Applied Mathematics is a way to use math to solve problems in the real world, like in economics, engineering, and medicine. 

An investigation was conducted on the Rubik's cube to determine if a relationship exists between the order of a move sequence (the number of times the sequence must be performed on a solved cube for the cube to return to its original state) and the cube's average variegation (degree of disorder).

What happens to a Rubik's Cube when you repeat the same set of moves over and over? The cube scrambles, then eventually returns to its solved state. The number of repeats needed is called the order of the sequence.

You write a computer program in QBASIC to track the cube's average variegation (degree of disorder) after each repetition. When you graph the results, the points fit a 4th-degree polynomial equation.

The data suggest that larger orders produce higher-degree polynomials. Further testing could confirm whether that pattern always holds.

CALIFORNIA STATE SCIENCE FAIR


2005 PROJECT SUMMARY

Project Title

Variegation and Repeated Sequences on the Rubik's Cube
Abstract

Objectives/Goals

An investigation was conducted on the Rubik's cube to determine if a relationship exists between the
order of a move sequence (the number of times the sequence must be performed on
a solved cube for the
cube to return to its original state) and the cube's average variegation (degree
of disorder).

Methods/Materials

A computer program was written in QBASIC to simulate a Rubik's cube and compute
average
variegation. Using this program, data was collected, revealing the average
variegation of a cube as various
sequences were repeated on it.

Results

In sequences with orders small enough to be analyzed, it was discovered that
when variegation was
graphed against the number of repetitions of the sequence, the resulting points
fit a 4th degree polynomial
equation.

Conclusions/Discussion

Based on this and the appearance of the graphs of the remaining sequences, it is
suggested that variegation
during repetition of any given sequence may always change according to a
polynomial expression of
varying degree. The results also suggest that the larger the order of the
sequence, the higher the degree of
the polynomial, although further investigation must be carried out before this
can be proven.

Summary Statement

This project explores the mathematics of how the average variegation of a
Rubik's cube changes as
sequences of moves are repeated on it.

Help Received

Fellow student helped debug QBASIC computer program.

Rubik's Cube Disorder and Repeated Move Sequences

Repeated Sequences and Variegation on the Rubik's Cube

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Math is all about numbers, patterns, and solving problems. Computer science is about using math and technology to create computer programs and solve real-world problems. Together, they help us understand how things work, how to solve problems, and how to use technology to make our lives better!

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Rubik's Cube Disorder and Repeated Move Sequences

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