Glossary¶

This is a glossary of terms used in the hypercubing community. We take a mostly prescriptivist¹ approach: terminology has a great effect on how we think about puzzles, and we try to be mindful when naming concepts and inventing notation to ensure that they encourage better understanding of puzzles and are useful in as many contexts as possible.

Before you propose new terminology …

We’ve wrestled in the past with poor terminology that actively hurt understanding. First, gain hands-on experience and intuition for the thing you want to describe, and then see what terms are actually needed. There’s no value in making up words for the pieces on a 7-dimensional puzzle, for example, if there’s no need to communicate about them.

Puzzle elements¶

A 1-dimensional turning axis is not always well-defined for higher-dimensional puzzles, because rotations generally happen in a plane, not around an axis.

Polytope elements¶

For an \(N\)-dimensional polytope: (Some of these terms are from Polytope - Wikipedia)

vertex = rank 0, single point
edge = rank 1, line connecting two vertices
face = rank 2, polygon constructed from edges
cell = rank 3, polyhedron constructed from faces
…
\(N\)-face = rank \(N\), polytope constructed from rank \(N-1\) elements
peak = \(N-3\) face
ridge = \(N-2\) face
facet = rank \(N-1\), polytope constructed from rank \(N-2\) elements

In 4D, we prefer facet rather than cell. In simple terms: on most puzzles, a facet is the thing with a single color.

Pieces¶

Basic definitions for an \(N\)-dimensional hypercubic puzzle:

corner = piece with \(N\) colors (4 colors in 4D)
edge = piece with \(N-1\) colors (3 colors in 4D)
peak or 3c = piece with 3 colors (5D+)
ridge or 2c = piece with 2 colors (4D+)
center or 1c = piece with 1 color

We prefer words rather than 1c, 2c, etc. because the words generalize better to other, non-facet-turning puzzles and describe how a piece behaves rather than how it looks.

Moves¶

axis or turning axis = ray start from the center of the puzzle, around which puzzle elements rotate during twists
twist or move or turn = movement of pieces that changes the puzzle state
rotation or full-puzzle rotation = rotation of the whole puzzle that does not change its state

Puzzle state¶

A puzzle’s state graph is the graph of all its states. Each state of the puzzle has a node, and the nodes are connected by single moves.
A piece’s attitude is the transformation from its solved state to its current state. For example, each piece on 3x3x3 has 24 possible attitudes.
A piece’s attitude can be decomposed into its permutation, the component that affects its grip signature, and its orientation, the component that does not. For example, each corner on a 3x3x3 has 8 permutations and 3 orientations.
Pieces are indistinguishable if swapping them never affects whether the puzzle is solved. For example, the blue center pieces on a standard 4x4x4 are all indistinguishable.
Pieces are distinguishable if swapping them can affect whether the puzzle is solved.
Piece orientations are indistinguishable if changing one orientation to the other never affects whether the puzzle is solved. For example, each center on a standard 3x3x3 has 4 orientations, all of which are indistinguishable.
Piece orientations are distinguishable if changing one orientation to the other can affect whether the puzzle is solved.

Revealing information that distinguishes indistinguishable pieces or orientations makes them no longer distinguishable, thus changes the puzzle.

Puzzle properties¶

Algebraic properties¶

A move is blocked in a particular puzzle state if there is some feature of the puzzle preventing the move from being applied. Generally this is because there is a piece that is partially inside and partially outside of the region affected by the move.
Bandaging is the process of combining pieces in order to block moves.
Unbandaging is the process of splitting pieces in order to make more moves possible.
A puzzle is doctrinaire or fully unbandaged if every move is always accessible.
A puzzle is bandaged if it is not doctrinaire, but can be finitely unbandaged to a doctrinaire puzzle.
A puzzle is jumbling if it has infinitely many grips. For finite puzzles, this simpler definition is equivalent: A puzzle is jumbling if it cannot be finitely unbandaged to a doctrinaire puzzle.

