The foundation of the trinagon system is symmetry.

    Symmetry is too big a word to explore here.
    It is found in nature, in mathematics, in 'chaos' and what we perceive as 'order', to name a few.
    it is a principle that can give us insights into complex mechanics or processes, be they of the real (material, physical) or virtual (philosophical, social, mathematical ..) kind, without having to understand the exact workings of these.

    And really just like that, we can enjoy symmetry. We can derive hypercomplex theories about them, or we can just ..

    .. PLAY !

    Dodecahedron0This is the game to give you that opportunity !

    Trinagon is a permutation puzzle system.

    Not just one puzzle, but many many puzzles that can be played on multiple playgrounds with miriad configurations / layouts / designs / patterns.

    Symmetry has many many expressions :) Here you see another one.

    Now, as of december 2019. with all the polyhedra in the game, and many clearly not so easy puzzles, finding the symetries is increasingly difficult.
    Therefore, many of these puzzles have been designed using rogue play ! 
    Once you find the perfect symetry, or partial symetries you will be able to solve them with a fraction of the original moves !


    Curious minds are not character attributes per se.

    Everyone is curious (about something, some more this and others that ..), but the interest to see and explore is in any being, human or otherwise.
    So we all have it, a curious mind.

    Curious minds want to see, they want to understant what's going on out there, want to touch and feel and DO.
    A curious mind gains satisfaction from experiencing first hand, and puzzling over things.

    For a curious mind the question is the nourishment (the answer is not).
    The process of pondering and getting involved in a question, is the sunlight that makes them thrive and grow.

    Now we can feed our curiosity more than ever before. Information is only a few fingermovements away.

    Something else needs to happen though.

    Spoon feeding supposed knowledge, repeating and simply believing someone elses conclusions without reflection, does not let our curious minds thrive.
    It deadens them to deeper reflections, makes them sluggish and bored.
    Then answers become more important than questions, easy answers mostly.

    Answers that we do not arrive at ourselves, answers that we buy or copy or repeat from sources, which we put our trust in, mostly only because someone else named those sources 'experts' !?
    Right or wrong, doesn not make a difference. then. 


    “What would come, would come...and you would have to meet it, when it did.” -  J.K. Rowling, Harry Potter and the Goblet of Fire.

    Who would meet it ?
    Meet it with what ?

    Answers by themselves are arrogant, questions by themselves humble.

    There is a choice.



    Update 2023 : 
    A short description of recent developments : 

    • Permutations of each Puzzle : 
      Now five years after the first launch of the game a few rather simple algorithmns have been used to count for any puzzle configuration how the triangle can actually be situated on the playgrounds.
      This leads to a simple deduction of the possible permutations of each puzzle. The numbers are from as small as one thinks on the smaller polyhedra to extremely huge for the larger ones.
      There are still some cases where the algorithmn has to be adjusted slightly, because the maths are actually tricky.
      So far it is still not clear when rotational states on the playground can be linked, like the are for example on the rubiks cube. I.e. it is not possible to rotate a cornerpiece of the cube without also rotating another. The rotations of those pieces are linked, which reduces the total amount of permutations. 
      Trinagon has a few small puzzle configurations (e.g. on the octahedron, tetrahedron), where the algorithmns have been off. They needed adjusting by the (puzzle specific!) linking factor of 2,3, or 6.
      It is rather easy to see, when playing, where no adjustment needs to be made (no link), but it's not easy at all to prove where one would be needed. A good guess can be made, but this is not proof.
      To check if there is no link, all one needs to do is play to a state where only one triangle is rotated from a previous position.
      This problem was solved at least to a certain degree : 
      For the small puzzles it is possible to simply count the number of permutations. The algorithm do check is quite easily written, but the numbers do rather quickly get huge (-E24 on the Icosahedron) on bigger polyhedra, where this is not possible without help from googles servers. Fortunately there the different setups allow for single triangle rotation permutations. 

