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This is an implementation of the full symmetric group S4 as a twisty puzzle. A puzzle without restriction like parity issues.

The idea came up when during a mathematics session at the university (twenty years ago), the inventor realized that a puzzle based on permutations could be easily made by using certain ways. In fact, such a structure could accommodate the creation of any algebraic puzzle based on a group! (including transforming Rubik's Cube into Rubik's... Tube(!), and many others).

There are at least two views onto this puzzle.

The mathematical view:

From the mathematical point of view most twistypuzzles can be viewed as permutations groups. For example there is the full symmetric group S54 which has 54! elements. The Rubiks Cube is a subgroup of S54. The 2x2x2 is a subgroup of S24. The skewb is a subgroup of S30.

Pantazis' Matrix however is an implementation of S4. This puzzle is not a SUBGROUP of S4 but the full permutation group.

In this puzzle two permutations (and there inverses) of S4 are implemented and can be applied to the state of the puzzle.

The technical view:

This puzzle is made of two cylinders, one external and one internal. The external cylinder has two sets of four capsules C1 and C2. Each capsule-set may accommodate the four marbles with colors blue, green, red and yellow. A set of stickers below the capule sets indicate the solved state. (S1 for C1 which is blue-green-red-yellow, and S2 for C2 which is blue-red-green-yellow).

The internal cylinder has two sets with four pipes each P1 and P2. By rotating the internal cylinder inside the external cylinder, each pipe-set takes the four marbles from one capsule set, rearranges their order, and sends them (with gravity) to the other capsule set. This corresponds to using a permutation matrix.

The goal is to place the marbles inside the capsules in such a way, that their colors match the solution-stickers S1 or S2. Another challenge is going from S1 to S2 and vice versa.

Note that, this puzzle can be expanded and/or blocked in many ways to create many more interesting designs.

This puzzle is a collaboration between Pantazis Houlis who developed the concept and Gr�goire Pfennig. Pantazis had the idea (see the original sketch in image 5), developed the concept and even built a few working prototypes. One of them is shown in image 4. Gr�goire Pfennig implemented a more accurate model through 3D printing, based on Pantazis' previous work.

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Contributors

Thank you to the following people for their assistance in helping collect the information on this page: **Andreas Nortmann, Pantazis Houlis**.

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