What makes a puzzle fun? People will usually say "that moment where you realize what you have to do".
What makes one puzzle better than another puzzle? Here are a couple of things that I think most people would say:
-"It's important that the puzzle solution is logical [for some definition of "logical"]"
-"It's nice if there's a simple solution to something that seemed complicated, so that your small realization becomes REALLY useful".
-"It should have taken me a long time to get the solution"
I want to people to become more aware of another way of valuing a puzzle. I want people to ask: "what does this puzzle communicate to me?".
Here's something that was tweeted by Marc Ten Bosch, who is making Miegakure, a puzzler about world with four spatial dimensions:
"Can people stop saying "once a solution finally reveals itself, there is an immense sense of satisfaction to be had" about puzzle games? "Once I successfully beat part of the game, I felt good." - No shit! They are implying they value the pleasure more than what the puzzle is about."
Marc has the same value system that I do. We say: the pleasure is pretty good, but it is only a pleasant side effect of the player learning something important, which is the bigger point. Miegakure is full of pleasure, but what makes it great is that it fills your brain with important and transcendentally beautiful facts about the nature of spaces and shapes.
My favourite game of 2012 was Incredipede, a puzzler where you design limbs for animals, and use those limbs in various ways. To give a small example of something Incredipede communicates: it made me realize things about how birds' wings manipulate air. I learnt that over the course of only two or three puzzles, so the game was communicating a lot in a very short space of time.
Most good puzzles involve some degree of communication; it's just that usually the thing that is communicated is a little more abstract or has a less immediate bearing on the world than in Incredipede's does. Communication doesn't necessarily mean you have to tell us something about the "real" world. There's a branch of mathematics called "number theory", which -for a while- was famed for having no practical application, even though the theorems it gave us felt extremely profound and surprising.
At the heart of it, the solution to a puzzle has to involve something unexpected. The next time you solve a puzzle (that you may feel very good about) ask: "what is the nature of this unexpected thing? Does it illuminate lots of different possibilities within the game's engine to me? Does it illuminate lots of different possibilities in the universe in which I live?"
There's another way of saying all this which I learned from Tim Rogers. Eiji Aonuma once said a good puzzle should "make the player feel smart". Tim Rogers says that a puzzle in Braid "demands that you become actually†smart". Which is to say: you must cotton on to something genuinely new that you didn't know before.
My joint-favourite game of all time is Portal. Portal lets you connect two distant points in space; that is a very unique thing to do, and the game's level design communicates its many interesting implications (the physics of it is mostly wrong, but that doesn't mean that it can't show you interesting mathematics). I want you to notice two things about Portal:
1. Portal's makers did NOT sit down and say "we want to communicate a load of stuff". They just said: "we want to make the most fun video game we can". Communication and fun do not seem to be opposed. In fact we have reason to cautiously believe that decent communication is always fun.
2. Portal's puzzles are not "hard", as in "they do not take most people a lot of time to solve". This is because the puzzles are "focused" - there are no red herrings, complex actions are broken down into simple actions, small details are used to give you clues about what you should do, etc. The point is that Portal communicates quite complex things, and so to make the communication clear they simplified things as much as they could without compromising the solution's coolness. This resulted in less intense "orgasms" of solution-getting; but attributing too much value to orgasms is a little immature, in light of decent communication.
If you'd like to know more about how communication works, I recommend this lecture by Marc Ten Bosch and Jonathan Blow about mechanics. I gave a lecture of my own about communicative level design, and that should appear on the internet soon. Personally I am looking at many different avenues for communication in level design; some of my results appear in my games Music of the Spheres and The Stranger Loop.