# Books on Abstract Algebra and its applications to solve puzzles

I am currently reading some introductory textbooks on abstract algebra ("Contemporary Abstract Algebra" by Gallian and "A book on Abstract Algebra" by Pinter). I am learning the concepts from these books but I am not yet confident about using them in solving real-world problems. I know that Abstract Algebra can be used to solve a lot of interesting puzzles( For example, 15-puzzle, Rubik's cube, etc). Are there any books or websites that includes usage of abstract algebra framework to solve some puzzles?

• – Shaun Sep 12 '18 at 16:48
• Are there any books or websites that . I stopped reading here because the answer to your question is to use a search engine. Any search engine. Why do you need to try to fire up a bunch of meat-search-engines to get highly incomplete and limited responses? – rschwieb Sep 12 '18 at 17:16

He outlines how you might set up the problems of describing the configuration space of the Rubik's cube and other puzzles. However, I suspect more thorough discussions might be open.

Wikipedia has the Rubik's cube group as $G=(\mathbb{Z}_2^7\times \mathbb{Z}_3^{11})\ltimes ((A_8 \times A_{12})\ltimes \mathbb{Z}_2)$. Can you think of an element of order $3$ in that group? All the twists, $\{F,B,U,L,D,R\}$ have order $4$. Perhaps there's a sequence where when you repreat it three times, preserves the identity?

Look at any speed-cubing page. The algorithms are slow much less than God's number of 20. What local features could allow you to speed up?

Rubiks.com's own solution page. Can you build your own sequence that preserves the top row? Top two rows? Corner swapping?

These kinds of questions may still have not been attacked theoretically.

• Thank you! "Adventures in Group Theory" sounds like the book I am looking for :) – Suhan Shetty Sep 13 '18 at 6:03

In this forum thread, several links to books (or PDFs) about twisty puzzles are provided. In the first few posts are papers (or books) which most people who searched for this genre know about. However in later posts (specifically, mine), I provided links to other works. For example, one of the best ones is Permutation Puzzles: A Mathematical Perspective written by Dr. Jamie Mulholland from SFU university.