    • Gods Number for each puzzle : 
      Within all this also a solution-finding algorithmn has been checking for the best possible solutions to many puzzles. The algorithmn uses brute, but intelligent, force :).
      The results of the best solutions are in my opinion : Totally astounding.
      This information and the possibility to also load 'gods solution' are currently being worked into the game.
      Gods solutions will only become available once a puzzle was solved. There is a learning curve and a self-rewarding curve there : It's much easier to find a really short gods solution if you know how many moves it takes. Especially on the smallest polyhedra. Therefore, let the surprise come slowly.
      Watch out for the next update :)

    • Ad Brute Force : 
      With n rotators active there is a maximum of (2*n -1) new configurations ('states') with every move. So the number of states increases rather fast by the power of the number of moves.
      Depending on the basic setup, size and rotationtypes, many states will be repeating themselves, which leads to at best a 1/3rd less that need to be stored. But this comparison is exponentially more effort.
      So it is pretty clear why finding gods number is not a matter of minutes on the harder & bigger puzzles, but would take days or years to cumpute (on a regular PC). Obviously the search for a perfect solution goes from the start up, as well as from the solution downwards, and will meet in the middle.
      There are more than one gods soultions, but only one needs to be found. 


    Trinagon puzzles are mathematically (and otherwise) an uncharted area ! 
    ( .. because its new :) )

    For the rubiks cube many solving systems have been created. Simulations were run and a lot of thinking & processing power got spent on figuring out the rubiks cubes 'Gods Number', and many mathematical papers have been written on this one single puzzle (one resource here ..), and other permutation puzzles.
    There are many apps that will autosolve it for you, and there a numerous tutorials where you can learn to apply solving patterns for the rubiks cube.

    Trinagon, being completely new, has not even begun to be explored !! (by proper mathematicians at least)

    If you are a mathematician specialized in group theory (that being the mathematical branch for a permutatiuon puzzle like trinagon or the rubiks cube) or becoming a mathematician, trinagon should hopefully give you a few hours ?, weeks ? months of work. Depending on how deep you wish to dive into it. 

    Apart from the simple beauty of the maths involved, maybe the matematically inclined can help with something else.
    We would definitely be interested in finding two (or three) algorithms :

    • One that can check whether a proposed solution pattern can actually be achieved in play, without the actual solution needed.
      (A basic theory (more an idea) for this exists already. Still an occasional work in progress. 
    • One algorithm, to find the least number of moves needed to solve a puzzle !? (Gods Number). 
      The small number of moves needed, to solve puzzles has been a surprise to thier own creators already often enough. More than once ! a single puzzle can have astounding elegant solutions.
    • and well maybe therefore .. how about an algorithm that can solve any trinagon puzzle ?
      Maybe even do it better than a good player ? 

    If funds can be generated selling this game, they will also be used to challenge the mathemmatically inclined to find some answer and win something apart from fame in the process.
    Currently the free puzzles have not been solved ! (Except by lumpazy, of course)

    promo codes only for the first correct answer

    Other mathematically interesting features within trinagon may hopefully be a source of inspiration for many students (of all grades) :

    For example :

    • platonic bodies (and other polyhedra) and many of their properties.
      Even just visual representations, getting a feel for it and learning to 'see' 3D, will greatly improve the ability to work with polyhedric shapes, be it in art or in architecture (and maybe even both) or others.
    • general symetries in physics, chemistry and mathematics,
    • vector calculus
    • group theory. (Trinagon may even be a nice diversion for  some particle physicists, using their understanding of dihedral groups ?)
    • probability theory

    A few simple 'ponderances' :

    • How many different solution patterns (only the end positions) are there to a given colors-only puzzle ? 
      How does this change when you add facemarkers, and directions ?  

    • ... and can all of them be achieved / reached in normal play ? (mostly yes, if no. Why not.).
      How many can be ? (That's where it gets difficult). 

    • Why are there exactly 6 rotationtypes ?
      Or could there be more ?  (Yes there could be more ! but only if ... ) 

    • How many turns does it take to move a triangle from A to B, and how many shortest paths are there, depending on the playgound ?

    • In the Icosahedron. Do 2 triangles that are as far away as they can be from each other have the same spatial orientation ? Why (not) ? 
      Is there a connection to how many moves it takes to move from A to B ?

