## Exploring 3x3x3-Like Puzzles (Part III—Part II Is Still Wrong)

*Last modified April 20, 2022.*

In this article, we refine yet again the definition of 3x3x3-like puzzles (333LPs), and touch upon the value of self-honesty.

### A recap of parts I and II

In part I, we stated

We define a 3x3x3-like puzzle (333LP) as a twisty puzzle that is “similar enough” to the 3x3x3 and the Megaminx.

For reference, the Megaminx is a dodecahedral variant of the 3x3x3.

We then described what “similar enough” meant using a somewhat simple set of requirements. In part II, those requirements became the following:

**Definition 1**: A 3x3x3-like puzzle (333LP) on an \(n\)-manifold is a twisty puzzle such that:

- The puzzle is shallow-cut.
- The puzzle is facet-turning and doctrinaire. If the puzzle contains a finite number of facets, then we require that each facet receive a unique colour. Otherwise, the puzzle contains countably infinite facets, and the facets must be coloured using a finite number of colours such that the coloration is periodic, with no two adjacent facets sharing the same colour, and such that the puzzle is not “limited in size”.
- There is exactly one 1-coloured piece per facet.
- The puzzle can be solved using Roice’s method. The puzzle does not have corners (\((n+1)\)-coloured pieces) with fixed permutations relative to each other.
- Let \(P\) represent the polytope, tiling, or honeycomb that admits the 333LP. Then, the vertex figure of \(P\) is a (potentially irregular) \(n\)-simplex, and \(P\) itself is vertex-transitive and convex.

### What’s wrong with the definition above?

Firstly, the definition above unnecessarily restricts shape modifications or sticker modifications from being 333LPs. Therefore, by the definition above, the Ultimate Cube would not be a 333LP, even though one can solve it just like a regular 3x3x3.

Definition 1 also restricts the Kite Trap from being a 333LP, even though it actually solves quite well using Roice’s method, which in this case specifically means an edges-first corners-second method. The restriction occurs because the Kite Trap is a jumbling puzzle, and because jumbling puzzles are non-doctrinaire by definition, they fail requirement 2 of the definition. Note that I only found out about the Kite Trap puzzle last December (2021), whereas part II was written in mid-2020.

### Refining the requirements for a 333LP

What I wanted all along was a label for puzzles that were like a 3x3x3 that can also be solved edges-first. But, knowing that my intent was to label puzzles with similar solutions, I could have stated the requirements more directly. For example, instead of requiring the puzzle not be “limited in size” (Definition 1, requirement 2), I could have written that the puzzle must be solved with a certain family of algorithms, and let that requirement do the heavy work.

**Definition 2**: Let * after a turn represent doing a turn by any amount. A three-dimensional 3x3x3-like puzzle (333LP) is a twisty puzzle that either satisfies the following conditions, or is a sticker and/or shape modification of another twisty puzzle that satisfies the following conditions:

- The twisty puzzle is face-turning and has at most one centre per face. The centres must also be fixed.
- The twisty puzzle lends itself to an edges-first corners-second solution.
- For the solution in requirement 2, all edges of the puzzle must be solvable using the commutators [R*, U*] and/or [R’*, F*], and all corners of the puzzle must be solvable using the commutators [[R*, U*], D*] and/or [[R’*, F*], D*].

We assume for requirement 3 that the puzzle has faces that can be labelled U, R, F, D, and be mapped onto the standard U, R, F, and D faces of the 3x3x3, respectively, with the same adjacency rules.

Definition 2 allows puzzles such as the following to be counted as 333LPs:

- 3x3x3
- Megaminx
- 2x2x2
- 3x3x2
- Ultimate Cube
- Latch Cube
- Tuttminx
- Kite Trap

Requirement 3 of Definition 2 rejects the following puzzles that would otherwise be counted as 333LPs:

- Halpern-Meier Tetrahedron
- Skewb, which is a shape modification of the Halpern-Meier Tetrahedron
- LanLan Star Pyraminx
- 4-Corners Cube, which is a shape modification of the LanLan Star Pyraminx

We can call those puzzles quasi-333LPs.

Definition 2 also disallows puzzles such as the following to be counted as 333LPs:

The Hexadecagon might seem like a 16-sided Megaminx, but in reality, it is a jumbling puzzle that contains two sets of edges and two sets of corners. One set of corners and edges solves like the LanLan Star Pyraminx, and the other set solves like a 3x3x2.

Lastly, we can generalize the definitions above to 333LPs in higher dimensions. This process results in a definition that is similar to Definition 1, but is more direct in its intention. It also correctly labels more puzzles as 333LPs, including all puzzles accepted by Definition 2.

**Definition 3**: A 3x3x3-like puzzle (333LP) on an \(n\)-manifold, where \(n \ge 2\), is a twisty puzzle that either satisfies the following conditions, or is a sticker or shape modification of another twisty puzzle that satisfies the following conditions:

- The twisty puzzle is facet-turning, with at most one 1-coloured piece per facet. The 1-coloured pieces must also be fixed.
- The twisty puzzle lends itself to a solution where one solves the 2-coloured pieces first, followed by the 3-coloured pieces, and so on, until they end the solve by solving the \((n+1)\)-coloured pieces.
- For the solution in requirement 2, all pieces must be solvable using higher-dimensional sledgehammers. In particular, \(m\)-coloured pieces for \(2 \le m \le n+1\) must be solvable using a commutator of the form \(\alpha_m = [[[[S, D_3], D_4], \ldots], D_m]\). Note that \(\alpha_2 = S\).

Definition 3 allows more puzzles such as the following to be counted as 333LPs:

Definition 3 disallows puzzles such as the following from being counted as 333LPs:

- MagicCube4D triangular duoprisms
- Magic Puzzle Ultimate cell-turning 24-cell

We can also call puzzles that satisfy all but the third requirement of Definition 3 quasi-333LPs.

In summary, I wanted a term to describe a subset of twisty puzzles, including the 3x3x3 and Megaminx, that I particularly enjoy solving. Defining that term precisely meant I had to be honest with myself, and state what I actually wanted in the first place.

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