Visual modifications¶

A sticker mod of a puzzle is a modification that involves changing the coloring of a puzzle. A sticker mod may have different indistinguishable pieces than the original puzzle.
A shape mod of a puzzle is a modification that involves changing the shape of the puzzle without changing the behavior of pieces. A shape mod may have different indistinguishable pieces than the original puzzle. A shape mod sometimes requires modifying the coloring as well.
A puzzle is shapeshifting if the visible shape of the puzzle depends on its state.

Note that shapeshifting has nothing to do with algebraic properties (doctrinaire/bandaging/jumbling).

Any puzzle can be made shapeshifting by changing the shape of one of the pieces.
Any single-core twisty puzzle can be made non-shapeshifting by carving it from a sphere.

Constructions¶

A solid is a construction of a puzzle by cutting a finite polytope, possibly with some pieces removed.
A tiling is a construction of a puzzle by cutting a filled space, typically with no pieces removed.
A soup is a construction of a puzzle by adding objects to an initially empty space.

Completions¶

A real puzzle is one with all interior pieces. For example, a real 7x7x7 has \(7^3=343\) pieces, compared to \(7^3-5^3=218\) pieces for a standard 7x7x7.
A complex puzzle is one with a piece for each possible grip signature using the grip-theoretic construction. These puzzles have \(2^n\) pieces, where \(n\) is the number of grips on the puzzle.
A laminated puzzle is one with a piece for each possible grip signature using a laminated construction. A laminated puzzle is a subset of the complex puzzle.
A multi puzzle is one with pieces from several different cut depths. An example is the Multidodecahedron. A multi puzzle is a subset of the laminated puzzle.
A circle puzzle is one with circles carved into the faces, where pieces inside one or more of the circles do not turn with their face. A circle where all circles behave equivalently is a subset of the complex puzzle. For example, see this video of a circle 3x3x3.
A super puzzle is one where all orientations are distinguishable.

Real, complex, and laminated puzzles are often implicitly super. For example, the super real 5x5x5 has 125 pieces, all distinguishable (and with all distinguishable attitudes). It is equivalent to the Double Circle Real 5x5x5 (video).

Cut depths¶

Cut depth terminology varies by community. Listed here are the definitions we use in hypercubing.

A shallow cut is any cut equivalent to the shallowest possible cut. An example is the Megaminx.
A deep cut is any cut deeper than a shallow cut.
A half cut is a cut that passes through the center of the puzzle into two identical halves. An example is the Pentultimate.
A to-adjacent cut is a cut that passes through the center of an adjacent face. An example is the Pyraminx Crystal. A to-adjacent cut is exactly the depth required to not have axis pieces (pieces that turn with exactly one grip).
A deeper-than-adjacent cut is any cut deeper than a to-adjacent cut. An example is the Curvy Starminx or Litestarminx.
A deeper-than-origin cut is any cut deeper than a half cut. An example is Deeper Madness, especially compared to Shallower Madness. Another example is the Enabler Cube.

Other cut types¶

A cut is accessible if there is some move that separates pieces along it.
A stored cut is one that is not accessible from the solved state of the puzzle. For example, the extra cuts present on a Curvy Copter Plus are stored cuts; compare to the Curvy Copter, which has no stored cuts.
A wedge cut is a cut comprised of multiple cut planes, where twists are parallel to both planes. This is only possible in 4D+. An example is the wedge-turning 3⁴.

Solving¶

Actions¶

An action is sequence of moves that preserves invariants of the stage. Usually, an action keeps certain pieces solved. For example, when a 4ⁿ has been reduced to a 3ⁿ using big cube reduction, outer layer moves are the only actions. In this case, the actions are reduced moves. Another common set of actions is RKT.

Parity¶

There is no community consensus on the definitions of parity. Below are some proposed definitions:

group theory parity = a case where the puzzle is in an unexpected coset of a subgroup of index 2
- It is often more broadly applied to a case where the puzzle is in an unexpected coset of a subgroup of any index.
cuber parity = a case that is difficult to solve that the solver didn’t expect
- Melinda’s definition: a local maximum, where the puzzle is largely solved but requires many moves to fix
- Hactar’s definition: a case which cannot be solved using the actions expected at this stage in the solve

None of these definitions are satisfactory. According to most of these definitions, RKT parity is not parity at all, but is more accurately called “RKT error.” According to Melinda’s definition, most PLL cases are parity. The first definition given for “cuber parity” is highly subjective, but is the only one that captures its current use.

