# A Collection of Cubing Curiosities

The standard T-perm RUR'U'R'FR2U'R'U'RUR'F' with R/R' replaced by r/r' is an A-perm. This is (probably) useless for speedsolving, but it's interesting nonetheless. The purpose of this page is to collect and preserve these cubing curiosities, which until now have existed as "folklore" with no proper home.

Considered for inclusion are

- Interesting algorithms, fingertricks, or ideas on the 3x3 or other twisty puzzles, not necessarily of any practical use, and especially oddities applying to one or very few cases and with no obvious explanation
- Anything else relating more or less to the cube itself that is particularly interesting

Please send suggestions to the author, with attribution (name and date) when known. Also see the accompanying Speedsolving thread.

Thanks to Clément Gallet and Stefan Pochmann for the inspiration.

## Variations on standard algorithms

- The standard T-perm RUR'U'R'FR2U'R'U'RUR'F' with R/R' replaced by r/r' is an A-perm.
- An E-perm: (LU'Ru2L'UR')2. This is N-perm with u2 instead of U2.
- If you replace U2 with E2, it becomes a Z perm. (Suggested by Robert Yau)
- and LU'Ru2L'UR' LU'RU2L'U(LE2L')R' is a V-perm. (Suggested by François Courtès)
- J-perm RUR'F'RUR'U'R'FR2U'R' with some thick turns RUR'F'rUR'U'r'FR2U'R' is an A-perm. This one is actually potentially useful from this direction.

## Algorithms

- The Sune RUR'URU2R' is a single commutator: [R U R2, R U2 R2]. (suggested by Lucas Garron)
- The ZBLL algorithm LUL'x'U'F'UFU2RUR'U has the following interpretation: pretend F is the cross color, and solve the cross while preserving three pairs; solving the last pair gives a last layer skip. (Suggested by Shotaro Makisumi)
- Another similar ZBLL algorithm. Setup : U' B U B2 R B R' B L U L' B'. Solve : consider F face as cross face, and solve cross then last pair. (Suggested by Clément Gallet)
- A superflip: ((M'U)4 x y')3
- (R U R' U') (L' U' L U) (U R U' R') (U' L' U L) is a 3-cycle of edges on the 3x3x3 and a 3-cycle of corners on the megaminx. (Found by Stefan Pochmann)
- Rotate the U center 180 degrees: (R'U'RU')5
- A hexaflip: (RUR'F)5
- A < R,U > 2-gen S'U2SU2: (R U R2 U' R') (U' R' U2 R U). A variation: R2 U R U R2' U' R' U' R' U2 R'. (suggested by Michael Gottlieb and Lucas Garron)
- An Mu-gen U-perm where all the M-moves and all the u-moves can naturally go in the same direction each time: M2 u' M' u2' M' u' M2' (suggested by Lucas Garron)
- (R U' R' U xy)3 flips 3 corners and 2 edges, so (R U' R' U [r] [u])6 flips 3 corners and (R U' R' U [r] [u])9 flips 2 edges. On the contrary, (R U' R' U y'x')3 = id. (Found by Clément Gallet)
- LL scramble: B2 D' F R2 F' D B U2 B U y2. RUR'U' leaves a J-perm on the R face. Similar case: scramble L R F2 R B L' U2 L B' R2 F L', oriented with r U' L'. (Suggested by ?)
- A 3-cycle: [FRBL,U] (suggested by Thom Barlow)
- The double-Sune can be rotated/mirrored/inverted to perform any 3-cycle of edges for a given corner orientation (keeping corners permuted). (Suggested by Lucas Garron)

## Pretty Patterns

- Cube-in-cube: (R'F'RURU'R'Fy'z')4 (rotation around the ULF corner). (Found by Stefan Pochmann) The base is a great F2L algorithm found by Joel van Noort and popularized by Erik Akkersdijk.
- Cube-in-cube: (R F2 L R F2 L' R2 [u'] [r'])4, rotation around URF. (Found by Per Kristen Fredlund)
- (R U R' U' y)7 is a nice pattern and can also be created by getting four center-edge pairs into one slice and turning it ([R B L R' F R : E]) or getting four corner-edge pairs into one layer and turning it ([R B L R' F R : D] U'). Note that the setup R B L R' F R is the same for both. (Found by Stefan Pochmann)

## F2L

- Set up with R'UR2U'R'y. Solve the FR pair in 4 moves: LF2L'F2. Actually useful from this direction, as rU2r'F2.
- There are 5 (RU-gen) F2L cases where the corner is facing up. Four of these have very nice algs, while the fifth somehow does not. Moreover, each of the four algs is its own self-inverse (by permutation), and preserves the orientation of all other LL pieces.
- (edge in slot)
- (edge clockwise once from corner)
- RU'R'U2RUR' (edge clockwise twice from corner)
- [No good alg] (edge clockwise three times from corner)
- RU2R'U'RU2R'URU2R' (edge clockwise 4 times from corner)

## Fingertricks

- The standard pure octaflip is usually done as (dDrR)3 (popularized by Shotaro Makisumi). Joel van Noort's fingertrick: ((E'U2')(M'L2))3.

## FMC

- All 10 move 2C2E swaps are cyclic shifts and/or inverses of each other. (Suggested by Erik Jernqvist)
- The "31 club": A surprisingly large number of famous/important cubers have 31 moves as their official personal best. The current list includes Dan Cohen, Gilles Roux *and* Lars Petrus, Lucas Garron, Michael Gottlieb, Mike Hughey, Stefan Pochmann, and Yu Nakajima. (And it was the NAR twice) (suggested by Michael Gottlieb)

## Cube Math

- The number of unsolved configurations to a 3x3x3, an 8x8x8, and an 11x11x11 are all prime numbers. (Chris Hardwick use 3+8=11 to remember this). Lucas Garron found that there is no other n x n x n before n = 54 where the number of unsolved configurations is prime. It is not known whether there are infinitely many such n. (Suggested and popularized by Chris Hardwick)
- There are 2*(8C1) + 8C4 = 86 = 2*43 "super-superflips." 43 is an unusual prime factor for a number related to the combinatorics of cubing. (Suggested by Chris Hardwick)
- There are exactly two states in the center of the cube group (identity and superflip), and they are the closest and farthest possible from solved (in HTM). (Suggested by Lucas Garron)
- UF and RBL generate the cube group (proof by Tomas Rokicki). Such pairs are surprisingly common; two randomly chosen elements seem to generate the cube group with roughly 50% chance (experiment by Martin Schoenert; communicated by Lucas Garron and Bruce Norskog)