These questions, of course, have many answers, and I am not the first one to ask them. I write this both to continue discussion of the topic, as well as get my own thoughts on order.
First of all, what is a puzzle?
A puzzle, in essence, is a problem designed to be solved by some measure of ingenuity, providing the solver with a degree of amusement. Solving a problem is at the very core of a puzzle, and the reason why people are drawn to them, but what distinguishes a puzzle from any other problem we encounter in our everyday lives? Arguably, a puzzle is supposed to be enjoyable to solve.
Puzzle quality does not depend on genre, as personal taste goes far in determining satisfaction. Some people hate logic puzzles and others jigsaw puzzles; can we objectively assess their quality when preference is so paramount? To a certain degree. (I would compare it to appreciating the craft behind a piece of music, even if it is of a genre one dislikes.) Context is one essential factor—one might like to solve a sudoku while commuting to work but dislike having to do one inside of an escape room. The issue of context however, might be more of a reflection on the design of an overall puzzle game rather than the individual puzzle. It is possible to ruin an otherwise enjoyable puzzle by inserting it into a scenario that would make it less enjoyable, but a bad puzzle will probably remain bad regardless of context. Context aside, I will try to avoid critiquing puzzle genres and instead focus on a few basic things that are less subjective: Logic, execution, and fairness.
A puzzle is logical when it has understandable reasoning behind the design choices made in its creation. At a very base level, 2+2=4 is more logical than 2+2=6, because it conforms to rules we are familiar with and understand. To make 2=3 make sense, you’d need to build a separate set of rules and make those rules apparent to your player.
Some puzzles require us to make connections between disparate objects. For example, you find a cylinder, and somewhere there is a circle, ideally the same diameter as the cylinder, inviting you to put the cylinder down on it. But let’s say there was an outline of a square instead - would you ever think to put the cylinder down on it? To some people, a less obvious connection implies a higher level of difficulty, and it may be true that a puzzle would be more difficult for it, but if the difficulty comes from faulty logic rather than something requiring deeper analysis, the added challenge may feel arbitrary.
The bottom line: How does a player feel when they haven’t solved a puzzle and then they hear the solution? Is their reaction “Ohhhhhh I should have seen that, that makes sense!” or is it further confusion? If they are left wondering how they were supposed to arrive at that solution, that’s not sign of solid logic.
A good puzzle requires a clear execution, meaning that difficulty is not due to ambiguity, and one should be able to extract a definite solution once the problem is solved. A couple examples:
You’re told that you’re looking for something in particular, but the answer is actually something else. A riddle says the password you’re looking for is a star, but it turns out to be the name of a constellation. They’re related, yes, but conflating the two creates ambiguity that sets a bad precedent for subsequent puzzles in the game.
Another common instance is a set of components that don’t quite fit or align properly, leaving you wondering if there is an error or if you’re doing something wrong. If you’re thinking, “I guess this is supposed to go here…?” then the puzzle is lacking a certain degree of precision. This is something fairly simple to correct and should not be ignored, as it will save your players a lot of frustration.
There should also be little room left for user error, especially when doing something that requires outside knowledge, like an internet research or a book cipher. A book cipher can be done in several different ways (chapters, pages, paragraphs, sentences, words, letters) so there needs to be some kind of instruction, otherwise your player will be left guessing through five different methods of extracting the solution.
The bottom line: When you perform the correct action or figure out an answer, you should be able to know that it’s correct. Avoid ambiguity, vague verbiage, multiple answer possibilities, and imprecision, especially with physical components.
How can a puzzle be unfair? It’s kind of feels like playing a game with a cheater. Lack of fairness often goes hand in hand with lack of logic and poor execution, but unfair puzzles often leave one with a special sense of dissatisfaction.
Intentional placement of red herrings: in-game items or pieces of information that seem like they are part of a puzzle but are actually just there to distract you, causing you to waste time chasing false solutions. It’s true that red herrings can sometimes occur unintentionally, but the biggest culprits are things that look very intentional and specific but in fact are just there to throw you off. If it’s an aesthetic choice, it should be clear that it’s not part of the puzzle.
Lack of information and general guessing: Let’s say you need to find 4 objects and place them in a line on a pedestal, but there are no clues relating to the order of the artifacts. It turns out you’re just supposed use trial and error. Without any feedback, like you might receive in a Mastermind game, this is complete guesswork, and guessing is different from solving - guessing turns the activity into a task reliant on chance rather than a puzzle.
Outside knowledge: Puzzles that expect you to know information that they’re not providing can be problematic. Tabletop games and puzzle hunts in which you need to use the internet for research usually state this explicitly, but if you are not expected to use outside resources and need to have knowledge of pop culture, history, or otherwise, this is basically just trivia; information to be found, rather than puzzles to be solved.
Unreasonable expectations: Things that cause failure because you simply can’t accomplish the task given to you. A puzzle that requires reading tiny letters in dim lighting feels unfair because ultimately it’s a measure of how good your eyesight is, not how good of a puzzle-solver you are. (It’s also not very fun.)
The bottom line: Eliminate red herrings. Include all relevant information, unless research is explicitly necessary. Don’t create difficulty through trial and error, guesswork, or unreasonable barriers.
So what if your puzzle is logical, fair, and well-executed - is it good? This is where subjectivity comes back in! On the bright side, it now has the potential to be good.
It’s hard to pinpoint exactly what makes a puzzle fun, but here are some questions that I like to ask myself: Does the puzzle rely on piecing together information or a process of tasks? If the latter, how long is the process? Does it get tedious? Does the puzzle introduce any new concepts? Does it teach you anything? Does it offer a sense of discovery? And finally, does completing it give you a sense of accomplishment?
It is hard to attain all or even most of these things. Like anything, it also take a lot of practice, and realistically, a puzzle will probably be liked by some people and not by others, and that’s okay! As a follow up, here’s a good watch about what makes for good video game puzzles.