{"version":"https:\/\/jsonfeed.org\/version\/1","title":"mathspp.com feed","home_page_url":"https:\/\/mathspp.com\/blog\/tags\/group-theory","feed_url":"https:\/\/mathspp.com\/blog\/tags\/group-theory.json","description":"Stay up-to-date with the articles on mathematics and programming that get published to mathspp.com.","author":{"name":"Rodrigo Gir\u00e3o Serr\u00e3o"},"items":[{"title":"Problem #043 \u2013 Rubik's cube scrambling","date_published":"2021-08-22T00:00:00+02:00","id":"https:\/\/mathspp.com\/blog\/problems\/rubiks-cube-scrambling","url":"https:\/\/mathspp.com\/blog\/problems\/rubiks-cube-scrambling","content_html":"

If I scramble a Rubik's cube for long enough,\nwill it solve itself?<\/p>\n\n

\"A
Photo by Serg Antonov on Unsplash.<\/figcaption><\/figure>

Problem statement<\/a><\/h2>\n

A Rubik's cube is a toy like the one you can see in the picture above.\nIt's a 3 by 3 by 3 cube, where each face has one of six colours.\nThe cube can be scrambled, and at that point the colours of the faces\nno longer match.<\/p>\n

The challenge I have for you involves proving something.\nI want you to prove the following statement:<\/p>\n

\n

“If you take a solved Rubik's cube and start scrambling it\nby following a fixed set of steps, you eventually end up with a solved\nRubik's cube again.”<\/p>\n<\/blockquote>\n

A silly example of how this is true is if you start turning the top face.\nYou rotate it once.\nTwice.\nThrice.\nA final turn, and it's back at the initial position!<\/p>\n

But this also applies to more complicated sequences of turns!<\/p>\n

If you have one, go grab a Rubik's cube and give it a go!\n(It's not the same thing, but you can also try this online simulator<\/a>.)<\/p>\n

For your convenience, here is a short GIF of me scrambling\na cube by repeating the same set of steps.<\/p>\n

First, here is an excerpt of the process, showing the set of steps I do repeatedly.\nIf you look closely, you can see that I only do two movements:<\/p>\n

\"\"<\/p>\n

Now, here is a sped up version from start to finish.\n(It took me around 3 minutes total to get back to the beginning\nthrough repeated scrambling!)<\/p>\n

\"\"<\/p>\n

I started out with a solved cube and ended up with a solved cube.<\/p>\n

Why?<\/p>\n

\n

Give it some thought!<\/p>\n<\/div>\n

If you need any clarification whatsoever, feel free to ask in the comment section below.<\/p>\n

Solvers<\/a><\/h2>\n

Congratulations to the ones that solved this problem correctly and, in particular, to the ones\nwho sent me their correct solutions:<\/p>\n