    • Is there a useful relationship between the distance of two triangles and the number of moves to bring one to the others place, or exchange them (in the hexagon) ?

    • And one out of the game : 
      In the tetrahedron puzzle. Use one flip-rotator (4,5 or 6) and 3 flat rotators.
      Is it possible have only one triangle face flipped ?
      Yes / No  &   Why ? :)   

      .. Are you sure :) ? I thought i was !

    • ... make up your own. There are enough questions for all and for everyone.

    After playing a while, a few good guesses might already answer some of these questions or give you an idea of how the get there.

    If you have come up with questions and maybe even followed through with some answers, you can sent them to us (for the resource center) or publish them in the forum.


    Trinagon first entered it's creators (my) imagination in 1988 or thereabouts.anno1988. Screenshot from using geogebra.

    At that time turbo pascal was the programming language we learned in school. There where none of those cool tools that make programming a breeze nowadays, but the base game could already run with VGA graphics (Anyone remember - 16 Colors at 640x480px).

    There were no animations programmed whatsoever. Instead the colors where interchanged. The game could be played using the keyboard. It really was more of a programming excercise, but i thought the concept was pretty cool and well.... New.
    The design was exactly like the base layout for the editor, 6 Rainbow colors around the hexagon grid with 9 triangles each.

    I played the game about 10 times.

    Then i decided, it was to easy to solve, and therefore sort of ... boring :/
    No further designs were tried or variations, and the idea discarded as too simple.

    Unfortunately the code (even though i do not have a turbo pascal compiler anymore) was not preserved throughout the years.



    Many many years later, in 2015 seeing some puzzle games around, and maybe playing 'monument valley' and other 3dimensional puzzles, the old idea came back in a very different shape :
    The same puzzle logic of adjacent interchangeable triangles, but wrapped around symetrical bodies (platonic ones especially).

    A very short search led to a wonderful program & learning tool : Geogebra, which allows for 3 dimensional vector calculations and graphical representations and is very easy to learn and apply. (The above picture was made using Geogebra) 
    With geogebra the basic 3D shapes could be understood and visualized, but it also became clear that the game idea would not necessarily make sense, and really the idea did not seem sooo great, as to make the effort of learning how to program it in nowadays programming environment.
    A short dabble into the unity editor made it clear that programming has become a very different kind of animal since 1988. Somehow much easier, with bigger concepts and structures in play that come out of the box, yet at the same time with a rather steep learning curve (unless you wanted to write Yet Another Monster Game).

    This year (2018) finally there seemed to be enough time and interest to learn something new. Unity and c#, and some blender modelling started a process that did turn out to be much more interesting than anticipated.

    The intention was really just the learning exercise, and the knowledge that, looking at the stupidity of some games out there, even an easy game can be an improvement. (Since then i have learned that it's not soooo easy after all).
    Not even 2 months later the game was already playable, but just like its original still lacked something, until ..

    the animations in 3D revealed a whole new world of possibilities !
    When triangles were allowed to have two distinguishable sides (faces), and then also a direction, things got interesting indeed.

    Suddenly a rather simple game evolved into a complex and diverse creation of many possibilities. Who would have thought of that !

    To discover this world was and is a pleasure and it's a decent amount of work to develop the details. The process is still going on. The mathematics (group theory) on that example will be a pleasure to see, as will be the creation of many puzzles, and solving them and getting a better idea of the principles.
    The actual idea from 2015, wrapping a 2D game over 3D bodies, was under construction throughout 2019 and has been published in Nov. 2019.



    You may have realized that there are no ads on this site, nor does any version of Trinagon show you advertising when you are puzzling.
    Also any ads between puzzles or to earn points have been banned for ever. It just doesn't fit the game.

    This game intends to create clarity of thought & focus. Instead of mindless entertainment, we wish for mindful entertainment. Our preference for anyone playing is to discover the game, and their ingeniousness solving the puzzles. Sometimes discovering may need a bit more time. 

