User:Mildsunrise/Puzzle groups

This page lists groups of popular puzzles and useful data about them.

Disclaimer: most of these things are bounds, and some of the facts may be incorrect or biased by the puzzles I've studied. I'm still learning group theory, be gentle uwu

Introduction
In order for a puzzle to be suitable for analysis as a group, it must be:


 * closed: the same set of moves must always be possible, regardless of what state the puzzle is in.
 * nondestructive (invertible): moves can always be reversed, returning the puzzle to the state before the move.

Elements of this group (we'll call it $$G$$ throughout this article) can be thought of as the various states that the puzzle can be in, but it's more accurate to think of them as manipulations (the result of a sequence of moves, often called algorithm) acting on the puzzle. The group operation would be the composition of manipulations, which is naturally associative, and the identity element is the null manipulation (do nothing to the puzzle; leave it in its current state).

The group will typically not be abelian, as that would make it a direct product of cyclic groups, which has a very predictable structure (making for an uninteresting puzzle). They can sometimes have an abelian part, but it's generally small relative to the whole group.

Because puzzle states are discrete, the group is finite and thus finitely generated. A common way to express it is by embedding in a larger group. That is, find a group (often a symmetric group) such that every move maps to an element of the group, and application of a move to a puzzle is equivalent to operation by its associated element on the left. The puzzle's group is then defined as the subgroup generated by (the associated elements of) the base moves.

As to what constitutes a manipulation (or, whether two puzzles are in the same state or not), manipulations to the whole puzzle (as a rigid object), such as rotations, are generally excluded from the group. So for example, rotating a Rubik's cube 90 degrees along the X axis (typically written "x" in notation) is not an element of the group, and the result of the rotation is considered to be the same cube as before. If such a structure is desired, it is usually easy to extend the puzzle's group with it (and is often a direct product with the rotation group of the puzzle as a single object, which in the case of a cube is $$S_4$$).

It is important to note that an element of the group (manipulation) is not an algorithm itself (sequence of moves), but the effect that applying this algorithm has on the puzzle. We could also say that an algorithm is a way to express the element of the group as a composition of moves (which are often base moves and their inverses). An element may be expressed as different compositions of moves. For example, in the Rubik's cube, algorithms F U R U' R' F' and F R U R' U' R U R' U' F' have the same effect on the cube, and thus correspond to the same element of the group. Mathematically, $$F^{-1} \cdot R^{-1} \cdot U^{-1} \cdot R \cdot U \cdot F = F^{-1} \cdot U^{-1} \cdot R^{-1} \cdot U \cdot R \cdot U^{-1} \cdot R^{-1} \cdot U \cdot R \cdot F

$$.

Reassembly group
A concept that I've found useful when analyzing many puzzles is the reassembly group $$R$$ of a puzzle. It's the extension of $$G$$ with additional manipulation that become possible when you're allowed to reassemble the puzzle. This means you're allowed to rotate and permute the different pieces or blocks of the puzzle freely.

Take the Rubik's cube as an example: rotating a single corner or edge isn't possible using legal moves, and thus, such a manipulation isn't an element of $$G$$, but it is an element of $$R$$ since you're definitely able to do that if you reassemble the cube.

Because (by definition) everything possible with legal moves is also possible with a reassembly, $$G$$ is a subgroup of $$R$$. In many puzzles, $$R$$ has a simpler structure and is often the result of wreath products of the different blocks of the puzzle, and $$G$$ captures the restrictions introduced on $$R$$ by the puzzle's mechanism and the legal moves. This often means introducing parity conditions, where the parity has to be nullified for the element to be part of $$G$$. This means $$G$$ is often $$R$$ but with "some of its commutative structure removed". In some puzzles, $$G$$ removes exactly all of the commutative part, meaning it is the commutator subgroup of $$R$$.

The notion of "reassembly" and "block" can sometimes be a bit subjective. Here it typically stops at the pieces that legal moves of the puzzle cause to move or rotate. For example, the individual stickers of a Rubik's cube are not considered blocks, since they remain attached to their edge or corner during legal moves. Likewise, the individual pins of a cross in Cross Teaser are not considered blocks; the 8 crosses themselves are.

