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Worlds of Design: The Tyranny and Freedom of Player Agency

“Player agency” refers to the player being allowed by the game to do things in the game that have real consequences to the long-term course, and especially the result, not just for succeeding or failing. Some campaigns offer a lot, some only a little. Are players just following the script or do they have the opportunity to make decisions that cause their long-term results to be significantly different from another player’s?

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Picture courtesy of Pixabay.

I play games to specifically be an agent in the universe that does effect things. I watch netflix or read books to be an observer. I have to be in control of something.” Kaze Kai

The subject of player agency is a controversial topic in game design. We have “rules emergent” games which are “open” versus “progressive” games which are “closed”; or “sandbox” which is open versus “linear” which is closed. The first of each pair can also lead to strong player agency, the second almost never does. I'll add a third one: games, which are open, versus puzzles, which are closed, because in a pure puzzle you must follow the solution devised by the designer.

Player agency is important because many long-time gamers want control, want agency, yet many game and adventure makers want control themselves, and take it away from players. It’s the difference between, say, Candyland or Snakes & Ladders(no agency), and games like Diplomacy and Carcassonne. For adults, Tic-Tac-Toe has no practical player agency, as it is a puzzle that is always a draw when well-played.

When a GM runs a particular adventure for several groups, do the results tend to be the same for each group (beyond whether they succeed or fail), or do the results tend to be “all over the map”? If the former, it leans toward being a linear adventure, while if the latter, it’s more “sandbox.”

Books can help us understand this. Most novels have no “reader agency”; the reader is “just along for the ride." Films offer no viewer agency. On the other hand, “Fighting Fantasy” and similar “you are there and you make the choices” books, where you choose what to do next from among about three possible actions, gives the reader-player agency over the short term. (Dark Mirror’s Bandersnatch is a more recent example.) Though in the end, if the player succeeds, there may be only one kind of success. Video games usually let players influence the small-scale/short term stuff a little, but not the large scale.

In between broad player agency and no player agency can be found games with false impressions of player agency, which you can recreate in an RPG adventure just as well as in a standalone game. The Walking Dead video game was often praised for the choices the player had to make, but in the end it all comes out the same way no matter what the player does (see this reference for a diagram of all the choices). Mass Effect is another game highly touted for player choices that ended up in the same place despite their decisions.

Full player agency creates story branches that don’t come back to the same place; the player’s choices just continues to branch. The reason this is rare in video games is because more choices and branches means more development, which costs money. In tabletop RPGs, a good GM can provide whatever branching is needed, on the fly if necessary.

The one place where player agency is seldom in question is in competitive tabletop games, especially wargames. Even there, many of the old SPI games more or less forced players to follow history. And many Eurostyle “games” are more puzzles than games, hence players must follow one of several solutions (“paths to victory”).

Why would a designer not provide Agency? I don’t understand it emotionally myself, but I can understand it intellectually. Some game designers are frustrated storytellers (or puzzle-makers) who have chosen not to use traditional forms such as novels, film, plays, oral storytelling. They want to provide “experiences." But in order to do so in a medium not as suited for it, they must introduce limitations on players in order to retain control of the narrative.

Only games (as opposed to novels or films) offer the choice of agency or not. There’s nothing wrong with a “lack of agency”, if that’s what players expect – as in a typical film or novel. I am not saying it’s wrong, just that most highly experienced game players don’t like lack of player agency.

I recommend you ask yourself a general question: “am I imposing my ideas and notions on the game, or allowing the players to use theirs?” Part of that answer is relevant to player agency. What you want the answer to be is up to you.

This article was contributed by Lewis Pulsipher (lewpuls) as part of EN World's Columnist (ENWC) program. Lew was Contributing Editor to Dragon, White Dwarf, and Space Gamer magazines and contributed monsters to TSR's original Fiend Folio, including the Elemental Princes of Evil, denzelian, and poltergeist. You can follow Lew on his web site and his Udemy course landing page. If you enjoy the daily news and articles from EN World, please consider contributing to our Patreon!
 
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Lewis Pulsipher

Comments

Jay Verkuilen

Dogsbody Waghalter
I mean the sand box style is often held up as the holy grail, and some players love it, totally taking control of the narrative, and I find that is cool, less work for me, then I can sit back and be the neutral adjudicator of the universe. Some players just want to set the world on fire, they don't care where or how. That working with the two directions between action and narrative play, sometimes takes real effort.
The sandbox style is indeed held up as the holy grail by a certain type of grognard, but for many groups I'm not sure it is.

