This is also how people can solve Rubik's Cubes blindfolded: use algorithms that only move a couple pieces without changing the rest of the state. It's suddenly way easier to keep track the state when only a few pieces are moving at a time.
When I was a speedcuber I could use three algorithms to solve a cube blindfolded after memorizing where each piece needed to go: one to flip two edges, one to rotate two corners, and one that switched two corners and two edges at the same time (T-perm for you cubers).
Then it was just 1. orient the edges and corners in a way that makes them easy to move around the cube and 2. move the pieces where they need to go.
This is a very rudimentary strategy, and there are MUCH faster ways to solve the cube, but this is all you need.
A commutator is any sequence of moves in the form of A B A' B', where A and B are sets of moves, and A' and B' are those sets of moves undone. So this example basically just restricts B to consist only of top-layer moves.
Terry Tao has a really great post where he depicts group actions (every Rubik's cube move sequence is a group action) as arrows in a graph, and with this picture, commutators can be seen as arrows that make a relatively small change: https://terrytao.wordpress.com/2012/05/11/cayley-graphs-and-... (see figure 14).
If you want to take the arrows very literally, you can think of the commutator as measuring the amount of curvature in the space the arrows lie in. The less of a difference the commutator makes, the less curved - with a commutator that doesn't do anything meaning the space is flat. Expressed in these terms, the two-part algorithm technique amounts to finding places in Rubik's cube algorithm space that are almost – but not quite – flat.
This is how one of my professors described it to me back in college, inspiring me to derive my own solution to the Rubik's cube. (And later for the "Square One" cube.) And I think the general concept is easy to describe to anyone: do something to the top and mess up the bottom, rotate the top and then do the exact opposite.
To me, it's much more satisfying than memorizing/learning existing solutions, even though I'll never be as fast as those that do leverage those solutions.
Either way, though, I'd recommend buying an inexpensive "speed cube" from Amazon, they are much less frustrating than the original design. (It doesn't bind up in the middle of a transformation.)
I recently re-visited the rubiks cube as a holiday project.
It really is quite accessible now with online guides and videos. I think about an hour a day for 5 days took me from never having solved a cube to being able to solve any starting configuration in a little less than 3 minutes.
If like me you ever had a cube you never beat as a kid, it is definitely worth revisiting.
Took me 5 days as well, though I might have spent more than an hour on it daily. I stepped away from the cube for about 4 years and was able to pick it up again just from muscle memory.
I now use the cube as a meditative practice. For 1-3 minutes, nothing in the world matters with the exception of solving the cube. Doesn't work well as a meditation tool when you start doing other cubes however, like the megminx. It just becomes a frustration device.
If you're solving using commutators now and are looking to upgrade to a faster method in the same spirit, check out the Heise method. It's also an intuitive method (no memorization required), which starts off with block-building and finishes using commutators. I made the transition from commutator-only to Heise and am enjoying it, and I'm still very far off the speed cap for Heise. (Its creator reports averaging ~30sec: http://twistypuzzles.com/~sandy/forum/viewtopic.php?p=45076&...)
One thing I recently realized about the cube. In a simulation, or in reality, the configuration of the cube can be completely described by the orientation of the 20 movable pieces. The location of a piece is actually determined by it's orientation.
For example, if you wanted to make a 3d model you might make a bunch of little cubes each centered in their own object space and then translate them to their location in the cube. If you do that, you have to track the orientation AND location of each piece. However, if you center the entire cube at the origin and then place each piece in its place relative to that origin, all "moves" simply rotate a piece around some axis which both changes their orientation AND moves them relative to the cube center. As such, position is redundant information.
I'm not sure how relevant this is, but to me it seems to point to alternative ways of finding a solution via computer.
But this is probably very old news to people who study the cube.
The book Winning Ways for Your Mathematical Plays[1] has three main algorithms that are in the vein of this basic idea. They were actually optimized for ease of remembering and executing on. With full documented solutions for moving specific cubes of the final layer. Really fun read.
I set a Scala assignment last year that was implementing a "human" cube-solver (ie, write a cube solver that searches for these sorts of moves, and then uses them to solve the cube the way people are taught to)
Fun problem; terrible assignment. I had to scaffold it so much that I was essentially giving most of the solution.
