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 !

This 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 !

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 !**!

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 or years of work.

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 calculate 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 ! on single puzzles,
- 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 be available for research purposes. Feel free to contact us in case this is your line of work/pleasure.

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
- probability theory

Just to make up a few :

- 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 no).

How many can be ? (That's where it gets difficult)

- Why are there exactly 6 rotationtypes ?

Or could there be more ?

- 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 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.

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.

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

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 : ....

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 : ....

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 : ....

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 : ....

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 :

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

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

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 :)

Trinagon first entered it's creators (my) imagination in 1988 or thereabouts.

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.