    The real value in all this : The joy of applying your intelligence instead of mindless tapping on the screen. The joy of understanding and achieving something. Not even necessarily proving yourself against someone else, but proving yourself to you. There is a value, that cannot be paid or bought !

    You can still simply buy the game for really almost a penny !

    We think this game is truly great, and we believe that we're selling it for much less than it can be worth to you.
    We believe that your gain from solving these puzzles by far outshines the money spent, and it's never paid in money anyways.
    It's value comes from effort and time and thinking. It is a value you create, and we thank you for it.
    Enjoy creating :)

    Trinagon was born in a different time (like 1988 -another millenium / century). 

    A time where the average attention span used to be more than 5 seconds.
    A time when immediate gratification was not the norm. Rewards had to be earned. (They actually still do - at least real rewards.)

    Games, regardless if in real life or in virtual, needed an effort to learn. Nothing ever was self explanatory, and you were lucky, if there were any explanations at all.
    Reading a book, getting deep into a story, and exploring with an active mind was necessary. Having to suss things and life out mostly on your own.
    Whenever we got a new game on our Commodore 64, we always quickly went through all the keys on the keyboard to find out if they had any functions :)

    Now compare that to the games kids aged 5 to 11 get delivered.
    The 'game', often nothing more than a simple-minded tapping or shifting / moving, likely will be boring to a dolphin within a few minutes. (because dolphins are sharp :))
    Yet, they offer easy entertainment and quick gratifications.  (Hurraai, you just tapped the red button. Whao ! Boooom. Level 58 solved.) Without those the 'players' will turn away within the first seconds. 
    Sure, this IS grossly exaggerated, but there is truth in it.

    This demand for easy entertainment and quick gratifications comes with a price.

    A high price.
    It is the expectation for everything to be easy, expectation to be lead every step on the way, and needing gratifications and rewards to keep going, long even before the going could get tough.
    It nourishes a specific mindsetting, one 
    that leads to dependence and usually someone else doing the thinking, learning and heavy lifting. And interesting enough, it also includes believing to be entitled to that service. 

    If this is your mindset & your expectations :
    Forget Trinagon.  It's not for you.

    Trinagon will give you a hard time on the occasion. Trinagon will also teach you a thing or two about focus & concentration, applying your brains, spatial awareness & imagination, abstractions &  thinking ahead in abstract ways.
    You can grow and learn. You can become.
    It's all there for the taking, ... if you reach out.

    We don't know, what we don't know.
    There are some really cool answers.
    But do we have the right questions ?

    Very very deep down somewhere in the inner workings of our being, we have assimilated the relationship between cause and effect as being one where cause comes before effect. Pretty much by definition.
    Just the same, sometimes later  we see our life as questions that need answering. Work that needs to be done. There is first  the task and then the effort.

    Let there be a question ! Give me an answer.
    Clearly this makes perfect sense :)
    But !
    Where is the FUN IN THAT !
    Sometimes beautiful circumstances come together, where we have answers, and only long after we find questions to use them for ! 
    Sometimes the right ideas will find new problems, that they already have solved.
    Trinagon was one such case, even in its creation, and it can be a place for this process to continue.
    When playing you will realise sooner or later, that intuitively you will have answers to questions you did not even know where there to ask.
    Questions may present themselves, for the answers you already have.

    hex4310Trinagon can be played at any age, unless you're kept too busy in kindergarden, or too busy getting through school chores, you will learn to think in a way that does give you insights into questions that you never knew you could have.
    Which ones will those be ? Let them surprise you, surprise yourself with the answers, surprise your friends, and enjoy the game.
    Games are never chores. Lets keep it that way.

    RubiOcta9 564784

    One of the playgrounds in Trinagon is the dual body to the cube, the octahedron.
    The octahedron has as many vertices (corners) as the cube has faces, and vice versa.

    If you visualise the triangles being the cornerstones of a cube you will see a 2x2 rubik's cube.

    A mathematically completely identical puzzle to the Rubik's cube can be seen above.
    The only difference is the absolute position of the triangles, whereas the corners of the 2x2 rubik's cube are solved relative to each other.

