Declension wrote:
For each of the three axes, define a certain rotation direction as positive. Then there are nine basic moves you can do, three on each axis, always turning one step in the positive direction. Let M be the set of these nine moves. From a given starting position, every possible position can be achieved in a finite number of steps, where each step is an element of M. One of these possible positions is the solution.
The set
Quote:
M union M^{2} union M^{3} union ...
is countable, so list all of the elements of this set in a certain order, interpret them as instructions, and try them one at a time. Reset the cube after each attempt by doing the instructions backwards.
There, it's solved. The rest is just optimisation.

WARNING: This method is guaranteed to solve the cube in a finite number of steps, but depending on the ordering you pick, it could be a very high finite number.
lol I tried it by doing a computer program. It could solve only 5-6 moves or so. So I had the idea to compute in advance the set of positions that are reached by 5-6 moves, and then do the program search for it. It could then solve 11-12 moves.
By the way, about the sets of solutions, it is interesting to note that at each move, you go from Mi to Mi+1 or Mi-1. And if you compute all the moves (from 1 to ~20), in fact you find positions you already encountered before.
There is one first position. Then, you have 6 faces and 4 quarters, so in M1 there are 6*3 = 18 positions. Then you don't need to turn the face you just turned, so in {M1,M2} there are 18*15 positions. Then, you don't need to turn the face you just turned, and if you turned the opposite face, you don't need to turn the first either, so in {M1,M2,M3} there are less than 18*15*(15 * 4/5 + 12*1/5) = 18*15*14.4 etc.
The more you go, the less new solutions, until you reach to point k where the number of new solutions in {M1..Mk} is less than for M(k-1). I made a program to compute every position in the Tetraminx, and after the point k, there was very few moves until every position is found.
btbnnyr wrote:
Are you talking about a trial-and-error approach, Mr. XXX? Playing around with something to figure out the underlying patterns is something that I've always preferred over a step-by-step read-the-instructions approach. Moar fiddling, less planning, for me. I didn't know that there were instructions for solving Rubik's cubes. For me, they would suck all the joy out of the playing.
Well, if you get anywhere from here...