There are a lot of terms in the theory of cubing that won't be at all understandable to outsiders, or even that are used in a different way in cubing than in the outside world. I'm going to define some of them; since I have an imperfect memory this list is not complete, but just a list of some things which I think are commonly or easily misunderstood and which I'd like to explain.

Algorithm: This just means a sequence of moves, whether it is made up on the spot or memorized beforehand. The common abbreviation is "alg". Typically it is a sequence that accomplishes something specific (such as solving part of a cube), but people sometimes talk about scrambling algs. A way to solve a specific puzzle is not an algorithm, but a method (see below). Some methods have many algorithms to memorize, and some require less or no memorization.

Bandaged: A puzzle is bandaged if some of its pieces are connected to what would normally be adjacent pieces in a way that restricts the movement of the puzzle depending on where those pieces are, which often increases the puzzle's difficulty. A bandaging is a way of connecting pieces together to make a bandaged puzzle.

Blindfold Solving: This is a way of solving a puzzle by memorizing the positions of all of its pieces, then putting on a blindfold and solving the whole puzzle without looking at it. The typical abbreviation for blindfold or a blindfold solve is BLD. There are really two types of BLD: solving all of the pieces like a speedsolve, and tracking where they go while you memorize, then memorizing what moves to do; or solving a few pieces at a time while not affecting the others, so you can memorize where each piece is supposed to go, which is often easier.

Block: This can either mean a group of adjacent layers on one axis, or a cuboidal set of pieces. A block turn is a turn of some group of adjacent layers. Block building is a type of intuitive step of a method where you assemble a block of pieces.

Center: Any piece which belongs to only one face. Some puzzles have a lot of these on each face, in many different types. Since they have only one color, though, if a puzzle is not a supercube (see below) they can often be interchanged with each other without making the puzzle unsolved. Centers typically do not have orientation, except sometimes on a supercube.

Corner: Any piece which belongs to more than two faces. Corners usually have from two to five possible orientations. Note that trivial tips (see below) are generally not considered corners, though, since they don't affect other pieces and are thus not really part of the puzzle.

Cube: Although formally this means a cubical puzzle, the Rubik's Cube is so popular that people will often casually refer to any twisty puzzle as a cube.

Cuber: Anyone who is interested in solving the Rubik's Cube or other twisty puzzles. If they are also interested in solving as fast as they can, they are a speedcuber.

Edge: Any piece which belongs to two faces. Edges usually have two possible orientations.

Face: One face of the puzzle. A face is solved when all of its stickers match (on a normally-colored cube this would mean they were all the same color). However, solving a face on many puzzles (such as 3x3x3) is useless, since it might still be necessary to permute the pieces on that face in order to solve the puzzle; it is much better to solve in terms of layers or blocks.

God's Algorithm: In general this term means a method to always solve a puzzle in the fewest possible number of moves. However, it can be used in a few different ways: sometimes it means a way to find this solution for any given scramble (or a list of solutions for every scramble), sometimes it means a solution for a given position, and sometimes it is used to mean the maximum number of moves that an optimal solution for a scramble can take.

Intuitive: A step is intuitive if you solve it by figuring out what moves to do during the solve, rather than by using algorithms you know (which is called an 'algorithmic' step). Typically people think of commutators as intuitive, since you don't memorize the moves that are in them, but instead learn the concept. Intuitive steps in a method are good because they require less memorization, take fewer moves, and are generally more fun to execute, but it does take more practice to be fast at them.

Lucky: A solve is lucky if one or more of the steps in the method is skipped; solves can be very easy without being lucky. Note that if a solver uses a different algorithm than normal to deliberately skip a step in their method, most people would say that does not make the solve lucky.

Method: A method is a predetermined way of solving a puzzle in a series of steps. Typically methods are named after their creator(s) or the steps themselves. A method created by Stefan Pochmann might be referred to as the Pochmann method, Pochmann, or even poch; abbreviations are common if the name is long. For example, there is a family of methods called Corners First (CF) because the first step or first few steps solve the corners of the puzzle without regard to other pieces.

