1 - What is a Rubik Cube?

In a world where robots are slowly tipping the scale of human workers to their robotic

counterparts, algorithmic automation has never been as big as it is currently and is

only set to get bigger. Robotics and automation present modern-day solutions to

tedious and unsafe jobs. But what if these ideas were applied to something that wasn't

tedious, but fun? What if it was applied to a Rubik's Cube?

1.0 The Rubik's Cube

Figure 1 - A Standard Rubik's Cube (Rubik’s, 2021)

The Rubik’s cube (R-C), the world’s most infamous puzzle. A 3x3x3, six-sided puzzle invented in 1974 by Hungarian professor Ernő Rubik. The puzzle's ultimate goal is to orientate the cubes such that each face coheres to a singular colour. Solving a Rubik Cube is generally considered a feat for humans, but a mere computational process for computers and their robotic counterparts.

As the creator, Erno Rubik, said:

If you are curious, you’ll find the puzzles around you. If you are determined, you will solve them.

With such determined minds of the likes of AI research and deployment companies and institutions such as OpenAI, MIT and enthusiasts online, they have showcased the brilliancy and complications involved with making Rubik Cube solving robots. From robotic hands (Akkaya et al., 2019) to sub-second solves (Swatman, 2017), it is clear to see that the Rubik’s Cube with its over 43 quintillion possible combinations, has clutched the intrigue of humanity.

But before understanding how robots can solve a Rubik Cube, it is important to have a rudimentary understanding of how a Rubik Cube functions. As such, this article will provide a fundamental understanding of the Rubik's Cube mechanical design, possible configurations and notation used to solve a Rubik Cube.

1.1 Mechanical Design

The 3x3x3 cube consists of 26 cubelets; the ‘27th cubelet’, in reality, does not exist (Chen, 2004) but is a centre shaft frame (Figure 2) that enables rotation of each face. The goal is to orientate each cubelet such that each face coheres to a single colour determined by the centre cubelet, which cannot move, only rotate.

Figure 2 - Centre Shaft of Rubik Cube (Zeng et al, 2018)

The cube consists of a total of 3 piece types (Figure 3): Six centre cubelets with one coloured facet, twelve edge cubies with two visible facets, and eight corner cubies that have three visible facets. One type of cubelet cannot exist within another types cubicle, e.g. a corner cubelet can never be in an edge cubelet space (Korf, 1997).

Figure 3 - Rubik Cube Piece Types (Rubik’s, 2021)

1.2 Possible Configurations

The total number of configurations of the cube is given by eight corner cubelets that can be arranged in 8! different configurations. Each corner cubelet has 38 possible orientations due to the three colour facets of corner pieces. The twelve edge cubelets can be arranged in 12! different ways, each with 212 possibilities due to the two colour facets of edge pieces. As the middle cubelets never move, they were not considered within the problem space (Chen, 2004). To account for illegal moves, the total number of configurations was reduced by a factor of 12, eliminating rotations that are theoretical but not physically achievable (Korf, 1997). The total possible configurations are then given by:

(8! ∙ 3⁸ ∙ 12! ∙ 2¹²)/12 = 43,252,003,274,489,856,000

Equation 1 – Total Number of Configurations

1.3 Singmaster Notation

The popularised standard for denoting rotations is known as Singmaster notation. Where: U (upper), F (front), D (downward), B (backward), L (left), R (right) denote the given face and a 90 degree clockwise turn of the face (Singmaster, 1981) shown in Figure 4 below.

Figure 4 - Singmaster Annotated Rubik Cube (Baumann, 1997)

When a letter is followed by an apostrophe (‘), it notes an anticlockwise turn, and a letter that is followed by a (2) or a (2’) notes a 180-degree turn of the face in the given direction (Kociemba, 2020). This notation can also be used to describe the scramble of any cube by representing the location of individual cubelets. Where each letter represents the location and colour of a cubelet for its given orientation. For example, the white, green, and red corner cubelet in Figure 4 can be described as UFR (Chen, 2004).

This notation proposed by Singmaster was adopted as the universal standard within the Rubik Cube world, which is proven as it is used throughout important books and papers within the field (Joyner, 2009). It accurately describes cubelet locations relative to the user from simple visual inspection, and a basis of understanding is crucial to understanding solving algorithm solutions featured in the next article.

Now an rudimentary understanding of the Rubik's cubes goal, mechanical configuration, and notation has been provided. The intricacies of the mathematical algorithms needed to solve a Rubik Cube can be further explored within the next article.

Thank you for reading.

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References:

Akkaya, I. et al (2019) Solving Rubik’s cube with a Robot Hand. Available at: https://arxiv.org/pdf/1910.07113.pdf

Baumann, J. (1997) A Step by Step Solution of Rubik’s “Magic Cube”. Available at: http://www.linkedresources.com/teach/rubik/solution.php

Chen, J. (2004) Group Theory and the Rubik’s Cube. Available at: http://people.math.harvard.edu/~jjchen/docs/Group%20Theory%20and%20the%20R ubik's%20Cube.pdf

Joyner, D. (2009) Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys. 2 nd edn. Baltimore: John Hopkins University Press. Joyner, W.D. (1997) Mathematics of the Rubik’s cube. Available at: https://www.fuw.edu.pl/~konieczn/RubikCube.pdf

Kociemba, H. (2020) The Mathematics Behind Cube Explorer. Available at: http://kociemba.org/cube.htm

Korf, R.E. (1997) ‘Finding Optimal Solutions to Rubik’s Cube Using Pattern Databases’, AAAI-97, pp. 700-705. Available at: https://www.aaai.org/Papers/AAAI/1997/AAAI97-109.pdf.

Rubik’s (2021) Rubik’s Cube 3x3. Available at: https://www.rubiks.com/en-us/rubik-scube-3x3.html

Singmaster, D. (1981) Notes on Rubik’s ‘Magic Cube’. Berkeley Heights: Enslow Publishing Inc

Swatman, R. (2017) Fastest Robot to Solve a Puzzle Cube. Available at: https://bit.ly/32tfH9A

Zeng, D. et al (2018) ‘Overview of Rubik’s Cube and Reflections on Its Application in Mechanism’, Chinese Journal of Mechanical Engineering, 31, p. 4. Available at: https://cjme.springeropen.com/track/pdf/10.1186/s10033-018-0269-7.pdf