## Maths

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 :)

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.

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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) 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.