Theory of puzzles
Nate here, and I need to let you know I am obsessed with puzzles. I have too many and I’ve spent so so many hours on YouTube watching them being solved. But that has led me to be able to create them. I designed the jack puzzle, Venn puzzle, tetra puzzle. I am incredibly proud of these things although I know they aren’t the most unique or best puzzles out there. Still, after laying awake at night thinking about these weird little things, I’ve began to put together a theory for what makes a good puzzle. This began as an internal document purely to understand how to do this tricky ass thing called designing puzzles. This is not a “How to design a puzzle” guide (although I’d like to write that). Nor is it a “what makes a good puzzle” rules for all people. It is simply my own theory for making puzzles and I am writing it out to help myself understand what puzzles really are and hopefully get better at making them. For those interested in learning how I design a puzzle, read on.
First off, I am not making these concepts, I am discovering them and putting them in words. My thinking, is if you took a reading of people’s enjoyment of a puzzle and compared to a checklist of these things you would see close correlation. But, the obscurity of mechanical puzzles has stopped anyone from ever putting down the rules clearly (to my knowledge). This is my attempt.
Nate’s personal puzzle rules:
Beautiful draws you in
Tactile draws you in
Just hard enough to solve keeps you there
Fun to solve makes you smile
Undeniably solvable calls itself a puzzle
Defines it’s own solution implies what to do
No words speaks another language
Underlying system holds secrets
Breadcrumbs gives hints
Red herring / Time changes course
If you can make an object that checks all of these boxes, it will be fun to do. The first four items are obvious (and very subjective) but they are also the most important. Many times in seeking a pass in one of the final rules, you can destroy the pass in one of the early ones. Before I print a prototype, I run it through these paces and see if I think it will pass them all. Design is a process, so there’s nothing wrong with a prototype that has some but not all of these attributes. But I’ve found that directly considering all of these helps me get back to the drawing board faster.
A good (mechanical) puzzle is visually beautiful.
True fact: puzzles mostly just sit on the shelf. Truer fact: most are bought based on a pretty image on the internet. If they look good there, they are more valuable because if they are out in the open and visible to people, they will be played. I want to make puzzles that are beautiful to just sit around. Many are symmetric and have a weird wizardy vibe to them. Or they seem to be based on a higher order of thought. Like a icosahedron with bars and pins sticking out. There is something about non-euclidean forms that confuse the shit out of me. A cube I can understand. x, y, and z, I get. But the pentagons on a icosahedron and the way a dodecagon can fit neatly within its faces is actual magic. The four axes of a triangular pyramid are too much to think about. But they do exist in our three dimensions. I can see them and touch them. But they’re slightly out of reach. These forms entice people to touch them which directly leads to…
A good puzzle is fun to touch.
I love the feel of rotating an acme screw in a vise. There is something satisfying about helixes doing their work. or feeling a planetary gear system smoothly twist itself in space. Or play with an trochoidal oil pump on an old motorcycle. Neatly formed shapes agreeing with one another and sliding ‘just so’ can be a very strong drug. Even without the “solve-the-puzzle” aspect, this can form the basis for a charming and satisfying toy.
A good puzzle is just hard enough.
Understanding degrees of freedom in a puzzle helps determine it’s difficulty before you produce it. This basically means, how many possible guesses are there for the player to guess. You’re looking for a number that is not too high and not too low. Imagine a chimpanzee simply moving this object. How many possible positions are there to try? You’ll want to control this number and keep it not too small and not too large.
Another concept that has helped me determine difficulty before testing is branching. Consider a maze. The number of splits in the path add to the difficulty. If a player can find themselves four steps along the way and they have to turn back and try again, that’s hard.
Degrees of freedom and branching help estimate difficulty but they aren’t foolproof. You have to test with random people. And in the end, the level of difficulty is not clear. You’re aiming for something of a goldilocks difficulty if you want normal people to enjoy the puzzle.
A good puzzle is fun to solve.
