Rubik's Revenge

Rubik's revenge in a state of chaos

Rubik's Revenge (German: Rubik's Rache ) or Rubik's Master Cube is a cube-shaped puzzle . It is a slightly larger and more difficult variant of the Rubik's Cube , also with six colors, but with four instead of three levels in each spatial direction. This increases the number of segments on each of the six side surfaces from nine to 4 × 4 = 16. It looks as if the surface were divided into 56 cubes, which change their position relative to one another by rotating 90 degrees around the nine axes of rotation. As with the classic Rubik's Cube, the aim is to move the cube from a disordered position into its ordered basic state, in which all segments of a side face have the same color. Compared to the Rubik's Cube, the Rubik's Revenge has the additional challenge that there is no fixed center piece.

history

The 4 × 4 × 4 version of the Rubik's Cube was invented in 1981 by the Hungarian mathematician Peter Sebestény and a patent has been applied for. He sold the rights to use the patent to the US-based Ideal Toy Company .

The axes that hold the cube together and realize the rotating mechanism are split into four parts and are held together by a core. The idea of ​​dividing the axles in this way was new and allowed the patent to be granted.

Ideal Toy marketed the quad cube together with the "Rubik's" trademark, sold it in the US as "Rubik's Revenge" and in Europe as "Rubik's Master Cube". In the 1980s, the market was soon so saturated with different puzzle variants that Ideal Toy stopped producing the four-cube after a few years. The implementation and marketing of other rotating mechanisms that were not patented have ensured that the four-cube is still produced and sold today.

Combinations

Parts of the Rubik's Revenge

The Master Cube consists of 56 moving parts, significantly more than the 26 of the 3 × 3 × 3 cube . As with this one, it is not a cube whose surface is broken up, but only cube segments. In detail these are:

Middle parts

• with one color each, 24 pieces, four per color
• In contrast to the classic Rubik's Cube, the middle parts are not fixed, which is why the correct position of the stones must be read off the corner stones.
• The middle sections can swap positions independently of each other, but since there are four of each color that cannot be distinguished, you get combinations of the middle sections. ${\displaystyle {\frac {24!}{4!^{6}}}={\frac {24!}{24^{6}}}}$

Edge parts

• with two colors each, 24 pieces, two per color combination
• The orientation of the edge parts depends entirely on their position in the cube. Although there are two of each variety, you cannot (by mistake) change places, as the orientation changes and the colors are exactly wrong.
• The edge parts can swap positions independently of each other, there is only one correct position and you cannot change the orientation, which leads to combinations of the edge parts. ${\displaystyle 24!}$

Corner pieces

• three colors each, eight pieces, one for each color combination
• The corner pieces behave exactly like those of the three-piece cube. You can swap positions independently, and each corner has three different orientations, which, however, influence each other: Due to the design, only a third of the theoretically possible combinations are accepted.
• Since the middle parts are movable, the Master Cube, like the Pocket Cube, does not have a fixed orientation in space: any corner can always be viewed as correct and thus as a reference point on which the further solution is based.
• This leads to combinations of the corner pieces. ${\displaystyle {\frac {(8-1)!\cdot 3^{8-1}}{3}}=7!\cdot 3^{6}}$

Overall there is with it ${\displaystyle {\frac {24!}{24^{6}}}\cdot 24!\cdot 7!\cdot 3^{6}={\frac {23!}{8^{4}}}\cdot 23!\cdot 7!\cdot 3^{2}=23!^{2}\cdot 9!\cdot 2^{-15}=}$

${\displaystyle 7.401.196.841.564.901.869.874.093.974.498.574.336.000.000.000}$ ${\displaystyle \approx 7{,}4\cdot 10^{45}}$Positions that the Master Cube can assume. That is far more than times as many as with the Rubik's Cube. ${\displaystyle 10^{20}}$

See also combinatorics , faculty

Level of difficulty

Although the Master Cube has many more possible positions than the three-way cube, it is not in itself any more difficult. Of course, more stones have to be sorted, but since a maximum of around 28.6% of the parts are moved with one rotation, in contrast to 40% in the classic variant, the parts have less influence on each other.

The middle parts in particular can be sorted with short and simple sequences of moves. Any known solution strategy for the corner parts can also be used for the master cube. Different with the edge parts: Here z. Sometimes new sequence of moves is required, as the behavior of individual curb stones has little correspondence in the three-piece cube. Pairs, i.e. two curb stones lying next to each other, can be sorted in the same way as individual curb stones with the triple cube.

Solution strategies

The approach of sorting the middle parts first and pairing the edges is widespread. From now on, the central axes of rotation are no longer used, which means that the master cube behaves like the three-part cube. Every known strategy for this then leads to the solution of the master cube, with two exceptions: It can happen that an edge pair is incorrectly oriented. As shown above, the two adjacent edge parts then have to swap their positions, which is only possible by also swapping two middle parts, four of which are always the same. The second possibility is that two pairs of edges are swapped, which is not possible with the normal 3x3 cube.

