A human being exploring the universe with our limited rationality is kind of like a blind man exploring Eurasia with a cane. Individually, it is almost pointless. But if the results are communicated well enough, the accumulation of knowledge can eventually become significant.

In the grand scheme of things, even the contributions of great geniuses like Isaac Newton or Albert Einstein turn out to relatively simple and incremental. It’s really the accumulation of knowledge (as well as other forms of cultural transmission) that leads to greatness. Before the widespread adoption of books and other forms of abstract representation (i.e., clay tablets, Wikipedia, etc.) this accumulation was represented in the form of cultural practices. And just like no single person really comprehends all of the science that goes into creating the things we rely on every day to survive, our primitive ancestors didn’t really understand all of the reasons for their cultural practices.

One consequence of this way of thinking is a certain form of humility. It is valuable to try and understand the world around us, but we simply do not have the mental capacity to create complex systems from scratch. Everything we do or think is, in some sense, an incremental change. And if it isn’t, it is probably too simplistic to be of much use.

Emotionally, when I come across a new idea it often feels as if the world is about to change. And that emotion is important. It helps keep me going. But recognizing the limits of our rationality gives us perspective. Our brains are simply not capable of seeing very far, so in our most bold moments we simply walk to the edge of darkness and take one more step.



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