“Fun” is a tough concept. But I have found a few things to be true. People don’t have fun flipping a cube for thirty minutes until the solution appears. People like feeling progress. Like a jigsaw puzzle. People want to feel themselves getting closer to understanding the puzzle and closer to the solution. If you can accomplish that, they will keep moving forward and keep enjoying the process. A key part of the puzzles I try to make is adding steps. Positions that feel more complete than before. Often times I’ve created puzzles that are “one monumental task”. This not fun because you are given no feedback about your attempts until you get them all right at once. The Venn puzzle is one of these but it makes up for it with tactile fun and low overall difficulty.
A good puzzle implies what the goal is.
While many people approach a mechanical puzzle with some back knowledge, I design as if they have never seen something like this before. Therefore the puzzle needs to define itself visually as a puzzle. If it is too simple they will assume it is a static object, too complex and it is fragile. The goal is to make them touch it. Make it look like it comes apart and it can handle a little jostling.
A good puzzle requires no printed or spoken rules.
Relying on printed directions makes this object a riddle, not a puzzle. A set of three coins with special directions can create a riddle. That doesn’t create a product you can sell and it’s not what I aim to make. These are the rules: “No external tools, no excessive force” That’s all you get. Everything else should be implied by the puzzle’s physical appearance.
A good puzzle is undeniably solvable.
If I see four cast parts of metal linked together I know someone linked them and they must be able to be solved. Materials play a strong role here in standing as proof of how a thing was made. A piece of wood with continuous grain is proof that it is one thing and solving a knot of pieces must be possible. The object itself should be proof that it can come apart. Screws, glued seams, all deter the player from continuing because they cast doubt on the possibility of a finish line. It is best to prove to the player with physicality that these parts must have a solution. Someone put them together, without external tools - so don’t give up!
A good puzzle requires the player to understand the system to solve it.
I would like to design puzzles that reveal their internal structures as you explore the form. You turn a part and look inside a corner and slowly you put information together that allows you to solve the thing. I think this is part of the magic of puzzles. They speak in rational terms that are extremely hard to put in words. The physical constraints of the shapes determine the way they can move. There is no compromising with it. You have to learn the rules and play along.
A good puzzle has just enough bread crumbs to find your way.
Bread crumbs are intentional hints given by the puzzle. It can be a seam between parts, a click of a ball bearing, the matched surfaces of two parts or the texture on a part. This lesson was hard won over many many designs. The truth is, if a puzzle has no breadcrumbs, it is not fun. Part of what Damein and I have learned over the last year of designing puzzles is how to best imbed information into an object. Seams and surfaces tell you how they would like to move. Better still, after wrapping your head around this, you can use a breadcrumb as a red herring and leed the puzzler down the wrong path (but don’t leave them there).
A good puzzle unveils layers of meaning over time.
A puzzle should take time to complete. The goal is to trap the puzzler with this object for a while and then release them. The best way to do this is to not reveal all of the information all at once. Dose it to the player as they dig into the puzzle. Try to tell a story. Give them clues that allow them to “understand” where they are going and then throw a curve ball. Require them to relearn what they think they know about the inner workings of the object. After finding success in the pervious list, I try to add this element.
CASE STUDIES and EXAMPLES
The Venn puzzle is a single mini-mental task. Most people can solve it without help in less than an hour. It keeps itself simple by having only three pieces. Better yet, three identical pieces. Usually it is given to the solver already completed as a sphere. So they are given a rule: it creates a sphere. There are also a number of rules that are inherent with physical puzzles. Objects cannot slide through one another, solid objects do not bend excessively, none of the pieces will disappear or reappear. These rules are not present in mathematics or in the digital world but they are so obvious in physical puzzles that they need not be stated with the puzzle.
There are also a number of rules understood in puzzle culture. No outside tools, no excessive force. These also usually go without saying unless you are giving your puzzles to complete newbies and the parts are somewhat fragile.