I've run a much more sandbox-y game and played in several. I like it a lot but it really needs to be run by a DM and players for whom it is a good match. One of my favorite kinds of game is the relatively small group of players with an ensemble cast of PCs and henchfolk, which I think is actually very consistent with the way the Lake Geneva crowd actually ran. In a game like this there isn't necessarily a ton of level advancement and characters are often picked to suit the part of the sandbox that's being explored. This doesn't work too well for folks who want a zero-to-hero with one PC experience, though.

All that aside, I've played or run for groups where the players just seem to flail around without some clear external pull provided by the DM. This doesn't make these groups bad, but they clearly need a lot more structuring.

Likewise, there are DMs who don't do sandboxes well. I have played with two who are good in many ways but just can't or won't do a sandbox game. One is capable of writing a really excellent narrative and comes up with some very cool stories and concepts, but it's been a real stretch for him to allow more player agency, though he's definitely gotten better over time. Another one is capable of running a pretty good game when he's got a solid script which he often adapts quite a bit, but isn't so great at coming up with his own stuff. Even when he's running a game that's been written by someone else, if it's too sandboxed, he doesn't seem to be able to pull the players along. He shines much brighter running a more narratively structured game.

What I often find problematic is that there seems to be some kind of total divide between sandbox and narrative railroad when, in fact, there's a lot of room between the two. It's one of the reasons I think games like Knights of the Old Republic were so well done. They combined an overall narrative structure but allow the player a lot of leeway in the mid-game to choose the pathway through it by giving the player the choice of four different planets to explore. There were some interactions between the different planets, too, and often reasons to return to ones you'd left. This same structure appeared in a lot of subsequent CPRGs, for good reason. Of course the hardcore "sanbox rulez!" folks don't like this but IMO it combines the best of both worlds and is very much how I tend to approach games now, although, again, it depends on the group of players.
 
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S'mon

Legend
As a capstone, A re-definition of the word "puzzle" to be "puzzle that has only one solution" to make the point a tautology -- "puzzles that only have one solution are closed because in a puzzle that has only one solution you must follow the solution devised by the designer"
I have never heard of a puzzle with multiple solutions. Do you have an example? All I can think of is a Gordian Knot puzzle with an intended solution that allows for out of context unintended solutions.

Personally I really hate puzzles in my RPGs, especially when the GM disallows unintended solutions like smashing the door to escape the trap room.
 

S'mon

Legend
The sandbox style is indeed held up as the holy grail by a certain type of grognard, but for many groups I'm not sure it is.
Yes - I like sandboxing and with proactive players it's great, but I have also seen players respond with blank stares to "So what do you want to do?" - some players want and need a clear path in front of them, while others love bouncing around instigating new stuff. The latter can be disruptive in a more linear campaign.

IME the best approach tends to be something like Skyrim with clear paths within an open world. I'm running a bunch of great Rich Baker modules & settings and seeing the different approaches along the closed-open scale that all work well:

Red Hand of Doom is a structured narrative that gives a lot of choice.
Princes of the Apocalypse is a good example of paths to follow within an open world.
Primeval Thule is an entire open campaign setting with sandbox plot-nuggets everywhere you look.

A hardcore sandboxer won't like Red Hand of Doom, and a rail-follower won't like Primeval Thule, but these are all good approaches IMO.
 

dragoner

Dying in Chargen
What I often find problematic is that there seems to be some kind of total divide between sandbox and narrative railroad when, in fact, there's a lot of room between the two.
I think there is too, I look at it as more of a number line, and it moves on the line. Games start out open, and then the players decide to follow a course of action, essentially they have signed on to railroading themselves. Sandboxes also take a decent amount of work for the setting to have depth, which works against new GM's.
 

Jay Verkuilen

Dogsbody Waghalter
I think there is too, I look at it as more of a number line, and it moves on the line. Games start out open, and then the players decide to follow a course of action, essentially they have signed on to railroading themselves. Sandboxes also take a decent amount of work for the setting to have depth, which works against new GM's.
Sandboxing certainly requires quite a bit of the DM and players both. If they don't have it---be it lack of experience, inclination, or time---it's not a good match for that particular group.