     In honor of the Rubik's Cube, we made a few puzzles using 8 colors, which correspond to the 8 corners of the cube, and therefore resemble the 2x2 cube the most, and named this collection after that fact.
    Shuffle those to play them rubik's-cube-style.

    Trinagon also offers easier variations and a few symetrical patterns that have secifically been made to resemble the typical problems coming up when solving a rubik's 2x2 cube.
    That is why they almost look repetitive, but aren't. Each offers a different 2x2 cube-dual-problem to solve.
    You might even learn some new tricks on these, and find more elegant ways to solve the cube.

    Here they are. From easy to hardest. More of them may be made.


    RubiOcta1 152532

    A simple color exchange. The triangles are exactly opposite their solved positions.
    With no faces or direction markers on, this puzzle can be solved rather easily.
    Puzzle Nr. 152532
    Difficulty : 20
    Rotator Type 1


    Best solution : To be seen... by  AnonyMouse, 23.11.2019


    RubiOcta2 183953

    The same as the previous puzzle but at the same time the faces have to be switched.

    Puzzle Nr 183953
    Rotator Type 4

    You shall find out how much more difficult this one is compared to the last. Maybe not by much ?


    Best solution : ....


    RubiOcta3 465970

    Here the pieces end up at the same spot, but with their faces switched.
    Only the solution (in the corner) is different.

    This puzzle is less obviously like the rubik's cube, because the cube would only use rotation type 1.
    Also for an extra challenge two rotators are missing !

    Puzzle Nr. 465970
    Difficulty : ??
    Rotator Type 1 & 4

    Best solution : ....


    RubiOcta4 379894

    The third step on the same theme, with face and directions active !
    The triangles sit exactly opposite their intended location. They are directed 'up and down' to the black rotators.
    Maybe a similar pattern to solving it exists as in puzzles 152532 & 183953 ?

    Puzzle Nr. 379894
    Difficulty : ??
    Rotator Type 4


    Best solution : ....


    RubiOcta5 623675

    Now it gets a bit harder. Here the red, green, purple & grey triangles are already there, but yellow <-> orange & blue <-> pink need to be exchanged.

    This is one case where a color-only puzzle is not easy anymore. The cube-duality is the reason for that.

    Puzzle Nr. 623675
    Difficulty : 40
    Rotator Type 1


    Best solution : ....


    RubiOcta6 048838

    Mathematically identical to the rubik's cube, with only rotation 1 in use, and directionmarkers active.
    The symytry is the same as puzzle 379894, i.e. the triangles simply opposite their intented spot.

    We're not sure if this one is harder or easier than the last. The specific symetry may make it easier ? (we know, but won't say :))

    Puzzle Nr. 048838
    Difficulty : ??
    Rotator Type 1

    Best solution : ....

    If you want to play 2x2 rubik's cube style, shuffle this one. It sould be possible to solve in 14 moves, at least by a god, or maybe by you.


    These next three puzzles have not been solved yet :

    RubiOcta7 616545

    Same as puzzle 623675, but facemarkers active and playing with mixed rotations.
    This puzzle would impossible, if not for the face-preserving rotationtype in it.

    As it is, it has not yet been solved !!!  but theory alone claims that it is solvable !
    Feel free to send in your solutions !

    Puzzle Nr. 616545
    Difficulty : actually unknown, since it has not been solved, but designed from theory.
    This one is likely harder than the two below. 
    Rotator Types 2, 4 & 6

    Best solution : tba


    RubiOcta8 605522

    Same as puzzle 048838 but with the color-symetry of 623675, i.e. four colors have been switched.

    Also unsolved. Theory says it is, and we'll be happy to put your name under it. 

    Puzzle Nr. 605522
    Difficulty : as hard as a shuffled 2x2 rubik's cube with absolute positions.
    Rotator Type 1


    Best solution : tba


    RubiOcta9 564784

    Another Variation on switching colors. This one switches 4 colors vertical (vertical = between the black rotators, one defined as 'up' the other as 'down') and the other 4 colors horizontally.

