The Math of Rubik's Cubes and Related Puzzles

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Course DatesLengthMeeting TimesStatusFormatInstructor(s)CRN
July 26, 2021 - August 11, 20217/26 - 8/112 WeeksOnlineOpenOnlinePatrick Liscio

Course Description

This course is focused on teaching students how to solve Rubik's Cubes and other similar "twisty puzzles." We will work through a series of puzzles, with a focus on developing general methods that can be applied to these, as well as plenty of other puzzles not discussed in the course. By the end, we will have developed some approaches and problem solving techniques that apply far beyond the world of cubing. Along the way, we will discuss the math that goes into making these solutions work. We'll introduce the field of Group Theory, which will allow us to analyze permutations of objects like the pieces on a cube. There will also be chances to talk about some other cool cube-related topics, like how to calculate the number of positions on a cube, and the math and computer algorithms behind finding "God's Number," the number of moves needed to solve the most messed-up cube possible.

The course will have 5 small modules, with each consisting of a math session, in which we cover a topic from group theory, and a puzzle session, in which we work through one or two puzzles that illustrate these concepts. We will first talk about permutations, some of the mathematics behind them, and how to manipulate them. This will be illustrated in two puzzles - the Ivy Cube and the Dino Cube - which can be solved largely through a series of 3-cycles. We will then introduce group theory and the idea of commutation with non-commutative moves. The Pyraminx and its 4x4 variant will show students how to apply these concepts to change the orientation of pieces on a puzzle. We can then discuss conjugation to show how to change one permutation into another. This will allow us to tackle the Rubik's Cube, allowing students to figure out the classic puzzle with techniques that they can figure out themselves. We can then discuss parity, which will allow students to combine their Rubik's Cube knowledge with some other techniques from earlier in the class in order to solve the 5x5 Rubik's cube. Mixed throughout the course will be discussions of other mathematical and computational aspects of puzzles and the Rubik's Cube, as the computation for the number of possible configurations and the work behind "God's number" - the number of turns required to solve the worst possible scramble of a Rubik's Cube.

Students will develop a range of skills and techniques for puzzles and problem-solving. By the end of the course, they should be able to solve many of the puzzles presented, as well as master techniques that they can use to solve other puzzles not presented in the course. Generally, students should gain skills and practice in taking a collection of rules or tools and using them to solve relevant problems. Mathematically, students will gain an understanding of group theory, especially permutations, commutators, and conjugation, which will prepare them for coursework and material in math, computer science, and physics.

Prerequisites: Open to all grades rising 10-12, with 2 years of algebra recommended

Course Information

  • Course Code: CEMA0931

Program Information


Brown’s Pre-College Program in the liberal arts and sciences for students completing grades 9-12 by June 2021.

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