Timeline for 3d slide puzzle group
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 27, 2022 at 16:40 | comment | added | Joachim König | I guess it may be even simpler to not allow rotations of the whole cube and just consider the group of "slides" bringing the gap back to its original position (even though depending on the exact rules, this may slightly reduce the positions considered "solvable"). Then there are 19 pieces which move in total (one corner as well as the centers of each face are fixed), and every move is a composition of 3-cycles, so the group is contained in $A_{19}$. Now looking at a few selected moves will give that it's actually $2$-transitive and contains some short cycle, forcing it to be $A_{19}$ exactly. | |
| Jan 17, 2022 at 14:31 | comment | added | Peter Taylor | If I'm correctly interpreting the results of my calculations with GAP then the group is $(C_2 \times C_2) \rtimes S_{19}$, but take that with some suspicion. | |
| Jan 17, 2022 at 14:08 | comment | added | Peter Taylor | Obvious optimisation for manual calculation: it only takes one (slide plus rotate cube to put gap back in position) and the rotation by 1/3 to generate the group. | |
| Jan 17, 2022 at 12:11 | comment | added | Peter Taylor | The gap starts in a corner of the cube, and effectively all slides move it to another corner of the cube. Then there are three legal slides from any position. For each, determine a permutation of the positions of the 26 external cubies generated by making the slide and then rotating the cube to put the corner back where it started. Then the group is the group generated by these three permutations (each chosen arbitrarily from three options) and the permutation corresponding to a rotation by 1/3 around the axis through the gap and the centre of the cube, quotiented by the latter permutation. | |
| Jan 17, 2022 at 10:55 | review | Close votes | |||
| Jan 22, 2022 at 3:07 | |||||
| S Jan 17, 2022 at 10:27 | review | First questions | |||
| Jan 17, 2022 at 10:39 | |||||
| S Jan 17, 2022 at 10:27 | history | asked | sno | CC BY-SA 4.0 |