How to go about solving these puzzles ?

Some basic principles & solving patterns derived from many hours of playtime :

'Rogue Play' below stands for : 

1. 'Rogue Play' (not really a principle) : 
   Solving a puzzle triangle by triangle, without regard for the particular symmetries of the playground or the rotators in play. 
   This is what you'll do when you first try the game, is perfectly fine.
   Naturally this method is clearly not very efficient.
   The fun you have solving trinagon puzzles will grow with your ability to avoid rogue play, and consider more and more of the puzzles properties at the same time.
   
   
2. Backwards exchange : 
   The first pattern. Always used in rogue play. Disregarding other triangles, the goal is to shift the one missing piece into position, by simple
   - moving it out of the way.
   - positioning the space,
   - move the triangle into the space and then turn the positioned triangle to its final position.
   This is similar to solving the first layer of a rubiks cube.
   
   
3. Line Dance : 
   It mostly works out more efficient to keep triangle grouped together. I.e Queuing them up. 
   This may also be misleading (HexS #8 - on the right) and therefore good to know, when NOT to use this pattern. 
   
   
4. Efficient Sorting (or advanced queueing)  :
   Some puzzles seem not much of a challenge, because the pathways to solve them are obvious. They often contain a few steps which could be left out, if the were just done in a different order.
   These Puzzles are often found in the Big Hexagon. What you learn there will help to find the coolest solutions on the smaller puzzles too.
   
   
5. Crankwork fit in :
   Similar to Nr 1 above, but thinking ahead a a few moves in advance, and thereby shifting colors into each other so that 2 queues are built instead of one.
   
   
6. Symmetry patterns :
   Trinagon puzzles are usually highly symmetrical.
   In those cases you can try and divide the board into symmetric partitions and repeat every move of one section in the other(s). 
   This may just bring you to the most elegant solution.
   But : There are many cases where specifically going against symmetry will be necessary !
   Then you need to find out how to best break the symmetry !
   
   
7. Strictly Ballroom : 
   Crankworking  around a symmetry and more than one position synchronously.  In 2D
   A few of the Small Hex Puzzles are especially made to learn this pattern.
   
   
8. Around a corner from both ends (3D crankworks) :
   The small puzzles can really be the most challenging ones. Space is so limited, that every move always includes some triangles which you wish could just stay where they are.
   This is similar to the last moves of the rubiks cube. By the way, the octahedron is really close to a 2x2 rubiks cube in it's properties. So if you can solve the 2x2 rubiks cube you'll be having a much easier time with the octahedron.
   The bigger polyhedra puzzles are a mix between efficient sorting & symetry patterns that go around corners. The first move on one end will have a symetric effect a few moves later when you reach the other side of the polyhedron.
   Combining these basics then often results in surprisingly 'easy' solutions.
   
   These patterns first apply to colors only, but are just as relevant to puzzles containing directions & faces. 

Direction & Face specific ideas to help :

1. Counting distances : (Faces) 
   Counting the distance and the rotationtype moves.
   
   
2. Single odd inversion : 
   Two triangles of a group  on an odd symmetry rotator. Obviously they can never oppose each other, but will always be at least by one closer to each other on one side than the other. An almost- half rotation (e.g  2 out of 5) can sometimes work wonders.
   
   
3. Conservation moves :  
   Prepositioning triangle around a rotator which will conserve their relative orientation,  or break it away on purpose.  Typically needed in rotationtype 5 & 6. 
   Also helps to rule out wrong moves.
   
   
4. Repetitive patterns : 
   .. are a direct application of a quality in permuatation puzzles that can be shown through group theory :
   Repeating a certain move pattern over and over leads, at some point, back to the original state !
   In between there are opportunities that can allow to interchange two or more triangles in a certain way, so that then going on with the move pattern will lead to the desired result.
   This is easy in principle, but not so much so in practice. To find the right exchanges in between often needs actually written down patterns (for mere humans at least).
   For puzzles with difficulties above the 100 this often becomes necessary.
   Just don't let it become work :)
   How this is done : Assume you have triangles of a specific group (mostly meaning : they look the same) which are already in thew right position but turned.
   Find 2 or more patterns that exchanges triangles within the same group. Then write the pattern down, including face & rotation states. From here you'll figure out the rest :)
   
   Btw. this is exactly how the rubiks cube patterns have been found / designed. I.e. Anyone solving the rubik’s cube with a known system is doing just that.