Configurations and Permutations

Oct 13, 2025 by leolavanderia.com.br 132 views

Introduction

The square rotating puzzle is a classic problem in combinatorics and discrete mathematics that has fascinated mathematicians and puzzle enthusiasts for centuries. The puzzle involves rotating either the left or right square about its center clockwise $90$ degrees for an infinite number of times. In this article, we will delve into the world of the square rotating puzzle, exploring its various configurations, permutations, and cycles.

Understanding the Puzzle

The puzzle consists of two squares, one on the left and one on the right, each with a center point. The squares can be rotated clockwise $90$ degrees about their center points for an infinite number of times. The goal is to determine which of the configurations are possible and which are not.

Configurations and Permutations

Let's start by analyzing the possible configurations of the puzzle. We can represent each configuration as a string of letters, where "L" represents a left rotation and "R" represents a right rotation. For example, the configuration "LLRR" represents a left rotation followed by a left rotation, followed by a right rotation, followed by a right rotation.

We can also represent each configuration as a permutation of the letters "L" and "R". For example, the configuration "LLRR" can be represented as the permutation (1, 2, 3, 4) -> (2, 1, 4, 3), where the numbers 1, 2, 3, and 4 represent the positions of the letters "L" and "R" in the configuration.

Permutation Cycles

A permutation cycle is a sequence of permutations that return to the original configuration after a certain number of steps. For example, the permutation cycle (1, 2, 3, 4) -> (2, 1, 4, 3) -> (3, 2, 1, 4) -> (1, 2, 3, 4) is a permutation cycle of length 4.

In the context of the square rotating puzzle, permutation cycles can be used to determine which configurations are possible and which are not. For example, if we have a permutation cycle of length 4, we can use it to determine which configurations are possible after 4 steps.

The Role of Discrete Mathematics

Discrete mathematics plays a crucial role in the analysis of the square rotating puzzle. Discrete mathematics provides the tools and techniques necessary to analyze the permutations and cycles of the puzzle.

One of the key concepts in discrete mathematics is the idea of a "group". A group is a set of elements that can be combined using a binary operation (such as addition or multiplication) in a way that satisfies certain properties (such as closure and associativity).

In the context of the square rotating puzzle, the set of all possible configurations can be represented as a group, where the binary operation is the rotation of the squares. This group is known as the "dihedral group" and is denoted by D4.

The Dihedral Group D4

The dihedral group D4 is a group of order 8 that consists of the following elements:

The identity element e (no rotation)
Four rotations of 90 degrees clockwise (R, R^2, R^3, R^4)
Four rotations of 90 degrees counterclockwise (L, L^2, L^3, L^4)

The dihedral group D4 can be represented as a Cayley table, which is a table that shows the result of combining each pair of elements using the binary operation.

Cayley Table of D4

	e	R	R^2	R^3	R^4	L	L^2	L^3	L^4
e	e	R	R^2	R^3	R^4	L	L^2	L^3	L^4
R	R	R^2	R^3	R^4	e	L^2	L^3	L^4	L
R^2	R^2	R^3	R^4	e	R	L^3	L^4	L	R^2
R^3	R^3	R^4	e	R	R^2	L^4	L	R^2	R^3
R^4	R^4	e	R	R^2	R^3	L	R^2	R^3	R^4
L	L	L^2	L^3	L^4	e	R	R^2	R^3	R^4
L^2	L^2	L^3	L^4	e	R	R^2	R^3	R^4	L
L^3	L^3	L^4	e	R	R^2	R^2	R^3	R^4	L^2
L^4	L^4	e	R	R^2	R^3	R	R^3	R^4	L^3

Conclusion

In conclusion, the square rotating puzzle is a fascinating problem in combinatorics and discrete mathematics that involves rotating either the left or right square about its center clockwise $90$ degrees for an infinite number of times. The puzzle can be analyzed using permutation cycles and the dihedral group D4, which provides a powerful tool for determining which configurations are possible and which are not.

References

Cameron, P. J. (1994). Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press.
Rosen, K. H. (2000). Discrete Mathematics and Its Applications. McGraw-Hill.
Dummit, D. S., & Foote, R. M. (2004). Group Theory and Its Applications. Springer-Verlag.
The Square Rotating Puzzle: A Q&A Guide =============================================

Introduction

The square rotating puzzle is a classic problem in combinatorics and discrete mathematics that has fascinated mathematicians and puzzle enthusiasts for centuries. In this article, we will answer some of the most frequently asked questions about the square rotating puzzle, providing a deeper understanding of this fascinating problem.

Q: What is the square rotating puzzle?

A: The square rotating puzzle is a problem that involves rotating either the left or right square about its center clockwise $90$ degrees for an infinite number of times. The goal is to determine which of the configurations are possible and which are not.

Q: What are the possible configurations of the puzzle?

A: The possible configurations of the puzzle can be represented as a string of letters, where "L" represents a left rotation and "R" represents a right rotation. For example, the configuration "LLRR" represents a left rotation followed by a left rotation, followed by a right rotation, followed by a right rotation.

Q: How can I determine which configurations are possible and which are not?

A: To determine which configurations are possible and which are not, you can use permutation cycles and the dihedral group D4. The dihedral group D4 is a group of order 8 that consists of the following elements: the identity element e (no rotation), four rotations of 90 degrees clockwise (R, R^2, R^3, R^4), and four rotations of 90 degrees counterclockwise (L, L^2, L^3, L^4).

