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Exploration in Modern Mathematics

Mike Janssen

Section 1.3 Moving Cubies

Motivating Questions.

In this section, we will explore the following questions.
  1. How can we move the corner cubies so they are in the correct location?
  2. How can we describe the movement of the middle “slice” of our Cube?
  3. How can we move the edge cubies so they are in the correct location?
Our goal in this chapter is to explore, and eventually solve, the Cube. To do this, every cubie must be in the correct location with the correct orientation. In this section, we focus on two moves which will allow us to put the cubies in the correct location. Once they are in the correct location, we’ll see moves in the next section which will help us orient them correctly, enabling us to (finally) solve the Cube.
We begin by considering Cube Move 1.3.1.

Exploration 1.3.2.

The Cube move described in Cube Move 1.3.1 moves some of the corners on the front face of the Cube.
  1. By performing this move several times, identify on the blank Cube below what is happening to the front face of the Cube.
  2. Using the cubie notation described in Definition 1.2.9, describe what happens to the front face of the Cube after performing this move.
  3. Given what happens to the front face, exercise your human creativity and suggest a short name/abbreviation for this move.
  4. What is the order of the move?
  5. Practice the move until you can reliably execute it.
Figure 1.3.3. A blank cube.

Activity 1.3.4.

Now that you have identified exactly what Cube Move 1.3.1 does, use it to solve as much of your Cube it enables.
In order to move the edges, it will be helpful to add one more type of fundamental Cube move to our repertoire.

Definition 1.3.5.

By \(S_R\) we mean a clockwise rotation of the (vertical) middle slice as we look at the right face.
\begin{equation*} \underrightarrow{\quad S_R\quad} \end{equation*}

Exploration 1.3.7.

The Cube move described in Cube Move 1.3.6 moves some of the edges on the up face of the Cube.
  1. By performing this move several times, identify on the blank Cube below what is happening to the up face of the Cube.
  2. Using the cubie notation described in Definition 1.2.9, describe what is happens to the up face of the Cube after performing this move.
  3. Given what happens to the up face, exercise your human creativity and suggest a short name/abbreviation for this move.
  4. What is the order of the move?
  5. Practice the move until you can reliably execute it.
Figure 1.3.8. A blank cube.

Activity 1.3.9.

Now that you have identified exactly what Cube Move 1.3.1 and Cube Move 1.3.6 do, use them to put every cubie on your Cube in its correct location.

Conclusion.

In this section, we focused on two moves that enable us to put our cubies in the correct location. We first found Cube Move 1.3.1, which affects corners on the front face, and no other cubies. Similarly, Cube Move 1.3.6 affects edges on the up face, and no other cubies. We were then able to put every cubie on our Cube in the correct location. Hurray!

Exercises Exercises

1.

Consider the state of Lila’s Cube in Figure 1.3.10. She is so close to solving it! Can you help her finish?
Figure 1.3.10.

2.

Sam is nearly done with his Cube; it’s pictured in Figure 1.3.11. Can you help him finish?
Figure 1.3.11.

3.

Thanks to your help, Lila solved her Cube in Exercise 1.3.1, but now she needs more help! Can you help her finish the Cube pictured in Figure 1.3.12?
Figure 1.3.12.

4.

Consider the situation in Figure 1.3.13. Why will Cube Move 1.3.1 alone be insufficient for putting the corners in their correct locations? Devise a strategy for putting the corners in the correct locations.
Figure 1.3.13.
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