1693 Words7 Pages

Group theory
A major branch of modern discovered by mathematician Evariste Galois, group theory is the mathematical language of symmetry. It is the concept that is most associative with Rubik’s cube due to its invariant symmetry and the interdependence of it components. In mathematics, a “group” is a set of elements and a rule to combine those elements which satisfies certain properties. The elements of a group can consist of numbers, symmetries of a shape, or the elements themselves can be the different ways of rearranging another set of elements. (“Group”)
An important part of understanding group theory, a binary operation is rule for combining elements of a set that will consequent in another element of the same type. For example, addition*…show more content…*

Each permutation symbolizes the result of a single element of the group. Different moves of the cube could correspond to the same final permutation and therefore to seemingly the same exact element of the group. However, a single element of the Rubik’s cube group can be expressed using different sequences of moves, just as a rotation of 4050 is the same as 450. Here, we can utilize the expression “followed by” to represent the operation in this group. Thus, one move followed by another move is itself a move of the Rubik’s cube. The number of elements in the Rubik’s cube group is, as we calculated earlier, approximately 43 quintillion because there are many distinct permutations of the cube that can be reached by legal move (without the disassembly of the cube). Any two legal moves can be combined to result in one of those permutations. To verify that this is a group: - There is an identity element, namely “not performing any move”. “Not performing any move” followed by a move x equals the move*…show more content…*

If the puzzle were commutative, it would not matter what order the move were implemented in, as long as each face was rotated an appropriate number of times. In this way, group theory provides deductive conclusions that would be applicable to all groups. Since the Rubik’s cube is a group, anything that is true for a group in general is true for the Rubik’s cube. (“Galois”) Furthermore, group theory is helpful in deducing certain algorithms. In particular, they are useful in deducing algorithms that possess a commutator structure, namely XYX-1Y-1(X and Y are moves or move sequences and X-1 and Y-1 are their inverses), or a conjugate structure, namely XYX-1 which is usually referred to as a “setup move”.(“Rubik’s Cube”) Both these concepts will be discussed

Each permutation symbolizes the result of a single element of the group. Different moves of the cube could correspond to the same final permutation and therefore to seemingly the same exact element of the group. However, a single element of the Rubik’s cube group can be expressed using different sequences of moves, just as a rotation of 4050 is the same as 450. Here, we can utilize the expression “followed by” to represent the operation in this group. Thus, one move followed by another move is itself a move of the Rubik’s cube. The number of elements in the Rubik’s cube group is, as we calculated earlier, approximately 43 quintillion because there are many distinct permutations of the cube that can be reached by legal move (without the disassembly of the cube). Any two legal moves can be combined to result in one of those permutations. To verify that this is a group: - There is an identity element, namely “not performing any move”. “Not performing any move” followed by a move x equals the move

If the puzzle were commutative, it would not matter what order the move were implemented in, as long as each face was rotated an appropriate number of times. In this way, group theory provides deductive conclusions that would be applicable to all groups. Since the Rubik’s cube is a group, anything that is true for a group in general is true for the Rubik’s cube. (“Galois”) Furthermore, group theory is helpful in deducing certain algorithms. In particular, they are useful in deducing algorithms that possess a commutator structure, namely XYX-1Y-1(X and Y are moves or move sequences and X-1 and Y-1 are their inverses), or a conjugate structure, namely XYX-1 which is usually referred to as a “setup move”.(“Rubik’s Cube”) Both these concepts will be discussed

Related

## Reflection On Trigonometry

1035 Words | 5 PagesA tangent graph is a little different from the first two. This graph has a period of π, no amplitude, and the zeros are multiples of π. This graph also has a vertical asymptote that is at π/2. The domain equals all real numbers except for the vertical asymptote and the range is from (-infinity, +infinity). The next three graphs are the reciprocal of the first three.