Legend
Fields listed for each puzzle:


 * puzzle group ($$G$$): group describing the structure of the puzzle. See Introduction above.


 * reassembly group ($$R$$): the puzzle group, but extended with manipulations to the puzzle's pieces, or blocks, when allowed to move or rotate independently. See Reassembly group above.


 * symmetric embedding: smallest symmetric group that $$G$$ embeds in. Unless stated otherwise, this is also the smallest symmetric group that $$R$$ embeds in, so $$G \leq R \leq S_n$$.


 * generators: minimal amount of elements needed to generate $$G$$, and a typical choice for such a set.


 * size: cardinality of $$G$$, i.e. amount of valid states the puzzle can be in (valid meaning it's reachable by a sequence of legal moves starting from a solved state).


 * reassembly multiple: cardinality of $$R$$ divided by cardinality of $$G$$. That is, how many possible assemblies exist for every valid state of the puzzle. If you were to reassemble the puzzle in a random way, this tells you the odds that it would be solvable with legal moves.


 * normal subgroup?: is the puzzle group a normal subgroup of the reassembly group?


 * diameter: the diameter of $$G$$, with a common choice of generators. In other words, the lowest possible amount of base movements that makes it possible to solve any state of the puzzle.


 * maximum order: maximum order of an element in $$G$$. Because the group is finite, every element has an order, that is, the puzzle eventually returns to the starting state after applying it enough times.


 * symmetries: characterization of symmetries appearing in the puzzle. A symmetry (in technical words, an automorphism of $$G$$) arises when you can map states / manipulations to other states / manipulations in a way that is consistent with the puzzle; for instance, solving the puzzle in the mapped state solves it in the real state. The characterization of these symmetries is the structure of the automorphism group of $$G$$.

Rubik's Cube

 * symmetric embedding: $$S_{48}$$ (8 non-center stickers for each of the 6 faces). The center stickers are fixed because we are excluding rotations of the whole cube. Can be broken down into a direct product $$S_{24} \times S_{24}$$ by separating into edge and corner stickers.


 * reassembly group: $$S(2, 12) \times S(3, 8)$$ or equivalently $$(Z_2 \wr S_{12}) \times (Z_3 \wr S_8)$$ or equivalently $$(Z_2^{12} \rtimes S_{12}) \times (Z_3^8 \rtimes S_8)$$. By taking into account the block structure (stickers belong to blocks that are permuted and rotated as a whole), both edge and corner permutations narrow down from $$S_{24}$$ to $$S(2, 12)$$ and $$S(3, 8)$$ respectively (where $$S(m, n)$$ refers to the generalized symmetric group).


 * puzzle group: $$[S(2, 12) \times S(3, 8)]' \rtimes Z_2$$ or equivalently $$[S(2, 12)' \times S(3, 8)'] \rtimes Z_2$$ or equivalently $$[(Z_2^{11} \rtimes A_{12}) \times (Z_3^{7} \rtimes A_{8})] \rtimes Z_2$$. It removes almost the entire commutative structure of $$R$$ (that is, the permutation sign + rotation sum for each of the branches) except one of the permutation signs, hence $$\rtimes Z_2$$.


 * reassembly multiple: 12. Three parity conditions (all of them normal subgroups) contribute to this factor:
 * Sum of edge rotation ($$Z_2$$) nullified (or equivalently: the orientation of the last edge is fixed).
 * Sum of corner rotations ($$Z_3$$) nullified (or equivalently: the orientation of the last corner is fixed).
 * Edge permutation and corner permutation must have the same sign ($$Z_2$$) (or equivalently: the sign of one of the permutations is fixed).


 * size: $$(2^{12} \cdot 12!) \cdot (3^8 \cdot 8!) \div 12 = 2^{27} 3^{14} 5^3 7^2 11$$, equal to 43252003274489856000 ($$\approx 10^{19.64}$$)

Picture stickers
TODO

Cross Teaser
TODO