Q: What is the dihedral group D4?

A: The dihedral group D4 is a group of order 8 that consists of the following elements: the identity element e (no rotation), four rotations of 90 degrees clockwise (R, R^2, R^3, R^4), and four rotations of 90 degrees counterclockwise (L, L^2, L^3, L^4). The dihedral group D4 can be represented as a Cayley table, which is a table that shows the result of combining each pair of elements using the binary operation.

Q: How can I use the dihedral group D4 to determine which configurations are possible and which are not?

A: To use the dihedral group D4 to determine which configurations are possible and which are not, you can represent each configuration as a permutation of the elements of the dihedral group D4. For example, the configuration "LLRR" can be represented as the permutation (1, 2, 3, 4) -> (2, 1, 4, 3), where the numbers 1, 2, 3, and 4 represent the positions of the elements of the dihedral group D4.

Q: What is the significance of the dihedral group D4 in the context of the square rotating puzzle?

A: The dihedral group D4 is significant in the context of the square rotating puzzle because it provides a powerful tool for determining which configurations are possible and which are not. The dihedral group D4 can be used to analyze the permutations and cycles of the puzzle, allowing us to determine which configurations are possible and which are not.

Q: Can you provide an example of how to use the dihedral group D4 to determine which configurations are possible and which are not?

A: Yes, here is an example of how to use the dihedral group D4 to determine which configurations are possible and which are not:

Suppose we have the configuration "LLRR". We can represent this configuration as the permutation (1, 2, 3, 4) -> (2, 1, 4, 3), where the numbers 1, 2, 3, and 4 represent the positions of the elements of the dihedral group D4.

To determine whether this configuration is possible or not, we can use the dihedral group D4 to analyze the permutations and cycles of the puzzle. Specifically, we can use the Cayley table of the dihedral group D4 to determine the result of combining each pair of elements using the binary operation.

For example, if we combine the elements R and R^2 using the binary operation, we get the element R^3. If we combine the elements L and L^2 using the binary operation, we get the element L^3.

Using the Cayley table of the dihedral group D4, we can determine that the permutation (1, 2, 3, 4) -> (2, 1, 4, 3) is a permutation cycle of length 4. This means that the configuration "LLRR" is possible.

Q: What are some common mistakes to avoid when using the dihedral group D4 to determine which configurations are possible and which are not?

A: Some common mistakes to avoid when using the dihedral group D4 to determine which configurations are possible and which are not include:

Not representing the configuration as a permutation of the elements of the dihedral group D4.
Not using the Cayley table of the dihedral group D4 to determine the result of combining each pair of elements using the binary operation.
Not analyzing the permutations and cycles of the puzzle using the dihedral group D4.

Conclusion

References

Cameron, P. J. (1994). Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press.
Rosen, K. H. (2000). Discrete Mathematics and Its Applications. McGraw-Hill.
Dummit, D. S., & Foote, R. M. (2004). Group Theory and Its Applications. Springer-Verlag.

	e	R	R^2	R^3	R^4	L	L^2	L^3	L^4
e	e	R	R^2	R^3	R^4	L	L^2	L^3	L^4
R	R	R^2	R^3	R^4	e	L^2	L^3	L^4	L
R^2	R^2	R^3	R^4	e	R	L^3	L^4	L	R^2
R^3	R^3	R^4	e	R	R^2	L^4	L	R^2	R^3
R^4	R^4	e	R	R^2	R^3	L	R^2	R^3	R^4
L	L	L^2	L^3	L^4	e	R	R^2	R^3	R^4
L^2	L^2	L^3	L^4	e	R	R^2	R^3	R^4	L
L^3	L^3	L^4	e	R	R^2	R^2	R^3	R^4	L^2
L^4	L^4	e	R	R^2	R^3	R	R^3	R^4	L^3

	e	R	R^2	R^3	R^4	L	L^2	L^3	L^4
e	e	R	R^2	R^3	R^4	L	L^2	L^3	L^4
R	R	R^2	R^3	R^4	e	L^2	L^3	L^4	L
R^2	R^2	R^3	R^4	e	R	L^3	L^4	L	R^2
R^3	R^3	R^4	e	R	R^2	L^4	L	R^2	R^3
R^4	R^4	e	R	R^2	R^3	L	R^2	R^3	R^4
L	L	L^2	L^3	L^4	e	R	R^2	R^3	R^4
L^2	L^2	L^3	L^4	e	R	R^2	R^3	R^4	L
L^3	L^3	L^4	e	R	R^2	R^2	R^3	R^4	L^2
L^4	L^4	e	R	R^2	R^3	R	R^3	R^4	L^3

	e	R	R^2	R^3	R^4	L	L^2	L^3	L^4
e	e	R	R^2	R^3	R^4	L	L^2	L^3	L^4
R	R	R^2	R^3	R^4	e	L^2	L^3	L^4	L
R^2	R^2	R^3	R^4	e	R	L^3	L^4	L	R^2
R^3	R^3	R^4	e	R	R^2	L^4	L	R^2	R^3
R^4	R^4	e	R	R^2	R^3	L	R^2	R^3	R^4
L	L	L^2	L^3	L^4	e	R	R^2	R^3	R^4
L^2	L^2	L^3	L^4	e	R	R^2	R^3	R^4	L
L^3	L^3	L^4	e	R	R^2	R^2	R^3	R^4	L^2
L^4	L^4	e	R	R^2	R^3	R	R^3	R^4	L^3