Glossary¶
This is a glossary of terms used in the hypercubing community. We take a mostly prescriptivist^{1} 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 handson 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 7dimensional puzzle, for example, if there’s no need to communicate about them.
Puzzle elements¶
A 1dimensional turning axis is not always welldefined for higherdimensional puzzles, because rotations generally happen in a plane, not around an axis.
Polytope elements¶
For an \(N\)dimensional polytope:
 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 \(n1\) elements
 …
 facet = rank \(N1\), polytope constructed from rank \(N2\) 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¶
Some of these terms are from Polytope  Wikipedia.
Basic definitions for an \(N\)dimensional hypercubic puzzle:
 corner = piece with \(N\) colors (4 colors in 4D)
 edge = piece with \(N1\) 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, nonfacetturning 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 fullpuzzle rotation = rotation of the whole puzzle that does not change its state
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 4^{n} has been reduced to a 3^{n} 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 broady 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 3^{3}, 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 = nonfree and nonside 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 3^{3},
D
is the only base axis,U
is the top axis (the only free axis), andR
,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 nonintersecting 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 3^{3} an F2L pair consists of a corner (the head) and an edge (the body).
 On the 3^{4}, an F2La 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 = resolving 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 3^{4}, 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 this, we simply add a hyphen and the rank of the thing you’re solving at the end. Examples:
 PLL4 is the PLL step on a rank4 object, which permutes a rank3 object. With CFOP on 3^{4}, it consists of permuting the 2c’s, then permuting the rest like a 3^{3}.
 For F2L, you put the number before the letter at the end e.g. F2L5a, F2L6d, etc.
 If you were solving a 3^{6} with pure CFOP and you were solving the F2L of the final cube with triple RKT, that would be F2L3 of PLL4 of PLL5 of PLL6.
Puzzle descriptors¶
This section is a workinprogress.
 Solid vs. tiling vs. soup
 Doctrinaire
 Reduced
 Bandaged
 Unbandaged
 Shapeshifting
 Sliding vs. twisting
 Circle
 Super
 Real
 Complex
 Stickermod
 Shapemod
 Cuboid terms (tower, brick, floppy, domino, pancake)
 Other common puzzle families: weirdling, bubbloid, rotategap, slidinggap (15puzzle), loopover
Cut depth¶
This section is a workinprogress.
 Shallow cut
 Half cut
 Deep cut
 Deeper than adjacent
 Deeper than origin
 Semideep cut?

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