Post AVB8B0csOMNyM4WhHs by mildsunrise@tech.lgbt
(DIR) More posts by mildsunrise@tech.lgbt
(DIR) Post #AVB8ArtupdfTIJQg64 by mildsunrise@tech.lgbt
2023-04-30T07:33:15Z
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✨ Cross Teaser, part 2: electric boogaloo ✨read part 1 here (summary below):https://tech.lgbt/@mildsunrise/110272249291679953
(DIR) Post #AVB8AtKtUoqHkH3jmq by mildsunrise@tech.lgbt
2023-04-30T07:39:25Z
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this is about analyzing the Cross Teaser (a very niche puzzle) using group theory.in the last thread we introduced the puzzle and concluded:➡️ the intuitive base moves (move the appropriate cross in one of the 4 directions) aren't useful bc they aren't always possible in all states➡️ we'll consider only the subset of puzzle states where the center spot is empty, because every other state is 1 or 2 trivial cross movements away from one of these states➡️ we'll thus only consider puzzle manipulations that take the puzzle between states in this subset
(DIR) Post #AVB8AuJrq7cCnNaQpE by mildsunrise@tech.lgbt
2023-04-30T07:50:43Z
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➡️ we determined 4 "base moves" (and their inverses), which we call U, L, D, R: these each entail 4 "real" cross movements➡️ the effect they have on the puzzle is cycling 3 neighboring crosses (while also rotating them in different ways, ofc)➡️ we proved these 4 base moves generate the whole set of possible manipulations to the puzzle (and thus, of puzzle states reachable through valid moves)
(DIR) Post #AVB8AvX1KhjMYSkRxw by mildsunrise@tech.lgbt
2023-04-30T07:53:34Z
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➡️ we proved there's a bijection between sequences of real cross moves (again, starting & ending in a state of our subset) and sequences of our composite "base moves", as long as both sequences are simplified (don't reverse a previous move)
(DIR) Post #AVB8AwhgyVrSBqkUEq by mildsunrise@tech.lgbt
2023-04-30T08:15:13Z
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➡️ because our composite base moves are closed under composition, they form a group 💙➡️ elements of this group can be modeled as a sequence of two actions: (1) rotate each cross in a distinct way, and (2) permute crosses in some way➡️ we thus chose to embed our group G into the group S₄ ≀ S₈ (since S₄ is the group of rotations of a cube)➡️ so, our group G is the subgroup of S₄ ≀ S₈ generated by {U, R, D, L}
(DIR) Post #AVB8Axz6DHNaA7tu0e by mildsunrise@tech.lgbt
2023-04-30T08:20:33Z
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➡️ S₄ ≀ S₈ contains all manipulations we can do by reassembling the puzzle➡️ the puzzle has the same symmetries as a non-square rectangle: we can rotate and flip 180 degrees, but NOT rotate 90 degrees (despite the appearance)➡️ these symmetries form the so-called Klein four-group K₄ and are expressible as elements of S₄ ≀ S₈
(DIR) Post #AVB8AzA7plnFoc4Dpo by mildsunrise@tech.lgbt
2023-04-30T12:29:41Z
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now we have the base framework to start analyzing this puzzle as a finite group.let's start by numbering the crosses 0-7. we'll do it clockwise, starting with the upper one:
(DIR) Post #AVB8B0csOMNyM4WhHs by mildsunrise@tech.lgbt
2023-04-30T13:03:28Z
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as we know, the rotations of a cube are permutations of 4 elements (the 4 diagonals of the cube).but here instead of expressing rotations as plain permutations, I'll express them in terms of axis quarter-turns {X, Y, Z} since I think it'll be more intuitive.each of them causes the top face to move in this direction:X ↘️Y ↙️Z 🔃
(DIR) Post #AVB8B2Ks2HAfeo7LV2 by mildsunrise@tech.lgbt
2023-04-30T13:08:26Z
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I'll use a prime (A') as notation for the inverse of an element, as it's more screenreader friendly than A⁻¹.remember rotations (and permutations in general) work like function composition: A B means (first apply B then A).I think I'm not forgetting anything so let's start ^^