Puzzles vs. puzzle games
originally written in 2021, but reposting as my old blog is gone
I've taken more of an interest recently in traditional pen and paper logic puzzles, and I think there are some interesting comparisons to be made with certain types of puzzle video games. The types of games I'm thinking about mainly here are games that usually take place on a grid or discrete space of some description and generally have only one, or at least a very limited set of possible solutions. Some examples include The Witness, Sokoban, Cosmic Express and more recently, A Monster's Expedition. I think these puzzle games are particularly interesting to think about in this context, because they look so similar to their pen-and-paper counterparts. However, what may be considered good design in one is considered the opposite for the other.
Bifurcation
Firstly, let's talk about bifurcation. The dictionary definition of the word 'bifurcate' is 'to split into two parts or branches', and in the context of puzzles it refers to splitting your solving path in to two. In real terms, imagine solving a sudoku puzzle. If you've ever attempted a sudoku puzzle before, you will inevitably have come to a point at which you have two possible candidates for a particular square. You might be hopelessly stuck at this point, unable to progress anywhere else on the grid using your usual techniques, and therefore may be tempted to guess which digit goes into the square. This is particularly tempting because even if you guess wrong and find a contradiction in your grid later on, assuming you've made no mistakes you can erase everything you did back to when you made your guess and be certain that the square must contain the candidate you didn't originally pick. This kind of logical reasoning is bifurcation, because it involves branching your solving path off in one direction, then branching in the other direction if that direction doesn't work out.
If a pen-and-paper logic puzzle requires bifurcation, that's generally seen as bad (or at the very least, inelegant) puzzle design. Why is that? Perhaps it's because guessing feels a bit like cheating - you don't get quite the same buzz from solving a puzzle through bifurcation as you do through pure clever logical steps. It feels like the 'brute force' approach to solving a puzzle. Perhaps it's also down to the format; it's not very pleasant to have to erase progress on paper, even with the lightest pencil marking and the most effective eraser. You'd need some kind of notation to even remember what to erase, unless your memory is (a lot) better than mine.
Now let's consider Sokoban. Sokoban takes place on a grid, and the aim is to move all the crates in the grid onto designated goal spaces. You can move crates by moving your character into them, but they stop if they hit a wall and you can't move over or through blocks. Sokoban is really the archetypical puzzle game in this particular genre - games like Stephen's Sausage Roll, A Monster's Expedition, and Baba Is You all trace their lineage back to Sokoban. Even in games of entirely different genres, one of the most common types of puzzle that might appear, love it or hate it, is the prolific block puzzle.
Now, imagine trying to play Sokoban without bifurcation. With every single move you made, you'd need to logically prove that that's the only possibility, just like you have to logically prove that a number goes in a particular cell before writing it in to a Sudoku grid (remember, we're strictly not bifurcating here). This might start off easy enough, especially if you find your character in a narrow corridor where there is only one obvious move to progress. However, I wager that in some situations you aren't going to be able to progress at all without thinking out at least 3 or 4 moves ahead, and at that point you may as well just be trying the actual moves out, right? Especially if you have an undo function!
Time travel
There are a few things to note here that distinguish the pen-and-paper puzzle from the digital one. Firstly, a digital game can remember your solving path for you, so if you reach a contradiction you can just back up a few steps effortlessly until you reach the point at which you had to make a guess at what to do next. That means that the format argument against bifurcation earlier on is easily mitigated. Secondly, there's very rarely facility in digital games for making notes, and it's difficult to even think of a way to make notes that makes sense. Sokoban has an additional hidden dimension over Sudoku, even though it's a 2D game - time. Blocks move around the grid and the state changes, but a sudoku puzzle is static. It's difficult to reason several moves ahead because the state changes with each move, almost like a game of Chess but more predicable.