## Four Stages Of Data Mining

922 Words | 4 PagesThe algorithm iterates to find the best rules for the different classes. A context-free grammar is used to specify which relational operators are allowed to appear in the antecedents of the rules and which attribute must appear in the consequents or the class. The use of a grammar provides expressiveness, flexibility, and ability to restrict the search space in the search for rules. The implementation of constraints using a grammar is a very natural way to express the syntax of rules. The rules must be constructed

## Mathematical Assignment: Mathematical Exploration: Sudoku

1066 Words | 5 PagesPattern recognition and logical deductions are also used in Sudoku. Counting

## The Loh-Hi's Magic Square

1236 Words | 5 PagesSince the magic square is a square, all sides are equal or there are the same number of rows and columns. The number of rows or columns is call the order which you can represent as the variable n. The number of small squares or the number of numbers in the magic square would equal n squared. N squared also means n multiplied by itself once showing how the rows and columns are multiplied to find the amount of numbers used in the square. A normal magic square would have the numbers 1 to n squared. Mathematicians figured out the magic sums by using a formula including variables that can be used to find every magic constant in every magic square.

## Bertrand Russell's The Theory Of Knowledge

929 Words | 4 PagesThe theory of types created a logical foundation for mathematics as presented by Russell in his 1908 paper Mathematical Logic as Based on the Theory of Types. Russell wrote his type theory in response to ambiguities he found in set theory. “Naive Set Theory”, as we now call it, defined sets vaguely and paradoxically allowed any variable to become a set without restriction. This confusion led Russell to publish his paradox explaining that no set consists of "all sets that do not contain themselves". Russell developed his type theory to be more precise than naive set theory.

## George Bole's Theory

700 Words | 3 PagesThe rules of Boolean algebra can be used to determine the consequences of the various combinations of these propositions depending on whether the statements are true or false. Two elements of Boolean algebra makes it a very important mathematical form for practical application. First, the propositions expressed in everyday language (like "I 'll be at home today") can become math expressions such as letters and numbers. Second, those symbols generally have one or two values: the propositions may be affirmative or negative. Operation can be a conjunction or a disjunction; and not only propositions, but also the result of combinations, are true or false.

## The Pros And Cons Of Sudoku

987 Words | 4 PagesThere are various approaches introduced to solve the Sudoku game till now. The solutions in the state of art are computationally complex as the algorithmic complexity for the solution falls under the class of NP-Complete. The paper discusses various solutions for the problem along with their pros and cons. The paper also discusses the performance of a serial version solution with respect to execution

## M. C. Escher Tessellation Essay

432 Words | 2 PagesEscher, honeycombs have something in common: they are consisted with repeating patterns of the same shape without any overlaps or gaps. This type of pattern is called tiling, or tessellation. “The word "tessellate" means to form or arrange small squares in a checkered or mosaic pattern”, according to Drexel University. Its Greek tesserae, a Greek word for "four." The first set of tilings were square tiles.

## Examples Of Inferential Statistics

793 Words | 4 PagesUnder inferential statistics, reliability analysis and a logistic regression has been done. Descriptive statistics are used to represent analyzed data in a meaningful and a clear way. 4.2 Reliability Test. Reliability analysis for this research allowed studying the properties of measurements and scales and the items that compose scales. Reliability analysis calculates number of commonly

## Proving De Moivre's Theorem Using Mathematical Induction

1396 Words | 6 PagesIt might not be Pythagoras theorem or Newton’s laws of motion. Nevertheless, this theorem is applied to complex numbers mostly whenever they are used. It may appear that complex numbers do not have a very straight application in real life mathematics; however, they can always be used to model a phenomenon. If this phenomenon is periodic and it is modeled with a complex number, De Moivre’s theorem would provide the solution to change the frequency of our model. In other words, this theorem would be the key to solve an ultimate issue with models, as we would use the exponent to do so.

### Reflection On Trigonometry

1035 Words | 5 Pages### Four Stages Of Data Mining

922 Words | 4 Pages### Mathematical Assignment: Mathematical Exploration: Sudoku

1066 Words | 5 Pages### The Loh-Hi's Magic Square

1236 Words | 5 Pages### Bertrand Russell's The Theory Of Knowledge

929 Words | 4 Pages### George Bole's Theory

700 Words | 3 Pages### The Pros And Cons Of Sudoku

987 Words | 4 Pages### M. C. Escher Tessellation Essay

432 Words | 2 Pages### Examples Of Inferential Statistics

793 Words | 4 Pages### Proving De Moivre's Theorem Using Mathematical Induction

1396 Words | 6 Pages