    Puzzle Nr. 564784
    Difficulty : most likely harder than the one before ?
    Rotator Type 1

    Also unsolved. Theory says it is, and we'll be happy to put your name under it. 

    Best solution : tba.

    With these three not being played (&solved) yet, please send your solution to lumpazy. The best solution shall be remembered right here.
    Some of the others have only been played once, so there is room to earn some genius points.
    How many moves the best solution really needs is unclear. With the symetries in play they mostly should be solvable in less than 14 moves. When some of the rotators have been left out or when playing mixed rotations, this number may be going up a bit - or not !
    You are invited to help find the best solutions :)

    Many indirect contributors have helped bring this vision along with tools & resources that make our life better, and maybe yours too.

    Here shall be named and linked (where possible & in no specific order) a few of them :

    • Blender ... Heavy (in ability and learning curve) 3D modeling and animation software. Open source ! 
      Special Thanks here to Andrew from Poliigon for the Tutorials. Makes learning something as freaky complicated as blender fun enough to follow through.

    • Visual Code alias VS Code  ... Coding text editor. Small nice and clean with a ton of options that do not get in your way. It's sooo nice to have an editor that shows you mistypes and automatically marks occurences of same words, or knows by itself how to comment something out. This tool makes coding pleasurably effective.
      Thanks to Kendra Havens :
      for the simple thumbs up and first baby steps with that great editor. I love it too.

    • Notepad++ ... clearly also great. Depends on the task.
    • JSTool ... Json-editing features for the two above editors.
    • Unity ... A free (at the beginning) tool to develop games with ! Very intelligent business idea. This makes it possible to start without much funds.
      You only have to work ... and try to figure out the zillion explanations (in the manual) that make perfect sense, ..when you already knew the answer !  Perfectly easy, once you know it. perfectly obscure if you're new.
      Fortunately there are some tutorials : 
    • Two Channels especially : 

      Brackeys - Delivers good info with enthusiasm and somehow is always smiling even as he's talking.
      It's a breeze watching and learning from the guy, never boring, never too fast, always on target. Yay !
      Thumbs up too for showing this extremely useful app : UnityRemote5 ! Yes it's in the MANUAL ! but did i find i there. Nope... because i did not know that i could look for it, since it's an 'advanced topic'. Hmmmm really ?!

      Apptly Creative Academy - Thanks for the Tutorial on persistent Scriptable Objects. The hat rocks !

    • Geogebra ... a free tool that can make mathematics and their 3D (or 2D) represantations a breeze. Visualising and constructing models not from the artistic, but from the mathematical viewpoint really is great with this software. Schools use it, and you can too. A very pleasant surprise to have the simplicity of vector calculus revealed on screen. With only very basic features much can be done already.
      Could we please reverse the update to Geogebra 6 ??  Geogebra 5 works sooo much better.

    • MathGV .. free open Source function plotting... 

    • Captura ... Record a video and snapshot of your work. Very useful, should you wish to make a youtube-tutorial onsolving trinagon puzzles :):)

    • Cake PHP ... Web-app programming framework using php. It's main credo "convention over configuration" can obscure meaning behind conventions sometimes too, but once you get used to think in the box as much as you (i) can, it works like a charm.
    • Joomla ... This Website is build with it. Thank you anyone, who ever contributed to it.
    • JDownloads extension for Joomla. Only a small part of it is needed on this site. It works like a charm though.

    • Free Music Archive ... Music that can be used for free.
      But yes, i paid (donated), and here are my favourite FMA composers and their music : 
      Alan Spiljak ... three of his titles were used in the free game, maybe check out Kai Engel there too, but there are many other good ones (and not so great ones). Unfortunately Alan cannot be reached for a proper license and no other music has yet presented itself. Therefore, just play your own music :)

    There are many many more resources that went into learning how to program a game like trinagon, and they will show up once the've been found or remembered :)

    Thank you to all contributors of codes, which i could just copy online (yes, with the approbiate license) and work from. Thanks to all, who can and do make code examples answer to the point. That is great skill. I try and do so on the occasion myself.




    ® Trinagon © Lucas Pradlwarter. Austria. All Rights Reserved. Impressum 2019