Back to the Venn puzzle. 1 it must become a sphere, 2 the parts are the same. These two facts are very clear immediately. The solver usually tries to place two parts together and then add the third. They are rebuffed and they sit back to try to understand what just happened. The third piece doesn’t fit. But how can that be? We can see the hole it fits into but it doesn’t fit. Sometimes they try entering from different directions using the infinite degrees fo freedom this puzzle allows. Many times they restart only to wind up at the exact same dead end. Rinse and repeat until they eventually find that all parts must be assembled at the same time.
Here enters the last bit of fun in the Venn puzzle: dexterity. It’s hard to hold all three parts at the same time and line them up perfectly. The weight of the metal and the rounded exterior refuse to sit on the table for you and help. Eventually you get the corners in and the puzzles slides itself together. You feel a moment when the puzzle accepts your answer and takes over the rest with a physical animation almost outside of your control. Dexterity is unintellectual. But that doesn’t mean it is a challenge undeserving of puzzles. Dexterity is the reason physical puzzles feel like a more substantial respite from daily life while doing a crossword on your phone doesn’t give you the same escape. We live inside screens and using your hands to hold something with weight is a pleasure people miss. See sign the real textures in the metal and feeling it’s coldness are luxuries for the modern man that commonplace in the past. This is the reason for puzzles.
The Venn and jack puzzle have a few breadcrumbs. Mostly, it’s that they are assembly puzzles. You see them disassemble with ease and then the challenge is undoing that. You’ve walked into the labyrinth calmly and now you must find your way back out. The other breadcrumb is the form. Curved cuts must mate with curved cuts. Flat with flat. The exterior of the Venn puzzle is a sphere. All twelve tips of the jack puzzle are exposed when it is completed. These are small bits of information delivered entirely without fanfare. But you know they must be true if you think about it for a minute.
Take apart puzzles are inherently secretive. The method of how to open them is obscured and you must try to find clues to open them. Usually those clues are in the form of feeling a movement, listening to a click or trying a few well known “secret” solutions. I don’t think these puzzles are worth making. Groping around in the dark and trying to find clues is not fun (for me). It is more fun to be confident that you have all the information you need and now it’s time to think. Additionally, I think it’s best to be given clues visually because visual clues can be trusted more thoroughly than an audible or tactile clue. Thus, visual clues can be used to create a mental model in a more complex way than audible or tactile clues.
Bread crumbs are best when they are inherent in the puzzle forms. You can also make breadcrumbs by adding visibility. Visual, aural, or mass. all kinds of things can be used to give clues. Lastly, clues can be placed on the exterior for the express purpose of telling how to unlock it. This is the worst kind of clue because it uses language, literal or symbolic. It takes you out of the form and reminds you this object is made and not found.
A puzzle should not feel like picking a lock or worse guessing a garage door combination. To pick a lock you already know the goal: to align all pins at the same time. If there is nothing else you’ve only created a poorly engineered lock. A puzzle has to have a curve ball in it. It has to surprise you somehow. Make you think you know the rules until you don’t. Make you think you’ve done it until you realize you haven’t. It must have a story built in.
A puzzle can be judged on its
Tactility - greatly improves replayability. Even if the solution is easy, tactility gives it the right to stick around.
Inventiveness - a new kind of puzzle always earns its right to exist. Combining that with a solid score in other categories is very good.
Surprise - some form of turn. Make the player make an assumption then take it away. Lead them down a path and make them confident then reveal a new depth. That extra depth multiplies the value of the solve. A second story to replace the first is a beautiful way to draw them in and inspire. A third story to replace the first two may annoy.
Beautiful - True fact: puzzles mostly just sit on the shelf. If they look good there they are more valuable because if they are out in the open and visible to guests, they will be played.
Goal - a puzzle always has a goal. Can you take me apart? Can you arrange my parts? Can you make a symmetric shape? It is best if that goal is conveyed implicitly rather than printed on instructions. When the goal is weird (not take apart put together) then the pets and forms should make every effort to convey the goal without words.
