a miracle in the number line
This game is a recreational implementation of, and homage to, the exceptional outer automorphism on Sym6, the group of bijections from six to itself. The rearrangement in the middle prompted by a transposition on the top is the action of the image of the top transposition under (an element of) the outer automorphismm.
I was motivated to start learning about finite groups in January 2025, inspired by nostalgebraist's sublime Christmas novel, The Apocalypse of Herschel Schoen. I first came across the exceptional automorphism in a proof where it was assumed without citation, prompting a fury of calculations before realising the fact being presupposed was, if true, quite nontrivial. This, in turn, prompted me to look into the theory of the automorphism. What I found gave the strange impression that the existence of the automorphism was, despite being a straightforward combinatorial fact about a specific quite low natural number and so inevitable if anything is, a fantastic stroke of good fortune. Similarly, despite hingeing on several arithmetic facts with no clear connection, it provided an answer to a very basic question about the group structure of the ever-relevant symmetry groups. A serendipity nevertheless aglow with necessity, and a coincidence with depth. I was astonished by it, and consumed for a few days with euphoria and an urge both to preach the good news and to understand the automorphism, as it were, "from the inside." This game is the consummation of that impulse.
Games and toys are a prime aesthetic medium for the presentation of groups and their properties, especially function groups like Sym6 which come with a canonical faithful group action. Actions, as the name suggests, are often readily understood as some transformation it is possible for an agent to effect in an object, and in a game these potential transformations are laid bare to the competent players as moves, affordances, Zuhandenheit, etc. Thus, a simple and elegant means of expressing the object suggested itself.
Mechanically, the game implements a construction of the automorphism along the lines of James Sylvester's "duads" and "synthemes" (in contemporary parlance, graph edges and perfect matchings). Any such explicit construction inevitably comes with the drawback that it selects just one of the 720 conjugate outer automorphisms (or, members of the nontrivial coset in the outer automorphism group) on Sym6 to implement. Sylvester's approach, while particularly straightforward to code, perhaps especially fosters an illusion of some primacy among the various automorphisms: it is interpreted as the induced action of a permutation of the six set on a certain set-of-sets bottoming out in the six set, which come equipped with a canonical (lexicographic) ordering as arrays that suggest a "natural" bjection with six itself. To dispel this illusion, as well as to provide extra challenge to the player, every new session of the game permutes by a random member of Sym6 the six "synthematic totalities" on which the permutations are acting through the automorphism, mimicking the left multiplication of an instance of the outer automorphism by the inner automorphisms of Sym6.
The restriction of input permutations to transpositions was chosen from the intuitive character of the generation of a finite symmetry group by its transpositions, the aesthetic need for some level of gameplay friction even for ideal players, and a desire to underscore the crucial transposition by the outer automorphism of the two conjugacy classes of 2-cycles in Sym6 (to wit, the transpositions and the 3-products of pairwise commutative transpositions). As one playtester observed to me, the effect is to sidestep a natural common approach to puzzle games, in which the player will first solve some restricted region of the board as an initial handhold and thence work outward. The pairing of all possible inputs to derangements across the board renders such strategies impossible.
The art for the game is almost entirely derived from base assets by Anima_nel on itch.io, with the exception of the bookshelf in the library window by GameDev Mum.