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 or piece-wise symmetries of graphs. Partial symmetries of restricted support are extended as identity on the whole of the graph and its set of sub-graphs. A partial symmetry of a graph or tiling has stabilizer all the graph (tiling) and acts on the support as permutations of a set of sub-graphs (sub-tilings). Therefore we get again groups that act as permutations of a set of sub-graphs (sub-tiling) of the initial graph (or tiling). To illustrate better the action of such groups of partial or part-wise or piece-wise automorphism of graphs , a tiling or lattices we introduce cellular automatons that represent the action of each generator and elements of such groups of piece-wise automorphims.
(For cellular automatons see e.g. http://mathworld.wolfram.com/CellularAutomaton.html ).
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 in at least the
case of the Rubik cube we give a sequence of arguments, that from the Scramble
generators, we discover and calculate the solution generators, which
in this case are either 3-cycles, or pairs of 2-cycles on fundamental regions
(=stickers) as the simplest possible action that can derive any other
action. Etc We must not confuse the simplest possible action on stickers
with the simplest possible finger-movement.
The situation can be compared with the 2nd-order algebraic equation. It is different for someone to give to him the formula of the two roots ( which corresponds here to give to him the recipe of how the solution generators, can be executed from the simplest finger or scramble turns) from giving him a logical way of how to discover the formulae of the two roots from the original equation with simple logical reasoning. (which corresponds here to showing a simple sequence of reasoning, of how to discover the solution generators).
The situation can be compared with the 2nd-order algebraic equation. It is different for someone to give to him the formula of the two roots ( which corresponds here to give to him the recipe of how the solution generators, can be executed from the simplest finger or scramble turns) from giving him a logical way of how to discover the formulae of the two roots from the original equation with simple logical reasoning. (which corresponds here to showing a simple sequence of reasoning, of how to discover the solution generators).
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 or piece-wise symmetries of graphs. Partial symmetries of restricted support are extended as identity on the whole of the graph and its set of sub-graphs. A partial symmetry of a graph or tiling has stabilizer all the graph (tiling) and acts on the support as permutations of a set of sub-graphs (sub-tilings). Therefore we get again groups that act as permutations of a set of sub-graphs (sub-tiling) of the initial graph (or tiling). To illustrate better the action of such groups of partial or part-wise or piece-wise automorphism of graphs , a tiling or lattices we introduce cellular automatons that represent the action of each generator and elements of such groups of piece-wise automorphims.
(For cellular automatons see e.g. http://mathworld.wolfram.com/CellularAutomaton.html ).
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 puzzle 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 of them through these rings.
We give a general definition of a puzzle or game on a graph or lattice with a group derived from groups of piece-wise symmetries of a graph , lattice or tiling, or polyhedron and we give a general solution for a wide class of them. This general solution method derives the solution of most of the known puzzles but also for many more that have not been materialized.
We give a general definition of a puzzle or game on a graph or lattice with a group derived from groups of piece-wise symmetries of a graph , lattice or tiling, or polyhedron and we give a general solution for a wide class of them. This general solution method derives the solution of most of the known puzzles but also for many more that have not been materialized.
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 Archimedian polyhedrons, and the Spherical Harmonic nodal patterns. All the above rings may be used also to create a large class of permutation polyhedral twisty and sliding puzzles (like Rubik;s cube and many more) that can be realized at least with software.
Finally it is given a necessary and sufficient condition for a planar or 3-dimensional polyhedral tile t to tile to a tiling T the 2-dimensional plane or 3-space, either in a periodic or in an a-periodic way (this is related to the 2nd part of the Hilbert problem 18) . This condition is with a dependence system D(t) of cycles of piece-wise congruence of parts of the boundary of the tile t and a corresponding group G(t) and ring R(t) . These are related to the corresponding groups G(T) and ring R(T) of all the tiling. Of course this does not mean that this condition can be easily used to discover such tiles in a computational or non-computational way, neither easily used to tile the space in to a tiling T once the tile t is known. It is simply a logical equivalent to the definition of a plane or space filler, with different more algebraic terms. The tiling and its piece-wise isometries define also a combinatorial specie (also a category) see e.g. http://en.wikipedia.org/wiki/Combinatorial_species and http://mathworld.wolfram.com/Species.html that transfers properties from geometry to action groups and permutations.
Finally it is given a necessary and sufficient condition for a planar or 3-dimensional polyhedral tile t to tile to a tiling T the 2-dimensional plane or 3-space, either in a periodic or in an a-periodic way (this is related to the 2nd part of the Hilbert problem 18) . This condition is with a dependence system D(t) of cycles of piece-wise congruence of parts of the boundary of the tile t and a corresponding group G(t) and ring R(t) . These are related to the corresponding groups G(T) and ring R(T) of all the tiling. Of course this does not mean that this condition can be easily used to discover such tiles in a computational or non-computational way, neither easily used to tile the space in to a tiling T once the tile t is known. It is simply a logical equivalent to the definition of a plane or space filler, with different more algebraic terms. The tiling and its piece-wise isometries define also a combinatorial specie (also a category) see e.g. http://en.wikipedia.org/wiki/Combinatorial_species and http://mathworld.wolfram.com/Species.html that transfers properties from geometry to action groups and permutations.
