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Solving Big Cubes by Logic
2. Structure of Big Cubes
Big cubes can be categorized into two types: odd cubes and even cubes. Odd cubes consist of an odd number of pieces on each side, such as 3x3x3, 5x5x5, and 7x7x7. Even cubes, on the other hand, have an even number of pieces on each side, like 4x4x4, 6x6x6, and 8x8x8.
Similar to the 3x3x3 cube, not all pieces in big cubes are interchangeable. In odd cubes, as well as in the 3x3x3, the pieces at the center of the faces always maintain their relative positions in space. This characteristic simplifies the solution of odd cubes.
In even cubes, however, none of the center pieces maintain their relative positions. Additionally, in both odd and even cubes, not all center pieces are always interchangeable. For example, in odd cubes, the edges located in the center of the sides can only be interchanged with each other.
Figures 1.1 to 1.4 illustrate the interchangeable pieces in the 4x4x4, 5x5x5, 6x6x6, and 7x7x7 cubes, indicated by the same color. Note that the centers of odd cubes are depicted as uncolored (black) since they preserve their relative positions in space and are not interchangeable. It is important to observe that despite rotating their layers, you cannot scramble the cubes because all interchangeable pieces have the same color in Figures 1.1 to 1.4.
It may appear confusing, particularly with larger cubes, but it can be simplified by considering symmetry.
We can categorize the interchangeable pieces into different "families."
For corners, regardless of the cube's size, there is only one family. However, when it comes to edges, all pieces equidistant from the center of the side belong to the same family. In the case of odd cubes, the edges located at the center of the sides also belong to the same family.
The "family" concept of centerpieces is slightly more complex. Let's consider a face of the cube. It belongs to a specific family consisting of four facelets that share a fourfold axis of rotational symmetry. Any other set of facelets with the same configuration on different faces will also belong to this same family.
But what exactly is a fourfold axis of rotational symmetry? Put simply, we say that a set of identical objects, distributed in space, possesses a fourfold axis of rotational symmetry if a 90-degree rotation around this axis results in the same arrangement. Refer to the example depicted in Figure 1.5.
If we were to rotate the set of objects in Figure 1.5 by 90 degrees around the imaginary axis that passes through the center point, we would observe an identical image. By repeating this rotation four times, we complete a full cycle around this axis. Therefore, the set depicted in Figure 1.5 possesses a fourfold axis of rotational symmetry.
Figure 1.5
Below are the families of centerpieces for larger cubes that adhere to this principle.
The 4x4x4 only has 1 family of centerpieces.
The 5x5x5 has 2 families of centerpieces.
The 6x6x6 has 4 families of centerpieces.
The 7x7x7 has 6 families of centerpieces.
The table below illustrates the number of families for each piece type in any given cube.
Key knowledge:
The centers of odd cubes remain fixed in their relative positions and are not interchangeable.
Edges that are equidistant from the center of the sides belong to the same family.
Sets of four centerpieces on faces with a fourfold axis of rotational symmetry form families, along with similar sets on other faces.
It is not necessary to memorize this information. By observing the symmetry of the cube's edges and faces, you can determine which pieces are interchangeable. Understanding these principles will help you solve big cubes without relying on memorized algorithms.
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