I suppose that a note I might want to take in Sokoban is something along the lines of 'if the player character is here, and a crate is here, then logically they have to move here'. Doing that would allow you to logically determine the necessary events out-of-order, and then piece them together into a set of logical steps that arrives at the solution. This is what the act of note-taking in pen-and-paper puzzles gives you - the ability to piece together a solution from several solved segments. I think this is most apparent in loop-style logic puzzles - puzzles that require the solver to draw a path through a grid that satisfies some amount of constraints. You don't solve these puzzles by drawing the entire path from start to finish - you place pieces of the loop and then join them together later. This sort of approach to solving would feel unusual in a video game with a time dimension, which the majority of puzzle games have.
Not all puzzle games have a time dimension, but often they don't have the facility to make notes anyway. Cosmic Express is a game about laying down track for a train that will only navigate the track when you're done, so in reality it's actually very close to a loop logic puzzle. However, while you can lay down disconnected track segments, you can't make any other notes on the grid, for example noting grid squares that cannot logically contain track. Even The Witness, a (nonetheless brilliant) game about solving grid puzzles by drawing lines through grids with various constraints that must be met, won't allow you to draw segments of the line and then join them up at the end - you have to draw the line all in one go. This encourages the player to either play with the grid, trying out several solutions until you find the one that works, or think out the entire solution in their head before transferring it to the grid. Interestingly, many players (including myself) transferred some of the puzzles to pen and paper and solved them there before going back to the game to enter the solution!
A game is a toy
So far then, we've seen that bifurcation is often essential to solving puzzle games. In fact, it's actively encouraged by the lack of note-taking support, the inability to join together partial solutions, and the existence of undo/reset tools. So does this make them bad? In some ways. I can definitely remember several times feeling unsatisfied after finding a solution in a puzzle game because I felt like I stumbled across the answer without really understanding why it worked, and I believe that this is a consequence of the encouragement of solving by bifurcation. This happens even in other sorts of puzzle games - I'm sure anyone who's played a LucasArts adventure game will have been in a situation at some point where you have no idea what to do next, so you just try every object in your inventory on everything you can interact with. It's never a satisfying way to solve a puzzle, so why is it often actively encouraged?
I think the answer to this is that a game is simultaneously a puzzle and a toy. A toy is something that is fun to interact with, explore, and experiment with. While it is possible to derive joy from finding unexpected consequences of the starting puzzle rules if you're really clever, a toy can provide additional unexpected consequences of interaction that couldn't possibly be known from the outset. The moments that elevate a good puzzle game above everything else are the moments where you are playing around with it and then something unexpected and surprising happens. Stephen's Sausage Roll and A Monster's Expedition are filled with these moments, and that's for me what makes them a cut above the others. These moments are impossible without encouraging exploration and experimentation with the mechanics. Encouraging bifurcation is an unfortunate side effect of this, but not a reason to avoid it entirely.
The trick when building a puzzle game then is to provide the ability to experiment and play with the mechanics while trying its best not to lead the player towards a state where they are just exhaustively trying every possible set of moves until they reach a solution. I do think that allowing players better planning tools, maybe even the ability to work backwards from a goal state, could potentially give players more avenues to a solution before they are brute forcing the puzzle. Perhaps digital puzzle games should lean much more heavily towards being a toy than a logic puzzle; defining a goal state but allowing several different possible solutions. Games such as Infinifactory and Opus Magnum take this kind of approach, allowing much more freedom and creativity to the player than a traditional puzzle game by asking the player to engineer a solution rather than find a predetermined one. These games could be even better in my opinion by rewarding experimentation with more surprising consequences to actions, although perhaps they already do and I just didn't experiment enough!
Conclusion
The bottom line is, games have the advantage over pen-and-paper logic puzzles that they can also be a toy that rewards experimentation. Leaning into this is what makes them great, but if there is a small number of possible solutions then this can be a double-edged sword, leading players to solve puzzles using brute force. I think that sometimes perhaps we can draw from pen-and-paper puzzles to remedy this - providing tools for partial solving could be a good way to do so if it's possible for the particular game. On the other hand, making the game even more fun to experiment with by adding more surprising interactions or opening up the potential paths to a solution could counteract the frustration and tedium of brute forcing a solution when a player feels they have no other choice.