But there are also various other options for solving the Master Cube.

notation

A graphic notation is rarely used in Rubik's Revenge, a continuation of the letter notation is widespread. 90-degree rotations of the outer sides in clockwise direction are denoted with capital letters, as with the three-way dice, according to their position front, back, top, bottom, left or right with . According rotations are provided in a clockwise direction of the inner levels with small letters: . Furthermore, a counterclockwise rotation means 90 degrees and a 180 degree rotation of the inner right plane. The clockwise direction refers to the plane to be rotated, while for example and have the same rotation orientation, it is between and exactly the other way around. Some use the lower case letters to designate a rotation of the outer and inner levels, but this is usually indicated because this is not the usual notation. ${\displaystyle V,H,O,U,L,R}$ ${\displaystyle v,h,o,u,l,r}$ ${\displaystyle r^{-1}}$ ${\displaystyle r^{2}}$ ${\displaystyle h}$ ${\displaystyle H}$ ${\displaystyle h}$ ${\displaystyle v}$

World records

The current world record in speed cubing for the Master Cube is 17.42 seconds and was set by Sebastian Weyer at the Danish Open 2019 .

The world record for the average time when solving the cube five times (the best and the worst of the five times are not included in the average) is held by Max Park with 21.11 seconds, set at Bay Area Speedcubin '21 2019 .

Stanley Chapel set the world record in blindly solving the cube in 1: 02.51 minutes at the Michigan Cubing Club Epsilon 2019 .

Remarks

• With the Master Cube (4 × 4 × 4) it is possible to emulate both the Pocket Cube (2 × 2 × 2) and the classic Rubik's Cube (3 × 3 × 3) . In the first case, only the middle axes of rotation may be used, in the second only the two outer axes of rotation.
• The master cube can be taken apart and reassembled. This applies to the versions of all manufacturers, but some require tools such as a screwdriver . Sometimes a couple of stickers have to be removed to access the mechanism.

literature

Introductions and instructions

• Douglas R. Hofstadter : On the magic of the Rubik's Cube . In: Mathematical games, spectrum of science . Edition May 1981, Heidelberg, p. 16ff. ISSN  0170-2971 .
• Kurt Endl: Le Rubik's cube, enigme du siècle. Würfel-Verlag, Giessen 1981, ISBN 3-923210-15-9 .
• Jérôme Jean-Charles: Master Rubik's Cube. Denoël, Paris 1982, ISBN 2-207-22790-1 .
• Michael Reid: Mastering Rubik's Revenge. Simon & Schuster, New York 1982, ISBN 0-671-45952-4 .
• Kurt Endl: Rubik's Master Cube. Würfel-Verlag, Giessen 1982, ISBN 3-923210-50-7 .
• Kurt Endl: The common strategy for all Rubik's cubes. Würfel-Verlag, Giessen 1983, ISBN 3-923210-80-9 .

mathematics

The following titles deal with the mathematical properties of the Rubik's Cube, but also contain instructions that may be easier to follow than the informal introductions:

• David Sing Master: Notes on Rubik's Magic Cube. 5th edition. Singmaster, London 1980, ISBN 0-907395-00-7 .
• Alexander H. Frey Jr., David Singmaster: Handbook of Cubik Math. Enslow, Hillside 1982, ISBN 0-89490-060-9 .
• Christoph Bandelow : Introduction to Cubology. Vieweg, Braunschweig and Wiesbaden 1981, ISBN 3-528-08499-5 ; extended English version: Christoph Bandelow: Inside Rubik's Cube and Beyond. Birkhäuser, Basel and Boston 1982, ISBN 3-7643-3078-3 .
• Ernő Rubik , Tamas Varga, Gerzson Keri, Gyorgy Marx, Tamas Vekerdy: Rubik's Cubic Compendium. Oxford University Press, London 1987, ISBN 0-19-853202-4 .

Web links

Commons : Rubik's Cube  - collection of images, videos and audio files

• Rubik's Revenge solution by Chris Hardwick
• Cube pattern
• German tutorial for the 4x4x4 cube

Individual evidence

1. ↑ Jaaps-Master-Cube-Seite (English)
2. ↑ World Cubing Association: Official 4 × 4 × 4 Single Results. Retrieved February 10, 2015 . 
3. ↑ World Cubing Association: Official 4 × 4 × 4 Average Results. Retrieved February 10, 2015 . 
4. ↑ World Cubing Association: Official 4 × 4 × 4 Blindfolded Single Results. Retrieved February 10, 2015 .