IMO a lot of the bad rep of railroading comes from the propensity to think that there "needs" to be a levels 1-20 overarching campaign. I don't think that's really true. I wasn't back in Ye Olde Dayes, where there were often series of modules but they lasted for a level range, say 1-5 or 6-10 with the assumption that the campaign would move elsewhere when this was done. You got some nice narrative structure but didn't feel the burden of trying to keep things interesting for more than about three levels.
 

dragoner

Dying in Chargen
IMO a lot of the bad rep of railroading comes from the propensity to think that there "needs" to be a levels 1-20 overarching campaign. I don't think that's really true. I wasn't back in Ye Olde Dayes, where there were often series of modules but they lasted for a level range, say 1-5 or 6-10 with the assumption that the campaign would move elsewhere when this was done. You got some nice narrative structure but didn't feel the burden of trying to keep things interesting for more than about three levels.
I remember those old modules, some of the best times were playing GDQ, I was 13 and the DM was 17, I used to think him so cool and mature, but now ... we were just kids. lol It was pretty sandboxy, had to be, the old modules were so thin, and there was an arc, by Queen of Demonweb Pits, we were stupid powerful. My Character was killed by a Succubus and Vampire, just a Human Paladin, then I started another, a Drow Assassin; there was a lot of hilarity in that game too. The DM depended on the random tables, and we wandered around killing stuff, like an Illithid outpost, (Mindflayers) it was all so new and weird. Most of the other modules we played singly until enough of our own lore had been built, that we just did whatever, usually rolling up new characters, AD&D had a sweet spot, from about Level 3-9.

I think one thing that newer modules did, was that in seeing some linear path was good, decided that more would be better. Except in that the more of material, it sort of locked the players in to doing it the one way. I feel expectations were about the same, though AD&D had some silly long combats, we had a tendency to streamline the rules. In time, both rules sets and adventures had morphed into the thick high crunch tomes that are probably what got the bad reputation. Then again there is no real accounting for plainly bad material, even then bad material in the hands of a good GM could be fun. That really seems to be the crux in my mind, the being a good GM part, it is where the game resolves.
 

Campbell

Relaxed Intensity
There's a severe category error in this thread. There are more ways to play than hex crawls and railroads. The idea that one needs to railroads in order to have a game that is compelling on a narrative level is misguided. One simple way is to require players play characters with hopes, dreams, and goals and then present honest defined antagonism and playing to find out what happens.
 

Lylandra

Explorer
I have never heard of a puzzle with multiple solutions. Do you have an example? All I can think of is a Gordian Knot puzzle with an intended solution that allows for out of context unintended solutions.

Personally I really hate puzzles in my RPGs, especially when the GM disallows unintended solutions like smashing the door to escape the trap room.
Trust me, there are puzzles that have more than one solution. But they tend to be more complex. One of the best ones that comes to my mind is the "rainbow room challenge" from the Zeitgeist adventure path where you'd have combat in a color puzzle room and a plethora of choices. Oh and it comes with a "hardmode" variant.

But if you got a GM whose only use of puzzles is to maybe combine Sudoku and riddles... well, you're screwed. And yes, I really dislike the "but it isn't intended to work that way" sentence. Ugh.
 

Jay Verkuilen

Dogsbody Waghalter
I remember those old modules, some of the best times were playing GDQ, I was 13 and the DM was 17, I used to think him so cool and mature, but now ... we were just kids.
For a while there WotC marketing hype was hardcore about creating that "shared experience" and whatnot. It was kind of sad in a lot of ways---trying to catch lightning in a bottle again, I guess, or the numerous attempts at "remaking" Woodstock.

lol It was pretty sandboxy, had to be, the old modules were so thin,
You could run them as narrative plot if the DM wanted to put that kind of structure in. In many respects, the modules were assembled with a vague assumption you'd be doing that, but you're 100% right they didn't put in a lot of RP weight. That was the job of the DM.

AD&D had a sweet spot, from about Level 3-9.
IMO that's not really changed a ton, though maybe it's broadened out a bit. The sweet spot of D&D is, in my view, somewhere between about 4th (strong enough to take it, but still with a lot of room to grow) and 13th or so levels (really starting to get insanely potent).