Metric: A specific and well-defined way to count the length of an algorithm or a puzzle solution. The most common metrics are htm or ftm (half turn or face turn metric, where every move of one face, or block containing a 1st slice, is counted as one turn), stm or btm (slice turn or block turn metric, where every move of one face, block, or slice is counted as one turn; some would say stm does not count blocks as one move but as separate slices), and qtm (like htm, except that only turns of the smallest valid angle increment (90 degrees on 3x3x3, for example) are counted as one move, and larger turns may be more than one move).

Orbit: This is the set of all pieces that can be moved to a specific location on the puzzle without rotating the puzzle. Sometimes there is more than one orbit of pieces that look the same.

Orientation: The way that a piece is positioned, independent of where it is on the puzzle; you can define a 'correct' orientation for each piece and location. On a Rubik's Cube, for instance, corners have three possible orientations. When you do moves to give a set of pieces the correct orientation, you are orienting them (not orientating). Pieces with the wrong orientation are typically said to be flipped (in the case of edges) or twisted (in the case of corners and centers) and algorithms to solve them are thus called flippers or twisters.

Parity: As I'll discuss later, pieces can normally be solved in cycles of three pieces, but if pieces cannot be solved like this then there is a (permutation) parity. There can also be orientation parities, if only one piece in an orbit is wrongly oriented, but these do not occur on all puzzles. The algorithm to fix any parity without disturbing other pieces is typically very long.

Permutation: Where a piece is on the puzzle, independent of the way it is positioned; if a piece is in the correct permutation it might still be misoriented (twisted or flipped). When you do moves to give a set of pieces the correct permutation, you are permuting them (not permutating).

Slice: One of the layers on an axis, not counting the outermost layer. A slice turn (or sometimes just slice) is a turn of one of these layers only.

Supercube: A puzzle where some or all pieces are marked so that the position and orientation of every piece must be correct for the puzzle to be solved.

Trivial tip: A piece on the puzzle which can always be brought to its solved position in one move without disturbing any other pieces; they are called trivial because they are so easy to solve that they are often not even mentioned in methods. They are also known as tips. A pyraminx has four of these.

Twisty puzzle: Basically a sequential move puzzle where the goal is to turn various parts of it until it reaches one of its solved states. A pretty good list can be found at the twistypuzzles.com website. These are often simply referred to as puzzles, or even cubes.

## 20080420

### Some Basic Vocabulary

### Notation

In this post I'll describe my generalized notation for twisty puzzles. I might stray from this notation at times but I think this does a pretty decent job of describing moves for puzzles. There are a few levels of notation built on each other, and although it might seem overcomplicated most applications of this (such as writing down specific algorithms) only use simple features of it: for example, the well-known Sune algorithm for the 3x3x3 is still written R U R' U R U2 R'.

Not all of this will necessary make sense or seem useful to you now. However, if I use some notation that you are not familiar with, it would be helpful to come back to this post as a reference.

First you give each face of your puzzle a letter (or more than one if you are careful about it). Fortunately, for algorithms on dodecahedral and icosahedral puzzles, you usually don't have to label all of the faces. The standard naming for cubical puzzles, which you have probably seen before if you are into speedsolving, is to call the faces U (for the upper face), D (for the down face), F (for the front face), B (for the back face), R (for the right face), and L (for the left face). Typically if you see three faces in a shot of a cube they are assumed to be U, F, and R. I'll assume that you know what faces I am using when I talk about cubical puzzles, but if I am discussing a different puzzle I'll probably describe how I am naming faces or at least make it so that you can determine the meanings from context.

Now you can label the axes that you turn pieces around. There are three types of axes: faces, edges, and corners. Check out the puzzles at gelatinbrain if you aren't entirely sure what this means; the most common puzzles with various axes are the Rubik's Cube (face), Bevel/Helicopter Cube (edge), and Pyraminx (corner). Face axes are labeled by their face, edge axes are labeled by the two faces they touch, and corner axes are labeled by the three faces they touch; the face's names can be in any order, but they should not have spaces between them. This means that if you see an axis label (say UF on a 3x3x3) you should be able to quickly determine what type of axis it belongs to, so this notation can still handle puzzles with more than one type of notation.

