On the adaptability of algorithms and the use of commutators

Introduction

You may have come across immense algorithms for certain puzzles which are so difficult to wrap your head around, you don't even consider learning them by heart. These (in many cases computer-generated) algorithms don't always show their structure and you aren't always sure why it does what it's supposed to do.

In this article, I'll talk about the adaptability of algorithms and the use of commutators to invent your own algorithms. In my opinion; something you've reasoned your way through is better remembered than something you've tried to memorize.

Algorithms

Anatomy

When you look up algorithms for certain cases, you may recognize some of its internal structure. For example: if you know some speedcubing PLL algorithms, you might have noticed that the F-perm is actually the T-perm with a set-up move. If you know what the T-perm does, than it's easy to see what the F-perm is for.

Another example is the opposite edge swap algorithm (R2 U2 R2 U2 R2 U2) which is just a modification of a Sune (R U R' U R U2 R') which also deals with edges.

Algorithms often consist of commutators and set-up moves. I'll talk about the use of commutators in algorithms in the next paragraph. Once you're able to recognize them, it can be easy to analyse the whole algorithm and understand precisely what it does and more importantly: why. Once you've mastered this aspect, making your own algorithms is just a walk in the park.

Adaptability

Simple algorithms like the Sune and the edge swaps (opposite and adjacent) can be applied to a whole branch of puzzles without any modification. However, some algorithms must be adapted to the kind of puzzle your dealing with. This adaptation can be a whole series of set-up moves, or just one turn extra.

A simple example: the algorithm which shuffles around corner pieces (U R U' L' U R' U' L) on a NxN cube, is easily adapted to a minx-type of puzzle by changing the U/U' into U2/U'2. This algorithm, keeping the modification in mind, has the same effect on a 12-sided puzzle. The same counts for the Sune and many more.

These are just small examples of simple well-known algorithms, but this stretches to more complicated longer algorithms too. The infamous OLL-parity algorithm of the 4x4, also known as the Redbull algorithm (R2 B2 U2 L U2 R' U2 R U2 F2 R F2 L' B2 L2), can be applied on almost every type of cuboid (with or without set-up moves, depending on the puzzle). This algorithm has even different effects when executed with different layers as R's, L's etc.

I believe this is an algorithm worth memorizing because it can be applied in so many cases and on so many puzzles.

Try for yourself if your favorite algorithms apply on different kinds of puzzles, you'll be amazed. Changing some moves to take the characteristics of the puzzle into account, or coming up with a set-up move, makes you remember just a handful of algorithms for your whole collection of puzzles instead of learning a specific method for each individual puzzle by heart.

Commutators

Commutators are a series of a handful of moves (often 4 turns). They affect a very specific part of the puzzle, which makes it useful to see precisely what it does and which pieces are left in peace. I'd like to call these couple of affected pieces the area of effect.

One of the most-used ones is F R' F' R (or down-down-up-up). This cycles around three specific edges and swaps two sets of two corners on a regular NxN cube. All the other pieces do not take part in the commutator

There are other well-known examples of commutators, but you really don't have to bother with them at all. They are very useful and adaptable indeed, but any set of moves which include just 2 faces can easily be used as a commutator.

The invention of new algorithms

Once you know how commutators work and how they can be useful, the next step to make algorithms for specific puzzles is not far away.

When a very specific part of the puzzle is affected by a commutator, it's easy to swap pieces of the puzzle. This can be done by replacing an affected piece by an unaffected piece. This way, you know that the specific piece you've chosen is now part of the area of effect (therefore, it will be affected) and the piece you've taken out, will be left in peace. I'd like to call this move which replaces just one specific piece in the area of effect, the middle move.

After replacing a piece, you can undo the commutator by doing its moves backwards. Now you can put the piece you've taken out back into the position where it left the area of effect and you'll see that a series of pieces has been swapped!

Let's discuss this strategy with a simple example on a regular 3x3 with the F R' F' R commutator:

1. After doing the 4-move commutator, you'll see that three edges have cycled around. There are also four corners in the area of effect, but I'll discuss them later.
2. Now we want to get 1 edge out of the area of effect. We can achieve this by doing, for example, an M-move. This replaces one of the affected edges with an edge which is not affected.
   Now we know two things: (1) the piece we took out won't be affected if we do the commutator again and (2) the edge we've put in will be affected.
3. Doing the commutator again backwards, R' F R F', and doing the middle move backards, M', will result in a nice 3-cycle of edges!

We could have chosen another middle move, for example L. This doesn't take out an edge, but a corner. Going over the same steps with this middle move, results in a neat 3-cycle of corners.

In general, an algorithm made using this strategy looks like the following:

comm. --- middle move --- undo comm. --- undo middle move

These algorithms may look useless on a 3x3, because I believe everyone who reads this article already has a favorite method for solving a 3x3. Nevertheless, I came up with a strategy to solve a 3x3 face by face, instead of layer by layer, with algorithms I invented using this approach. This way of thinking, however, can be applied on any puzzle!

I've solved my Tuttminx (a 32-sided puzzle) by using nothing more than commutators and middle moves.

Another personal example: the Octo Star Cube is a puzzle which was almost unique in its kind. I bought it specifically because I couldn't compare it to any other puzzle I own and I was ought to come up with my own new method. Using commutators, it has become one of my easiest puzzles!

Commutators are a very strong tool when it comes to puzzles you solve for the first time because they affect just a handful of pieces and you can easily see and analyse what it does to the puzzle.

Try for yourself! What are the characteristics of your puzzle and which moves are unique? Which moves can you do to create a small area of effect from which a piece can be isolated?

Once you've got the hang of it, generating your own algorithms won't be something unbelievable to do and it gives you a chance to explore the puzzle before diving into your first solve.