I think one thing that newer modules did, was that in seeing some linear path was good, decided that more would be better. Except in that the more of material, it sort of locked the players in to doing it the one way.
Exactly.

In time, both rules sets and adventures had morphed into the thick high crunch tomes that are probably what got the bad reputation. Then again there is no real accounting for plainly bad material,
Sure, a lot of that is not hard to understand either. Many of the rules that got added were simply put in place to make things more fair, clearer, give players more opportunities, and so on. They weren't done with bad motives in mind, but things got out of hand. 4E was unquestionably the biggest push in this direction and, for some players and DMs, it was great. They loved having it all laid out in front of them and how solidly balanced the system was. (I didn't---I found 4E fun to play in some respects, though still mostly in the "sweet spot", but really disliked DMing it.)

even then bad material in the hands of a good GM could be fun. That really seems to be the crux in my mind, the being a good GM part, it is where the game resolves.
Unquestionably, but game designers can't write games under the assumption that you first need to find a kicking DM.
 

Jay Verkuilen

Dogsbody Waghalter
Yes - I like sandboxing and with proactive players it's great, but I have also seen players respond with blank stares to "So what do you want to do?" - some players want and need a clear path in front of them, while others love bouncing around instigating new stuff. The latter can be disruptive in a more linear campaign.
This is a very important point. A group that has some plot followers and some sandboxers is going to have a recipe for discontent as the poor DM tries to keep both happy.

IME the best approach tends to be something like Skyrim with clear paths within an open world.
That's pretty sandboxy!
 

Jay Verkuilen

Dogsbody Waghalter
Trust me, there are puzzles that have more than one solution. But they tend to be more complex.
Yes, definitely. I think the skill challenge is a better replacement for a straight out puzzle, or else the DM should allow for different methods of some sort to achieve the goal, with the puzzle being one obstacle. The puzzle could be one way, but there may be others, such as a sequence of athletic feats, or a complicated lock. That gives players agency but still challenges them. It can also be for interesting RP if the character trying to solve things by wits gets in a race with the one trying to solve it by muscle. That's totally genre-appropriate in a more swords and sorcery style.
 

dragoner

Dying in Chargen
Many of the rules that got added were simply put in place to make things more fair, clearer, give players more opportunities, and so on. They weren't done with bad motives in mind, but things got out of hand.
It became having a rule for everything, simultaneously promoting a rules lawyer type of play, and making it more difficult for the GM to get to the point of rules mastery; plus a creep towards so many rules, that some didn't work as well, or were not play tested. Then there was the one player who wanted to use the additional rules, making the GM have to put in more work for that one player, often without much interest from the rest of the group, for example Psionics from AD&D or Cybernetics from Traveller.


Unquestionably, but game designers can't write games under the assumption that you first need to find a kicking DM.
The admonition for writers is to write for yourself, write what you know, it is true for games too. Some writers do capture lightning in a bottle, that's why they get the following they do.
 

Malrex

Explorer
Yes - I like sandboxing and with proactive players it's great, but I have also seen players respond with blank stares to "So what do you want to do?" - some players want and need a clear path in front of them, while others love bouncing around instigating new stuff. The latter can be disruptive in a more linear campaign.
I think that's fixed with an array of adventure hooks--some can be a clear path, some can just be vague rumors. But I agree, it depends on what sort of campaign is being run.
 

Malrex

Explorer
There's a severe category error in this thread. There are more ways to play than hex crawls and railroads. The idea that one needs to railroads in order to have a game that is compelling on a narrative level is misguided. One simple way is to require players play characters with hopes, dreams, and goals and then present honest defined antagonism and playing to find out what happens.
I think there is a lot of truth to this. However, I would never 'require' a player to do anything--I would read the player during play and if the character has goals or hidden agendas, then I add wood to that creative fire. This is how my group plays...it's great! Sometimes it can lead to inner party turmoil as goals conflict--but this usually takes a very looooong time to come to a head (a year or 2 in our case).

I was also going to add about the article...there is also the tool of the 'timeline'. In a way, a good timeline can almost combine a linear and sandbox playstyle. Key word--almost. The timeline MUST be loose enough to allow character agency--i.e. they can ignore it or partake in it if they want. The actions of the characters can change the timeline to slow things down or speed things up, or the timeline provides an ever changing environment that characters make choices to adapt. For linear, the timeline can cover that overarching plot and be used as tool to get a party 'back on track' of the story----I'm guessing about this as I'm a sandbox DM, but would assume it would work.
 

