Two group-theory statements that sound contradictory about the 15-puzzle

Question

In the wiki article: 15 puzzle, it reads

The symmetries of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed); this groupoid acts on configurations.

However, in this note: An Analysis of the 15-Puzzle, it has

Lemma 2.2. $L(K)$ is a group.

Here, $L(K)$ consists of all the permutations of the form $(i_k = 16, i_{k-1}) (i_{k-1}, i_{k-2}) \cdot (i_2, i_1) (i_1, i_0)$ where $i_s$ is a neighbor of $i_{i+1}$ for $0 \le s < k$.

How to understand these two statements that sound contradictory to me? Specifically, what do you think are the "symmetries of the fifteen puzzle" in the wiki article referred to?

Answer 1

The two articles are referring to different things.

The wiki article and two blogposts it cites

In a case like this, where references are available, it's good to dive into them. Right next to the claim you saw about symmetries being a groupoid, there are two superscript links, both leading to blog articles by the same person. To quote one of the articles:

A heated discussion went on a couple of years ago at sci-physics-research, starting with this message. Lubos Motl argued that group-theory is sufficient to analyze the problem and that there is no reason to resort to groupoids. [...] I’m mostly with Lubos on this. [...] At the same time, if one wants to present this example in class, one has to be pretty careful to avoid confusion between permutations encoding positions and those corresponding to slide-moves.

So there is the difference: in the blog articles, the author wants to use groupoids as a way to encode legal starting positions (which may have the empty slot anywhere) to the solution (where all the numbers are sorted and the hole is in the lower right hand corner, say.) To support this, the transformations have to be allowed to move the hole around the board. Then clearly sometimes you can't compose two such transformations, since each transformation begins with the hole in some special spot.

The note by Chapple, Croeze, Lazo & Merrill

What the second article does for transformations is this: take a board with the hole in the lower right-hand corner, then make any move sequence which finally results with the hole in the lower right-hand corner: this transformation is a symmetry. The problem with composition is totally eliminated. You will be able to compose any two transformations. The set of these transformations is a group.

Conclusion

So I would say that I understand what the first author is doing by using groupoids, but in my reckoning, it is silly to call these "symmetries" as done in the wiki article. Lubos Motl expressed it pretty well in the link in the source given above:

It is not clear to me what would be a notion of "symmetry" good for if its action on some elements were ill-defined. Mathematics has simply defined the word "symmetry" to be equivalent to the word "group"; in my opinion, it is the only reasonable definition. If you can't apply an operation - defining a symmetry - to an object, then this object simply does not have this symmetry. Period.

Answer 2

I think the wiki statement is a bit misleading (it is based on the article in references) - firstly, the notion of groupoid is different from the classical (here, a groupoid has all inverses, but the composition is only partial) and "symmetries" are not geometrical symmetries, but slide-movements of the puzzle, actually.