Finally, we get to the turns themselves. Turns are all around an axis, so you should modify the following basic types of turns based on the axis. For the examples I'll use the axis UR. We're going to suppose there is more than one slice around the axis (like in big cubes) for the purpose of the notation. Label the slices with integers starting at 1 for the outermost slice and ascending as you move deeper into the puzzle. There are five types of moves; all moves are clockwise turns as seen from a vantage point outside the puzzle.

- 3UR is a clockwise turn around the UR axis, turning the 3rd slice only (a slice turn). If the number is 1 (1UR) then it is allowed to just omit the 1, since this is a very common type of move.

- 3ur is a clockwise turn around the UR axis, turning all slices up to the 3rd slice (a block turn). If the number is 2 (2ur) then it is allowed to omit the 2, since this is the smallest block turn that can't be just written as a slice, and is also very common.

- 2-4ur is a clockwise turn around the UR axis, turning slices from the 2nd to the 4th (a block turn). This isn't a common type of move, but it's useful for writing patterns in block turn metric.

- *ur is a clockwise turn around the UR axis, turning basically half of the slices on the UR axis (and its opposite axis) rounded down. On a 4x4x4 or 5x5x5 cube this would be 2 slices; on a 6x6x6 or 7x7x7 it would be 3. Note that since it is lowercase it should be a block turn. This is useful for writing algorithms in a more general form.

- cUR is a clockwise cube rotation around the UR axis. Note that it's unambiguous because having lowercase and uppercase letters together is unique. Observe that sometimes we might need cube rotations around axes that you can't turn; for example on a 3x3x3 cube sometimes a rotation like cURF is useful.

Separate moves should always be written with spaces separating them. A group of moves can be enclosed with parentheses (i.e. ( and )). You can modify a move or a group of moves by putting a number after it (to indicate it is to be done that many times) or a ' (to indicate that its inverse is to be performed instead). Thus combinations such as 3UR' and r2 are possible. If we want to use both of these, either R2' or R'2 is acceptable, but personally I will tend to use R2' because I like it more.

Rarely, in generalized patterns or commutators, variables are needed to show that any slice or any amount of turn can be used. Typically any letters that are not in use to describe axes are acceptable. As in the notation we have already seen, if a variable is placed before the axis name (such as nr) it describes a slice number, and if a variable is placed after (such as Rx) it describes a number of times a turn should be done.

Finally, I'd like to say that it's true that this notation differs a bit from the conventional notation, especially when it comes to large NxNxN puzzles (bigcubes, that is), but I think that it is better generalizable to other puzzles and more consistent than the standard notation. Moves such as (rm') or Rw, as written in standard notation, are difficult to expand to different puzzles and often unwieldy. I hope my notation is reasonable enough to be understood by most visitors here.

## 20080416

### Introduction

This is a blog about twisty puzzles themselves. I aim to explain the way they work and the way they are solved so that the reader should eventually gain a deep understanding of the way these puzzles work, a thing which is sadly lacking in many of today's speedcubers. I also hope to put some of the methods I use or have discovered on here. Some of the material here might be original research and some may not; it's safe to assume that I've created anything that I explicitly credit to me, but I don't necessarily claim authorship of all that I write here.

The first couple posts will probably concern themselves with the mathematics of the cube, to get a feel for why things work as they do. Although I'm considering majoring in Math I'm not going to be too formal or theoretical, so anyone with an early high school understanding of mathematics (or better) should be fine. Then after that I'll start to go into the more interesting stuff, or perhaps some more complicated mathematics. Don't think of this as a kind of textbook where every section builds on the one before it: the material in these opening sections should be enough to understand all or most of the rest. If you are confused about something you can always ask me, or the friendly folks on speedsolving.com.

So what's with this name "Notes on Twisty Puzzles"? It is based off of the title of an early Rubik's Cube treatise by David Singmaster which he called "Notes on Rubik's Magic Cube" and which was a great leap ahead in understanding how the standard 3x3x3 cube works. Despite not having read his books about the Cube I do recommend that you read them if you have a chance, as I've heard they are excellent.

Finally, all opinions expressed herein, and all writing, are mine unless stated otherwise. Opinions can and do change, so if you have a problem with anything here feel free to tell me about it.

Have fun, and good luck!