Jay Verkuilen

Dogsbody Waghalter
I think that's fixed with an array of adventure hooks--some can be a clear path, some can just be vague rumors. But I agree, it depends on what sort of campaign is being run.
For some players more choices that "left or right down the hallway" is about as much as they can/want to handle. Three or four plot hooks is too much for them. I've seen choice cycling, where they start looking at option A, then B, then C, then back to A. This can happen for a variety of reasons, not wanting to be the guy/gal who makes the choice and offends the other players being one, but there are others. For a group like this, choices need to be pretty clear and laid out, for instance, A -> B or C (choose which order to do) -> D.

I've come to make use of this kind of structure in adventure designs a lot. I'll start in one area, say A, with the eventual goal to get to a particular other area, say G, but give a choice among different paths. It's kind of a "subway map" style dungeon.

A: Starting point
B: Plant monsters in a chasm
C: Ghouls
D: Troll
E: Phase spiders
F: Attack by soldiers of their main adversaries
G: Goal

Valid paths:

A -> C -> F -> G
A -> D -> F -> G
A -> B -> E -> G
etc.

I just chart these out on a piece of paper and budget XP accordingly so they're suitably beat up by the time they get to G, though it depends on the choices they made. This allows for player agency, but still gives me the DM enough structure to be well-prepared. I think it also helps to let exploration- or skill-oriented characters really benefit the group by letting the PCs make good choices or even bypass some of the encounters. Too often in a very narratively structured game they don't really get a chance to be useful.

Of course, the same kind of structure can be used for adventure design. Yeah, the total hex crawl or else grognard will hate this as still not providing enough player agency, but at some point I just stop listening to them.
 

Jay Verkuilen

Dogsbody Waghalter
It became having a rule for everything, simultaneously promoting a rules lawyer type of play, and making it more difficult for the GM to get to the point of rules mastery; plus a creep towards so many rules, that some didn't work as well, or were not play tested.
Yes, and of course game companies do need things to sell. I totally understand the desire to provide a very comprehensive set of rules, but, like anything else, it can go too far and become abused. Not enough rules? It's left to the DM and DMs vary. Too many rules? The rules lawyers and power gamers have a field day and everyone else just gets bored.


The admonition for writers is to write for yourself, write what you know, it is true for games too. Some writers do capture lightning in a bottle, that's why they get the following they do.
I was referring to the fact that there's so much nostalgia for modules like GDQ or the like and WotC seems to want to try to capture that lightning over and over. Many of those modules really aren't that good. It's like listening to album tracks by classic bands. A lot of those songs weren't that great and the recordings were not good either, but in the context of the times they were important.
 
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There's a severe category error in this thread. There are more ways to play than hex crawls and railroads. The idea that one needs to railroads in order to have a game that is compelling on a narrative level is misguided. One simple way is to require players play characters with hopes, dreams, and goals and then present honest defined antagonism and playing to find out what happens.
If I had a dollar for every time I've posted this on these boards . . .

Here's a post I made about this back in 2014:


Lewis Pulsipher was a prominent contributor to White Dwarf in its early days. The following quotes are from his article in an early number of White Dwarf (my copy is in Best of White Dwarf vol 1, 1980):

D&D players can be divided into two groups, those who want to play the game as a game and those who want to play it as a fantasy novel . . . The escapists can be divided into those who prefer to be told a story by the referee, in effect, with themselves as protagonists, and those who like a silly, totally unbelievable game. . . In California, for example, this leads to referees who make up more than half of what happens, what is encountered and so on, as the game progresses rather than doing it beforehand. . . . [T]he player is a passive receptor, with little control over what happens. . .

Gary Gygax has made it clear that D&D is a wargame, though the majority of players do not use it as such. . .

The referee [in a skill campaign] must think of himself as a friendly computer with discretion. Referee interference in the game must be reduced as much as possible . . . Effectively, this means that the referee should not make up anything important after an adventure has begun. He should only operate monsters encountered according to logic and, where necessary, dice rolls. . . . Occasionally an adventure will be dull, because players take the wrong turns or check the wrong rooms, while others may be 'milk runs' because the players are lucky. Referees must resist the temptation to manipulate the players by changing the situation. Every time the referee manipulates the game on the basis of his omniscience, he reduces the element of skill. . .