Open questions

Is there a definition for “cuber parity” that captures the way it’s naturally used?
Is there a catchy term we can use instead of “RKT parity”? Melinda proposes “RKT error.”

F2L¶

F2L is a very general solving strategy that works by building a small block of pieces and then inserting the block into its solved position. F2L stands for “first two layers” because it was originally developed to solve the first two layers of 3³, but in hypercubing we use it for many other puzzles.

F2L axes¶

free axes = axes which affect only unsolved pieces; can be turned freely
side axes = axes which affect some unsolved pieces and some solved pieces; can be turned, but must be turned back to restore solved pieces
base axes = non-free and non-side axis that is not completely solved; usually mostly solved, rarely turned during F2L
top axis = the free axis currently being worked on

Examples

In F2L on a 3³, D is the only base axis, U is the top axis (the only free axis), and R, L, F, & B are all side axes.
When beginning F2L on a megaminx there are, 6 free axes, 5 side axes, and 1 base axis.
Near the end of F2L on a megaminx there are, 1 free axis, 5 side axes, and 5 base axes.

We use the letter T to represent the top axis, R & F to represent intersecting side axes, and R & L to represent non-intersecting side axes.

F2L blocks¶

An F2L block or pair is a group of pieces that is paired and solved as one unit. There’s usually a head and body, where the head intersects with more twisting axes than the body.

The base sticker of a head is the sticker which will be facing the base axis when it is solved. The facing direction of the head of a block is whatever direction its base sticker is facing. The facing direction of the body of a block is the same as the head, when they are paired. This notion of which direction a head or body faces gives a way to describe edge orientation before the pieces have been paired, which is helpful especially in 4D+ where edge orientation is otherwise difficult to define.

Examples

On the 3³ an F2L pair consists of a corner (the head) and an edge (the body).
On the 3⁴, an F2L-a pair consists of an edge (the head) and a ridge (the body).

paired = fully assembled
split pair = one move away from paired, or can be paired as part of inserting the block

F2L action terminology¶

breaking the base = unsolving some pieces that were solved
restoring the base = re-solving some pieces
push = a twist of a side axis that breaks the base and puts new pieces on top
pull = a twist of a side axis that restores the base and puts new pieces on top
overpush = push again after pushing (e.g., R U R U R2’)
overpull = push as a continuation of a pull (e.g., R U R2’ U’ R)
push pair = formation of a pair via a push
pull pair = formation of a pair via a pull
hide = to remove a piece from the top (using a push or pull)
reveal = to bring a piece to the top (using a push or pull)
rebase or reorient = to reorient a piece to face a different direction (i.e., change where its base sticker is facing)
cap = to twist T to form a pair (where the head is on top and the body is not on the top)
uncap = to separate the head and body of a pair by twisting T

Open question

What should we call a move like RT on 3⁴, which doesn’t change the set of pieces on T and might or might not unsolve some pieces?

Methods in higher dimensions¶

In higher and higher dimensions, it gets annoying to have to say stuff like “permuting the last cell of the last cell of the…” etc. To avoid the verbosity, we simply add a hyphen and the rank of the thing you’re solving at the end. Examples:

PLL-4 is the PLL step on a rank-4 object, which permutes a rank-3 object. With CFOP on 3⁴, it consists of permuting the 2c pieces, then permuting the rest like a 3³.
For F2L, you put the number before the letter at the end e.g. F2L-5a, F2L-6d, etc.
If you were solving a 3⁶ with pure CFOP and you were solving the F2L of the final cube with triple RKT, that would be F2L-3 of PLL-4 of PLL-5 of PLL-6.

To-do¶

This section is a work-in-progress.

Sliding vs. twisting
Cuboid terms (tower, brick, floppy, domino, pancake)
Other common puzzle families: weirdling, bubbloid, rotate-gap, sliding-gap (15-puzzle), loopover

Go ahead, run us over with the descriptivist bus. ↩