A few months ago, I (finally) finished Douglas Hofstadter's legendary book, Gödel, Escher, Bach. In one of the early chapters, Hofstadter introduces a deceptively simple puzzle called the MU–puzzle. It's a wonderful little puzzle that sets the stage for the rest of the book, and I think about it often. On this site, you'll get to explore it yourself!
The original puzzle uses the letters M, U, and I, but for simplicity, we'll just use two symbols: ■ and □ (the puzzle is the same). You'll begin with a single ■ in your chain. As you go along, you'll add and remove boxes from your chain. The goal of the puzzle is simple: to wind up with a single □. Of course, there are rules for how you're allowed to manipulate the chain:
When you scroll down to the next screen, you'll be able to click on available rules (or use the indicated keyboard shortcuts) to manipulate the chain. On the screen below that, you'll find an inventory of all the chains you've discovered.
Once you've solved the puzzle (or given up), scroll down to the last screen to read the conclusion.
Finally, while this site works on mobile, you'll probably have a better experience on desktop.
Have fun!
As you may have begun to suspect, it is impossible to produce □. But why?
I'll indirectly answer your question with another question. What do these chains of boxes and the four rules represent? The most obvious answer is "nothing". But we can choose another, surprisingly satisfying interpretation.
Suppose a filled-in box (■) represents one "tally" and an empty box (□) is just a placeholder. Additionally, each chain as a whole can be taken to mean "This number of tallies isn't divisible by three". So, with this interpretation, ■ represents the statement "1 isn't divisible by three", and ■■□■■□ is interpreted as "4 isn't divisible by three".
Why would we choose this interpretation? Without thinking too hard, you might go back and check your inventory and notice that every chain, when interpreted, is true! That's because there's an isomorphism between our system (by "system", I mean the rules and our starting chain put together) and the concept of divisibility by three.
You might be skeptical. How do we know that this interpretation is valid? In this case, we can use some basic math and logic: if you take a look at the four rules, you'll notice that only Rule 2 and Rule 3 actually change the ■-count. And if you start with a ■-count that's not divisible by three, then doubling it (Rule 2) won't turn it into a number that is, and neither will subtracting three from it (Rule 3). So there you have it: this system only generates chains with a non-divisible-by-three ■-count.
Armed with this new interpretation, we immediately know the answer to the question, "Can you make □?" The answer is no, because it represents the statement "0 isn't divisible by three", which is false. We could also immediately discount the possibility of ■■■□ , ■■■□□■■■, and countless other chains.
What's cool about this example is that it illustrates the fact that sometimes, the explanation for a certain fact does not exist at the lowest level. In this case, that "low level' is mechanically producing new chains, like you were probably doing at first. As humans, though, we can't help "jumping out" of that system and begin wondering about the system itself. It was only at this higher level, that of math and logic, where we were able to solve the puzzle. Actually, not only did our higher-level interpretation allow us to solve this particular puzzle, it enables us to immediately solve any "Can you make _?"-type puzzle within the system.
Here's an especially weird question: did our interpretation of the system cause □ to be un-producable?
Obviously, we're mixing up explanation with causation; our interpretation of the boxes didn't cause □ to be un-producable, it's just not. The rules are the only "real" thing, and we just based our interpretation off of it.
But interestingly, in other cases, it feels very natural to explain low-level events as being a consequence of higher-level interpretations. For example, we explain the motions of objects as a consequence of the laws of physics (deduction), even though our invention of the laws of physics was really a consequence of the motions of objects (induction). When we notice our friend is getting left out of a friend group, we might attribute it to the enigmantic "group dynamic". How can a "group dynamic" cause the thing it describes? And, of course, we have the most tangled hierarchy of all: me (and you). I often attribute my actions to, well, myself. But my concept of "I" is really just a high-level interpretation of the low-level neural activity that actually causes me to do anything. In a very strict sense, my wants and desires can't cause me to do anything any more than my interpretation of some colored boxes can cause □ to be un-producible.
I feel like I see this kind of "downward causality" (as Hofstadter calls it) everywhere, which is why it caught my interest. In a way, it's just an illusion; there's no actual tail-eating paradox until we interpret things on a high level and use it to explain things on a lower level. But since we like to do that often, I find it worth paying attention to.