Hook - draws you in with something. Many puzzles have no hook and are only played by puzzle nerds. If you want to make some money on puzzles you need to draw in a larger group. Try using a hook. Getting someone to touch it is almost all it takes. Reveal a moving part early on.
Branches - many puzzles have branches. Like a maze it splits one way and another and the solution is making the right decision over and over. The best puzzles have secret doors and avenues that were there all along but you didn’t see them. It can’t always be feeling around in the dark. This is why the best goal is left unwritten. Unwritten goals allow you to change the rules as you go without making the player feel cheated. Since the rules are only implied they understand that they don’t get it yet. I think it’s this sense of exploring the unknown that gives puzzles their wonder. Not quite knowing the rules and discovering them then discovering you were wrong and NOW you know the truth.
Inherent solvability. - materials
3D printed parts are terrible for this because they can be printed in place. Cast metal is good because it is nearly proof of each units singular ness. Concrete or stone is the same as cast metal. Fasteners quickly break the impression of inherent doability because I don’t know when that screw was put in place. And I need verbal instructions not to undo the screw. Plastic offers some inherent proof of the puzzles makeup but not as definitely because of its weld ability, flexibility and o really cheapness. Better to use a classier material as some justification that we wouldn’t try to trick you.
DOF
To crack a four digit key pad, you have 10000 codes to guess. 0000, 0001, 0002 … 9999. There are ten degrees of freedom on the digit and four digits. So 10^4 makes 10000. Without any clues to find the answer, this is a terrible task and very few people derive any pleasure from this experience.
Another key pad with only a 1 and a 0, simplifies things. If you know the code is only two digits long. So you can try all 4 options. 00 01 10 11. Solved. But little fun was had.
The Venn puzzle has only one degree of freedom to solve. It is a ‘fall apart puzzle’. Putting it back together it has infinite degrees of freedom but it limits that infinity (Venn by clearly having only three identical pieces means it essentially has infinity/3 degrees of freedom and the jack puzzle begins to force you down a path with its notches. Causing you to place them at 90 degrees to one another and slowly understanding the orientation.)
A sequential movement puzzle may only have one degree of freedom throughout it’s disassembly. Moving one piece at a time and then only one other piece is available to move next. Sometimes simultaneous movement maybe necessary which is not quite one degree of freedom.
Sudoku has 81 squares and 10 digits. The number of possibilities is a 10^81. But there are rules. No digits repeat in a row. No digits repeat in a 3 by 3 box. This limits the number of solutions to “only” to about 10^9 or 10 billion. We knocked out 99.999…9999% (seventy one 9s) of our original options with two simple, elegant rules.
A blank 9 by 9 grid of squares and two rules doesn’t create a fun challenge for most people. It’s actually very easy to do and absolutely no fun. Thats where bread crumbs come in. Every sudoku has a handful of digits in specific positions which guide the player to a solution. A few clues gives you a trajectory. A cascading series of small puzzles and solutions. One leading to another that give you a sense of accomplishment. Each small event is proof that time is passing while a singular monumental task deters any attempt to peel away the layers.
A puzzle is like a joke. It is impossible to fully define and understand as an external theoretical thing. But I think the analogy is helpful to understand that a puzzle is experienced by someone who doesn’t know the punchline. The lines before that must lead to it without giving it away.
Part of the beauty of mechanical puzzles is that they give you the same challenge as sudoku or crosswords but they don’t use symbols. No letters or digits. No external rules defined by words. That also
Like a joke in that the punchline by itself is kind of crap. Anyone can slide some pieces of metal together and say that’s the punchline. But a good joke leads you into it and then reveals itself. Comedians iterate more than any other creative. Any of them will tell you that they write a joke and it bombs. And they change it and it gets little laugh. They change it and change it until it’s great. Puzzles are the same. You can’t design it and keep designing until it’s good. You have to make it and try it and get fresh people to try it. Only then can you learn what works and what doesn’t. If a clue you gave us a good clue or too obvious. If the reveal is impressive or more like a jigsaw puzzle and lands flat.