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 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)}

There are 500 € waiting for the (2) winners of the noo-fools-challenge.

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 is a nice diversion for a 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.

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 :)