# Megaminx

A 6-color Megaminx, solved
A 12-color Megaminx, solved
A 12-color Megaminx in a star-pattern arrangement

The Megaminx or Mégaminx (/ˈmɛɡəmɪŋks/, /ˈm-/) is a dodecahedron-shaped puzzle similar to the Rubik's Cube. It has a total of 50 movable pieces to rearrange, compared to the 20 movable pieces of the Rubik's Cube.

## History

The Megaminx, or Magic Dodecahedron, was invented by several people independently and produced by several different manufacturers with slightly different designs. Uwe Mèffert eventually bought the rights to some of the patents and continues to sell it in his puzzle shop under the Megaminx moniker.[1] It is also known by the name Hungarian Supernova, invented by Dr. Cristoph Bandelow.[2] His version came out first, shortly followed by Meffert's Megaminx. The proportions of the two puzzles are slightly different.

## Description

The Megaminx is made in the shape of a dodecahedron, and has 12 faces and center pieces, 20 corner pieces, and 30 edge pieces. The face centers each have a single color, which identifies the color of that face in the solved state. The edge pieces have two colors, and the corner pieces have three. Each face contains a center piece, 5 corner pieces and 5 edge pieces. The corner and edge pieces are shared with adjacent faces. The face centers can only rotate in place, but the other pieces can be permuted by twisting the face layer around the face center.

There are two main versions of the Megaminx: one with 6 colors, with opposite faces having the same color, and one with 12 different colors. The 12-color Megaminx is the only type legal in official WCA competitions, and is therefore much more popular than the 6-color version.

The purpose of the puzzle is to scramble the colors, and then restore it to its original state of having one color per face.

## Solutions

The 6-color Megaminx comes with an additional challenge which is not immediately obvious (and which does not occur on the 12-color puzzle). Its edge pieces come in visually identical pairs, because of the duplicated colors of opposite faces. However, although visually indistinguishable, they are nevertheless mathematically bound in a parity relationship. In any legal position (reachable from the solved state without disassembling the puzzle), there is always an even number of swapped pairs of edges. However, since swaps may be between visually identical edges, one may find that having solved almost the entire puzzle, one is left with a pair of swapped (distinct) edges that seems to defy all attempts to exchange them. The solution is to swap a single pair of 'identical' edges to resolve the parity issue, and then restore the rest of the puzzle.

This property is absent in the 12-color Megaminx, because all its edges are distinguishable, and it would be immediately obvious that there is another pair of swapped edges besides the pair one is working with.

Besides solving a Megaminx the regular way, patterns can be made on it just like a Rubik's Cube. Examples of these include a star, checkerboard, and pentagon in a pentagon patterns.

## Variations

There are many similar puzzles with different numbers of layers, most of which change the "mega" in the puzzle's name to another metric prefix. They are the Kilominx (2 layers), Master Kilominx (4 layers), Gigaminx (5 layers), Elite Kilominx (6 layers), Teraminx (7 layers), Petaminx (9 layers), Examinx (11 layers), Zettaminx (13 layers), and Yottaminx (15 layers).[3][4] The highest order mass-produced variant of the Megaminx is the Petaminx, which was released by MF8 and then later by ShengShou, and the highest order variant of the Megaminx ever made to date is the Yottaminx, created by Matt Bahner using 3D printing. It is the dodecahedral equivalent to a 15×15×15 Rubik's cube.

Alexander's Star is equivalent to solving only the edges of a six-color Megaminx.

The Impossiball is equivalent to solving only the corners of a Megaminx and is available with either six or twelve colors.

The Pyraminx Crystal is a modified Megaminx with deeper turning planes.

Tony Fisher has produced a shape modification of the Megaminx into a cube form which he called the Hexaminx.[5] Another variant is the Holey Megaminx, which has no center pieces, like the Void Cube. It is being produced by Mèffert as of July 2009. Other variants include the Flowerminx, Megaminx Ball, and Crazy Megaminx.

 A Holey Megaminx, with black body Kilominx, Megaminx, Master Kilominx, Gigaminx, Elite Kilominx, Teraminx

## Number of combinations

Ernesto González solving a Megaminx at TLP Tenerife 2017

Both versions of the Megaminx have 20 corners and 30 edges. In both cases, only even permutations are possible, regardless of the position of the other set of pieces. Thus, while it is possible to have a single pair of corners and a single pair of edges swapped on a Rubik's Cube, this is impossible on the Megaminx. There are 20!/2 ways to arrange the corners and 319 ways to orient them, since the orientation of the last corner depends on that of the preceding ones. There are 30!/2 ways to arrange the edges and 229 ways to flip them.

${\displaystyle 20!\times 3^{19}\times 30!\times 2^{27}\approx 1.01\times 10^{68}}$ The full number is 100 669 616 553 523 347 122 516 032 313 645 505 168 688 116 411 019 768 627 200 000 000 000 (roughly 101 unvigintillion on the short scale or 101 undecillion on the long scale).

The corners are distinguishable on a 6-color Megaminx because two corners with the same three colors will be mirror images of each other. There are 15 pairs of identical edges. It would not be possible to swap all 15 pairs, since this would be an odd permutation of the edges, so a reducing factor of 214 is applied to the preceding figure.

${\displaystyle 20!\times 3^{19}\times 30!\times 2^{13}\approx 6.14\times 10^{63}}$

The full number is 6 144 385 775 971 883 979 645 753 925 393 402 415 081 061 792 664 780 800 000 000 000 (roughly 6.1 vigintillion on the short scale or 6.1 decilliard on the long scale).

For the larger size variations (gigaminx, teraminx, petaminx etc), the general number of combinations is ${\displaystyle {\frac {30!\times 20!\times 60!^{n^{2}-1}\times 2^{28-n}\times 3^{19}}{5!^{12n(n-1)}}}}$ where ${\displaystyle n=1,2,3,4,...}$ respectively for megaminx, gigaminx, teraminx, petaminx, etc.[6] The number of combinations evaluates to ${\displaystyle 3.65\times 10^{263}}$ for gigaminx, ${\displaystyle 1.15\times 10^{573}}$ for teraminx, and ${\displaystyle 3.16\times 10^{996}}$ for petaminx.

## Records

Speedsolvers completing Megaminxes at the Estonian Open 2011.

The world record time for a Megaminx solve is 27.81 seconds, set by Juan Pablo Huanqui of Peru on 28 July 2018 at CubingUSA Nationals 2018, in Salt Lake City, Utah.[7]

The world record average of five solves (excluding best and worst) is 32.03 seconds, set by Yu Da-Hyun (유다현) of South Korea on 24 February 2018 at CWR Winter 2018 in Pyeongtaek, South Korea, with the times of 30.94 31.24 (35.78) (30.12) and 33.90 seconds.[7]

### Top 5 solvers by single solve[8]

Name Fastest solve Competition
Juan Pablo Huanqui 27.81s CubingUSA Nationals 2018
Yu Da-Hyun (유다현) 30.12s CWR Winter 2018
Amos Nordman 30.81s Myyrmäki Spring Challenge 2019
Nicolas Naing 31.19s Penn Cube Day Fall 2018
Yunliang Zhang (张赟量) 32.44s Hefei Spring 2019

### Top 5 solvers by average of 5 solves[9]

Name Fastest average Competition
Yu Da-Hyun (유다현) 32.03s CWR Winter 2018
Juan Pablo Huanqui 32.33s CubingUSA Nationals 2018
Nicolas Naing 34.53s No Place for DNF Spring 2019
Amos Nordman 34.63s Pikkujoulut 2018
Feliks Zemdegs 36.65s Weston-super-Mare Open 2018