Explanation Definition: The binary operator * is a sequence of cube’s move or rotation of a cube face. Eliminate the symbol * . For example, here interpret F B as F * B . This operation has inverse property. Group condition of Rubik’s cube: A group G consists of a set of objects and the binary operator * on those objects satisfying the following conditions. Closure property: The operation * is closed, so any elements h and g in G, h * g is also in G. Associative property: The operation * is associative, so any elements f, g and h in G, then ( f * g ) * h = f * ( g * h ) . Identity property : There is an identity element e ∈ G such that e * g = g * e = g . Inverse property : Every element in G has an inverse g − 1 relative to the operation * such that g * g − 1 = g − 1 * g = e . Commutative property: The set of n × n non-singular matrices from a group under multiplication is an example of non-commutative group or non-abelian group and it is also known as Rubik group. Approach: Here, R, F, L, B, T and U denotes the rotation of the Rubik’s cube 90 ∘ clockwise to the right, front, left, back, top, and under faces, respectively. The 90 ∘ clockwise rotation position of Rubik’s cube of R, F, L, B, T and U is given below. Convert the Rubik’s cube into standard position by making the given rotation in reverse order and reverse direction. Calculation: Prove that each move on the Rubik’s cube has an inverse move by using the definition mentioned above. Every element in G has an inverse g − 1 . That is, g * g − 1 = g − 1 * g = e

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ISBN: 9780357127193

Author: SMITH

Publisher: CENGAGE L

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Chapter 12.4, Problem 54PS

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Each move on the Rubik’s cube has an inverse move.

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Chapter 12.4, Problem 53PS

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Each move on the Rubik’s cube has an inverse move.

Solution Summary: The author explains that each move on the Rubik's cube has an inverse move.

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To find:

Each move on the Rubik’s cube has an inverse move.