Here we describe in simple details how one could solve the Rubik's cube, if e.g. he was Rubik himself, and no one had solve it before, by using only logical reasoning not memory or much time to spent. Let us assume that one cannot use the web or youtube and forums to find ready-made recipes and algorithms to solve the cube. In addition let us assume that he does not have much time to spent by experimenting randomly. And finally let us assume that he can use existing simple not too complicated mathematics for this purpose. Actually as Jaapsch remarks in his page http://www.jaapsch.net/puzzles/thistle.htm , there is the mathematical solution of Morwen B. Thistlethwaite who is a mathematician who devised a clever algorithm for solving the Rubik's Cube in remarkably few moves. It is a rather complicated method, and therefore cannot be memorized. It is only practical for computers and not for humans. Furthermore even the fundamental theorem of the permutation group of the Rubik's cube as formulated e.g. by W.D. Joyner in (http://www.permutationpuzzles.org/rubik/webnotes/rubik.pdf ) in one of its directions is utilizing findings of difficult algorithms of practitioners, that cannot be found easily by just playing with the cube. Can we find a mathematical logical reasoning solution without having to utilize computer calculations and complicated computational group theory? Is it possible with the above assumptions to solve the Rubik's cube? The answer is yes, and we will show how. The general concept is to start from the initial "scramble generators" of the permutation group of the puzzle (e. g. rotation of one only face in Rubik's cube ) which are in general moving many elements of the puzzle, thus are simple global action , and are easy to scramble the puzzle , but not easy to solve the puzzle through them. Then we try to find by reasoning and simple mathematics the "solution generators" that have simple local action (e.g. a 3 cycle for all even permutations or a transposition 2-cycle for both even and odd permutations) but are still generators of the permutation group of the puzzle, and then solve the puzzle with the obvious and no-thinking way through the solution-generators (and their conjugates). Furthermore this strategy and method can solve simultaneously most of the other twisty or sliding and permutation puzzles. I consider this as the true solution of Rubik's cube and the other permutation puzzles, as puzzle of logic, otherwise current attitudes make it a skill of fingers.
Also we introduce here a rather new class of finite non-commutative rings, based on the action of any finite permutation group and in particular of groups of partial symmetries of graphs. Partial or piece-wise symmetries of restricted support are extended as identity on the whole of the graph and its set of sub-graphs therefore we get again groups. The elements that the permutation acts , the permutations themselves and all subsets of them are exactly the elements of such a ring. The elements that the permutation acts on and their subsets constitute a normal commutative sub-ring. The boundary topological operator induces a corresponding to the algebraic entities. Thus it defines a new type of homology. In particular we define such permutation rings for any Graph cycling puzzles as introduced by R.D. Wilson in " Graph puzzles homotopy and the Alternating group" JOURNAL OF COMBINATORIAL THEORY (R) 16, 86-96 (1974) http://www2.informatik.uni-freiburg.de/~ki/teaching/ss12/readinggroup/private/wilson-combtheo1974.pdf
We solve most well known such games of them through these rings.
(See also http://www.jaapsch.net/puzzles/graphpuzz.htm )
(See also http://www.jaapsch.net/puzzles/graphpuzz.htm )
Permutation rings can be defined also for any polyherdron through the graph of their edges/vertices and faces. In particular we present some such rings for the Platonic the Archimedian polyhedrons, and the Spherical Harmonic nodal patterns. All the above rings may be used also to create permutation polyhedral twisty and/or sliding puzzles (like Rubik''s cube and many more) that can be realized at least with software.