The referee who, for example, schemes to take a magic item away from a player is incompetent. If the player doesn't deserve the item he shouldn't have obtained it in the first place. Don't lie to the players when speaking as referee. If players can't believe what the referee tells them they are case adrift without hope. . . .​

I tried to implement this advice in my early GMing (around 1984). I was not very good at it - I'm not a particular effective "computer with discretion", and my players didn't (and don't) like the occasional dull adventure. But I do like the advice about not manipulating the players. It was around 1986, with original Oriental Adventures, that I started to discover a way of GMing in which the GM would make stuff up on the spot, while still allowing players the scope to make choices which are genuine in their consequences, thereby avoiding the railroading that Pulsipher warns against. (More than 15 years later I discovered that this approach to GMing had been refined and theorised by Ron Edwards and others at The Forge.)

I don't hold it against @lewpuls that his framework for thinking about RPGing doesn't seem to have developed a great deal since the early 1980s. He know what he knows and generally writes about it well.

But as a RPG community I think it is worthwhile to pay at least some attention to developments in RPG design and play over the past 30-odd years. It's not as if the sort of RPGing you (Campbell) have posted about is particularly esoteric.
 

aramis erak

Explorer
This is a misunderstanding of agency:

"The wider the field of choices, the harder to make those choices it becomes for the players. The rule system itself is a constraint on both player and GM agency... and it works because it reduces near-infinite choices to a manageable few."

Agency isn't about number of choices, it's about the ability of players to influence and change the outcome. That can be achieved without offering players myriad choices. But the choices must be important (in so many games, they're not, they're choices for the sake of choice).
At the level Mr. Pulsifer is talking, and that I' pointing out he's ignoring, the difference between agency and choice of action is moot.
The range of meaningful actions must be reduced to a level where the GM can handle it; the ability of players to affect the story to a range of meaningful outcomes also is reduced, but the value of that is, in fact more playability, in that the GM has less to think of and has scaffolding upon which to hang, the actions of the players and judge their impact.

Agency and player choice, at that level, are still a unified function. The feel of agency is enhanced by prior expectation of probable success, and then that success having tangible in-fiction consequences.

When one gets down to the play level, the reduced agency implied by acceptance that the nature of who can define what and who can refute what becomes separated by the question, "Does this roll really matter to the fiction?" - and, largely, they do. The restriction on action set is also a restriction on agency, but it'a also a restriction on player ability to go afield.

If Player A has total agency, no one else has any. If all players have equal agency, none have total agency. While I hesitate to describe group total agency as a zero-sum-game situation, in many ways it is.

If A isn't allowed to make meaningful changes to B, then A's agency is limited. If A is required to use mechanics to affect meaningful changes to B, that's still restricting A's agency... but it does so in a manner that makes it potentially more practical.

Let's look at a non-agency rule by comparison... In Star Trek Adventures, all stress damage is recovered at end of scene. Unless and until a wound is inflicted, the character hit has no lasting change. Stress damage is a non-agency tool for affecting a sum of actions that creates a lasting impairment (at least until getting to sickbay or a doc with a surgical kit)... that impairment is proof of agency, in every way that stress isn't. The decision to shoot in that system also includes options to spend metacurrency to take the shot that hits and push the damage up to wounding... (adding 5 momentum almost always guarantees a wound, and thus imposing a wound-type disadvantage, which has lasting story and mechanical impact.) the rules restricting the "I shoot and kill" level of agency result in a less certain outcome, invoking the game aspect, and making the choices fewer, but thus easier to identify which choices are meaningful, and which are more/less likely to succeed.

Perceived agency versus actual story agency - yes, they are different. But they're so conflated that restricting one axiomatically restricts the other, but also increases the ease of use.
 

aramis erak

Explorer
I have never heard of a puzzle with multiple solutions. Do you have an example? All I can think of is a Gordian Knot puzzle with an intended solution that allows for out of context unintended solutions.
the simplest case to exemplify: Mazes. We think of mazes as a singe solution. But a maze can have two ways through. Or even three.

riddles often have multiple correct answers. EG: What has many teeth, but cannot eat? Comb, zipper, rake.

many "park the car" sliding item puzzles have 2 or more solutions; there is usually one optimum one and one or more longer ones.
 

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