The mathematics of Rubik’s cube - Solutions and group-theory ideas

Sep 22, 2026

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The talk explains the mathematical structure behind Rubik’s Cube, starting from the cube’s pieces and the orientation/placement goal. It analyzes counting questions by treating edge and corner cubies as permutations with independent orientation choices, then frames solving as a probability problem. It outlines a practical solving strategy and introduces key group-theory concepts—group structure, commutators, conjugation—along with standard cube move notation and inverses, emphasizing non-commutativity.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages35

Reading Time1 h 28 min

AccessFree to download

Elements of Rubik’s Cube

Counting ways to disassemble and reassemble

Procedure for solving the cube

Goal of the talk

Group-theory framework

Commutators and conjugation

Notation and move operations

Combining moves

Inverses and non-commutativity

The mathematics of Rubik’s Cube Michael Hutchings Department of Mathematics University of California, Berkeley BAMC Colloquium UC Berkeley December 11, 2022 Michael Hutchings (UC Berkeley) The mathematics of Rubik’s cube Bay Area Math Circle Colloquium 1 / 35 Elements of Rubik’s cube The cube contains: six center pieces which do not move relative to each other. twelve edge “cubies” each with two stickers on them. eight corner “cubies” each with three stickers on them. The goal of the puzzle is to move all of the edge cubies and corner cubies into their correct positions with their correct orientations. “It was a big challenge for me. Back then there were no solution manuals or YouTube tutorials, and I was on my own... I knew that a solution existed in theory, but I wasn’t sure if one could actually solve the cube in practice.... It was hard and took me a long time – several weeks, that’s for sure.” —Ern Rubik on solving the cube for the first time (Der Spiegel, 2010) Michael Hutchings (UC Berkeley) The mathematics of Rubik’s cube Bay Area Math Circle Colloquium 2 / 35 Warmup questions 1 If you disassemble the cube, how many ways are there toreassemble it (placing the cubies in any positions and orientations)? 2 If you randomly reassemble the cube, what is the probability that it can be solved (without disassembling it again)? Michael Hutchings (UC Berkeley) The mathematics of Rubik’s cube Bay Area Math Circle Colloquium 3 / 35 Answers 1 If you disassemble the cube, how many ways are there toreassemble it (placing the cubies in any positions and orientations)? 12! ways to permute the edge cubies 8! ways to permute the corner cubies 212 ways to orient the edge cubies 38 ways to orient the corner cubies Answer = 12!8!212 38 = 519 , 024 , 039 , 293 , 878 , 272 , 000 2 If you randomly reassemble the cube, what is the probability that it can be solved (without disassembling it again)? Answer = 1/12 Proof??? Michael Hutchings (UC Berkeley) The mathematics of Rubik’s cube Bay Area Math Circle Colloquium 4 / 35 Procedure for solving the cube A typical cube solving procedure: 1 Solve the lower two layers of the cube. (There are various clever tricks to speed this up.) 2 Orient the cubies in the top layer, so that the top face of the cube is the correct color. 3 Permute (i.e. interchange) the cubies in the top layer so that they are in their correct positions. (To do this fast requires memorizing many shortcuts for dealing with various cases.) Fun fact: The cube can always be solved using at most 20 face turns (proved by computer) . (Human solves usually need 50-60 turns.) Michael Hutchings (UC Berkeley) The mathematics of Rubik’s cube Bay Area Math Circle Colloquium 5 / 35 Goal of this talk Our goal is to introduce some mathematical ideas with which one can solve Rubik’s cube (and other similar puzzles such as larger cubes and other shapes), not so fast, but understanding each step. The moves that one can perform on Rubik’s cube form a mathematical structure called a group. One can solve Rubik’s cube using two basic ideas from group theory: commutators and conjugation. Michael Hutchings (UC Berkeley) The mathematics of Rubik’s cube Bay Area Math Circle Colloquium 6 / 35 Some notation Basic moves on Rubik’s cube: F = rotate the front face one quarter turn clockwise B = rotate the back face one quarter turn clockwise U = rotate the top face one quarter turn clockwise D = rotate the bottom face one quarter turn clockwise R = rotate the right face one quarter turn clockwise L = rotate the left face one quarter turn clockwise Additional moves (which move the center pieces): Fs = rotate a vertical “slice” one quarter turn clockwise as viewed from the front. Us = rotate a horizontal “slice” one quarter turn clockwise as viewed from above. Similarly Bs, Ds, Rs, Ls . Michael Hutchings (UC Berkeley) The mathematics of R

FAQs

How many ways are there to disassemble and reassemble a Rubik’s Cube?

The talk separates the problem into permutations and orientations: 12! ways to permute edge cubies, 8! ways to permute corner cubies, 2^12 ways to orient edge cubies, and 3^8 ways to orient corner cubies (with the provided computed product as the final number).

What is the probability that a randomly reassembled cube is solvable without disassembling it again?

It gives an answer of 1/12 and labels the proof as “Proof???”, indicating the exact reasoning is not fully provided in the visible text.

What mathematical ideas are used to understand solving steps for Rubik’s Cube?

Moves form a group, and the talk states that commutators and conjugation from group theory provide the two basic ideas used to solve the cube while understanding each step.

Sep 22, 2026

Upload by Fez