14. A simple mathematical logical reasoning solution of Rubik's cube. Groups of piece-wise symmetries of graphs, (aperiodic) tilings and polyhedrons ( twisty and sliding puzzles). A general method of solution.

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 )

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. 

13 The Jupiter-Earth game

In this game, it is experienced the interaction of the earthly civilization with another civilization in one of the satellites of planet Jupiter.
Nevertheless the non-obvious here is that this satellite and  its  civilization is on the 2nd ethereal reality not on the ordinary material reality as in earth. In this higher frequency material reality (ethereal) , the planet Jupiter is already a small  bluish sun. The way that the two civilizations (earth's and Jupiter's) interact and cooperate inside the solar system is a unique mode of entertaining but also spiritual imagination. 

12 The Planetary Life Ecological Sustainability Game

How the energy model, the cities/villages distribution, the forest preservation  the sea desalination and other innovative modes regulations and practices can balance and create a prosperous fair and sustainable life on the planet? This game is all about the above. 

11 The Old Millennium and New Millennium Energy models Game

In this game its is experienced the pros and cons of the classical fossil fuels energy model and a 21th century renewable solar energy model. Solar energy from infrared solar radiation is stored in the gravitational field which can be extracted all 24 hours by electromagnetic devices . The production, grid, distribution and of the social and macro/micro economic modes are different. The old model supports a centralized or oligarchic control of the energy production while the renewable energy model a more equally distributed democratic control of the energy.

The abundance and cheap energy in the new renewable energy model, permits easy desalination of the oceans and therefore abundance of drinking water too. 

10. A Moral Decisions Game

A very good similar game is the Scruples, invented by Henry Makow

http://en.wikipedia.org/wiki/Scruples_(game)

http://www.scruplesgame.com/main.html

9. The Hollow Earth Game

In this game the classical vision of Julious Vern of the hollow earth goes one level further.

The earth in this game is hollow (from the 2/3 of the radius till the center)  has internal bluish gaseous sun fixed in its  center, and vegetable and animal life together with an internal atmosphere with clouds sunshine and rains. There are no seasons and stars. The gravitation reverses twice once inside the solid shell (at 5/6 of the radius from the center , and about 1/3 of the radius from the center). There is a large city of 20 million people called Kalipolis or Esopolis . In the outer earth there is another large city called  Exopolis.

There are two openings of about 70 kilometers diameter at the poles , at the bottom of both oceans that link the inner and outer earth through water. Also there are about 12 tunnels of about 2000-3000 kilometers that you can go from the inner to the outer earth through land , through the solid shell. The game is about the interaction , interplay of the two civilizations the inner-earth civilization and the outer earth civilization.

Here is an 3D animation video, which gives the above assumed concept (in the video the width of the shell is shown half of what mentioned here)

http://www.youtube.com/watch?v=ubKFmIDBMjU

8. The Public and Private Property Game

In this game it is experience the pros and cons of the balance of the size of the public sector with its democratic functioning and the private sector which is mainly functioning through the oligarchy of the equities owners, and top managers. 
It is also experienced new modes of economic functioning of the public sector mainly though public services and private sector investments, plus other innovative ways. 
The issues of public debt and monetary system are also involved