This looks very similar to the method I used back in high school, and the fastest I ever got with it was 66 seconds, which is abysmal by speed-cubing standards but it impressed my friends and family.
[+] [-] patejam|9 years ago|reply
When I was a speedcuber I could use three algorithms to solve a cube blindfolded after memorizing where each piece needed to go: one to flip two edges, one to rotate two corners, and one that switched two corners and two edges at the same time (T-perm for you cubers).
Then it was just 1. orient the edges and corners in a way that makes them easy to move around the cube and 2. move the pieces where they need to go.
This is a very rudimentary strategy, and there are MUCH faster ways to solve the cube, but this is all you need.
[+] [-] logicallee|9 years ago|reply
[+] [-] Orangeair|9 years ago|reply
http://w.astro.berkeley.edu/~converse/rubiks.php?id1=basics&...
A commutator is any sequence of moves in the form of A B A' B', where A and B are sets of moves, and A' and B' are those sets of moves undone. So this example basically just restricts B to consist only of top-layer moves.
[+] [-] adeledeweylopez|9 years ago|reply
If you want to take the arrows very literally, you can think of the commutator as measuring the amount of curvature in the space the arrows lie in. The less of a difference the commutator makes, the less curved - with a commutator that doesn't do anything meaning the space is flat. Expressed in these terms, the two-part algorithm technique amounts to finding places in Rubik's cube algorithm space that are almost – but not quite – flat.
[+] [-] dunham|9 years ago|reply
To me, it's much more satisfying than memorizing/learning existing solutions, even though I'll never be as fast as those that do leverage those solutions.
Either way, though, I'd recommend buying an inexpensive "speed cube" from Amazon, they are much less frustrating than the original design. (It doesn't bind up in the middle of a transformation.)
[+] [-] alex-|9 years ago|reply
It really is quite accessible now with online guides and videos. I think about an hour a day for 5 days took me from never having solved a cube to being able to solve any starting configuration in a little less than 3 minutes.
If like me you ever had a cube you never beat as a kid, it is definitely worth revisiting.
[+] [-] TheOneTrueKyle|9 years ago|reply
I now use the cube as a meditative practice. For 1-3 minutes, nothing in the world matters with the exception of solving the cube. Doesn't work well as a meditation tool when you start doing other cubes however, like the megminx. It just becomes a frustration device.
[+] [-] blueblimp|9 years ago|reply
If you're solving using commutators now and are looking to upgrade to a faster method in the same spirit, check out the Heise method. It's also an intuitive method (no memorization required), which starts off with block-building and finishes using commutators. I made the transition from commutator-only to Heise and am enjoying it, and I'm still very far off the speed cap for Heise. (Its creator reports averaging ~30sec: http://twistypuzzles.com/~sandy/forum/viewtopic.php?p=45076&...)
[+] [-] phkahler|9 years ago|reply
For example, if you wanted to make a 3d model you might make a bunch of little cubes each centered in their own object space and then translate them to their location in the cube. If you do that, you have to track the orientation AND location of each piece. However, if you center the entire cube at the origin and then place each piece in its place relative to that origin, all "moves" simply rotate a piece around some axis which both changes their orientation AND moves them relative to the cube center. As such, position is redundant information.
I'm not sure how relevant this is, but to me it seems to point to alternative ways of finding a solution via computer.
But this is probably very old news to people who study the cube.
[+] [-] taeric|9 years ago|reply
[1] https://smile.amazon.com/Winning-Ways-Your-Mathematical-Play...
[+] [-] waqf|9 years ago|reply
[+] [-] uberstuber|9 years ago|reply
http://www.ryanheise.com/cube/
[+] [-] wbillingsley|9 years ago|reply
Fun problem; terrible assignment. I had to scaffold it so much that I was essentially giving most of the solution.
[+] [-] guipsp|9 years ago|reply
[+] [-] iamatworknow|9 years ago|reply
[+] [-] TheOneTrueKyle|9 years ago|reply
[deleted]
[+] [-] unknown|9 years ago|reply
[deleted]