In this section, we will explore the following questions.
What are faces and layers, and what methods exist for solving them?
How can we describe a complex series of Cube moves?
What is the order of a Cube move?
In this section, we’ll define a few terms and work on an initial exploration of the Cube’s white layer. Then, we’ll discuss the need for a precise method communicating about our Cubes, and introduce some standard notation.
Subsection1.2.1Faces and Layers
We begin with a definition.
Definition1.2.1.
A face of the Cube refers to one of its sides. We say a face is solved if all the stickers on that side are the same color.
There are many ways to solve the Cube, and we’ll explore one approach later in this chapter. A crucial step in learning to solve the Cube is learning to solve one face.
Challenge.
Scramble your Cube. Then solve the white face. If you have not done this before (or even if you have), this may take hours or days, not minutes! Avoid relying on any outside resources (including websites, friends, etc.). When you solve it, congratulations! Then scramble the face and solve it again. Repeat this process until you can reliably do it in just a few minutes. Once you can reliably solve the white face, describe, in writing, your methods as clearly and precisely as you can. What challenges did you have to overcome? How did you overcome them?
Congratulations on solving a face of your Cube! But a question comes to mind. When your white face is in its solved state, are all the cubies on the white face in the correct location?
Definition1.2.2.
A layer of the Cube consists of a face and all of the stickers on Cubies which compose the face. A layer is solved if all of the Cubies in the layer are in the correct location with the correct orientation.
Question1.2.3.
What is the difference between a face and a layer? How is solving a layer different than solving a face?
Challenge.
Scramble your Cube and then solve the white layer. As before, this may take hours or days. When you solve it, congratulations! Scramble it and do it again. Once you can reliably solve the white layer, describe, in writing, your methods as clearly and precisely as you can.
Subsection1.2.2The Need for Notation
Discussion1.2.4.
Find a partner. Take turns describing verbally and without pointing to your Cube your method for moving a corner cubie to its proper position. Similarly, verbally describe your method for moving an edge cubie to its proper position.
Discussion1.2.5.
As you probably noticed, it is challenging to orally describe complex/technical ideas in much detail without becoming lost. Is it easier to describe your cubie movements in writing? Why or why not?
Every area of inquiry has an associated collection of terminology and symbology. At its best, this terminology enables efficient and clear communication. However, terminology can often be a barrier. Thus, we want to avoid introducing unnecessary or confusing jargon whenever possible. However, as I hope Discussion 1.2.4 and Discussion 1.2.5 underscore, a lack of terminology or symbols severely impairs our ability to communicate deep technical ideas. It is with this background that we introduce the following notation, which has become standard in the Cubing community.
Definition1.2.6.
Hold the Cube in such a way that you are looking at one of the faces (your choice).
The face you are looking at is referred to as the front (F) face.
The face on the side opposite the front is referred to as the back (B) face.
The face on the right side is referred to as the right (R) face.
The face opposite the right is the left (L) face.
The face on the top of the Cube is the up (U) face.
The face on the bottom of the Cube is the down (D) face.
List the colors of each of the six faces in Figure 1.2.7.
As we we will see momentarily, the names described in Definition 1.2.6 not only help us refer to faces, but also moves of the Cube which help us solve it. In order to understand what a given move does to the Cube, we will need to refer to certain cubies by location. The following definition enables this.
Definition1.2.9.
A cubie is named by the face(s) on which it sits, using lowercase letters.
How can you tell whether the cubie in the named location is a corner or edge cubie?
How many letters are required to name a center cubie?
Where is the edge cubie that is blue and orange?
Figure1.2.11.A scrambled cube.
We are most interested in using our new notation to describe moves of the various faces. We thus make the following definition.
Definition1.2.12.
A given face name, e.g., \(R\text{,}\) describes a \(90^\circ\) clockwise turn of that face, as you look at the face. A given face name with prime symbol, e.g., \(R'\text{,}\) denotes a \(90^\circ\) counterclockwise turn of that face, as you look at the face. A sequence of face moves is written multiplicatively, left to right. We may use parentheses and exponents to write our moves more compactly.
Exploration1.2.13.
Consider the following questions in the context of a completely solved cube as pictured in Figure 1.2.14.
After performing the move \(R\text{,}\) what color(s) will be on the Up face?
Again starting from a solved cube, perform the move \(L\text{.}\) What color(s) are on the Up face?
Explain the difference in your answers to the first two questions.
How are \(F^3\) and \(F'\) related?
Consider the Cube move \(RF^2 U (LDR)^3\text{.}\) Which face will you turn first: the left face, or the front face?
Again consider the Cube move \(RF^2 U (LDR)^3\text{.}\) How many times in this move will we turn the right face?
What would it mean for two cube moves to be equal? Test your definition by determining whether \(LD = DL\text{.}\)
Suppose \(X\) and \(Y\) are two cube moves. When, if ever, is \(XY = YX\text{?}\)
Figure1.2.14.A solved cube.
Subsection1.2.3Order
A useful algebraic concept for describing Cube moves is that of order.
Definition1.2.15.
The order of a cube move \(M\) is the least number of times \(n \gt 1\) that \(M\) can be repeated before a solved Cube is solved again. We write \(|M| = n\text{.}\)
Example1.2.16.
The order of \(R\) is 4, since 4 clockwise quarter-turns of the right face of a solved Cube results in a solved Cube, and no fewer number of turns results in a solved Cube.
Exploration1.2.17.
What is \(|U^2|\text{?}\)
Calculate \(|R^3|\text{.}\) Why does this make sense?
Calculate \(|RUR' U '|\text{.}\)
Conclusion.
In this section, we began a systematic exploration of the properties of the Cube. First, we described the need for a notational shorthand in solving the white face and layer.
The standard notation refers to each face as you look at it, ignoring color, and describes a \(90^\circ\) clockwise turn of the face. So, \(FR\) means to first turn the front face \(90^\circ\) clockwise, then the right face \(90^\circ\) clockwise. We then noted that we can refer to a cubie by describing the face(s) on which it sits (using lowercase letters to distinguish cubies from faces): three letters refer to a corner, as it sits at the intersection of three faces; two letters refer to an edge, and one to a center.
We concluded by introducing the idea of order, which will help us analyze and understand more complex sequences of Cube moves in the next section.
Exercises1.2.4Exercises
1.
Calculate \(|RL|\text{.}\)
2.
Consider the Cube pictured below. What are the colors of the cubie is in the \(r bd\) position?
3.
Do some research online to find a sequence of Cube moves which gives the highest possible order. What is the move, and what is the order?
