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Aspects of simplicity and elegance:
Solution uses only 3 transforms. Solves first the corners, then the edges. This also solves the 2x2x2 cube with 2 transforms. Transforms use only the easy Right and Back turns. Transforms are short, simple, similar, repetitive. Transforms have simple, direct effects. Transforms are easy to apply. Description needs no images. Description takes 156 words. |
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The above Solution Method text consists of 44 words of Notation,
156 words of Solution, and 166 words of Notes.
The (verbose) Easy English version has 200 words altogether.
The Thinkers version has 90 words.
If you can solve one layer, you are ready for my solution method. Please read the solution method text with a Cube in your hand, preferably a 2x2x2 cube to start.. The edge transforms are easy, but you need to set up each edge swap, and undo the 1, 2, or 3 set-up turns afterward. Soon you should be able to solve the whole cube, at a leisurely pace, in 10 minutes: Corners: about 50 turns, 1-3 minutes; Edges: about 100 turns, 3-9 minutes. It's about simple elegance, not speed. The following discussion of the Solution Method assumes that the reader has read it! Thoughts on simplicity and elegance. Few transforms. Short transforms. Simple transforms. A solution consisting of just "the quarter-turn", has the fewest, the shortest, and the simplest transforms. A superintelligent entity could apply this transform and solve the cube in 26 quarter-turns, or 20 face-turns. But the solution process would require superhuman intuition. At the other extreme, we can imagine a list of all the 43 quintillion scrambled states, with the 20 face-turn solution sequence listed for each state. The solution process would be as simple as looking up the state of one's cube and applying the turns. But this solution process would be impossibly nonportable and unimaginably clumsy. Elegance Metrics: requirements: little intuition, expertise, deep thinking, memorization, dexterity, struggle. method: simple solution strategy; few concepts; simple concepts; clear; symmetric; transforms: few; short; simple; easy; clear; symmetric; uni-functional; few side-effects; description: concise, understandable, robust, effective; good notation; Which Transforms are sufficient to solve the cube? Corners: We need a corner mover and a corner twister. If we do corners first, our corner transforms can upset edges. If we move before we orient, the movers can upset orientation. Edges: We need an edge mover to place an edge cubie into a cubicle. We do not need a separate edge flipper, since the edge mover can insert the cubie from either side. If we do corners first, our edge transforms must not upset corners. I have found 2 Sufficient Transform Sets, plus a third which has elegance of function, if not form: Simple Elegant: cs, ct, es. cs swaps 2 corners; ct twists 3 corners; es swaps 2 edge pairs. Minimalist: c, es. c swaps 2 corner pairs, twists 2 corners; es swaps 2 edge pairs. Minimal: cs, nct, es. cs swaps 2 corners; nct twists 2 corners; es swaps 2 edge pairs. Transforms compared (Turn-Sequence, Face-Turns, Effects, Side-Effects): Simple Elegant (just beautifully symmetric R & B turns): cs (RB)7 R 15 swap 2 corners: lfu-fld (4 Edges) ct (R[u])12 12 twist 3 corners: rbd,lbd,lfd:R (12 Edges) es (R2B2)3 6 swap 2 edge pairs: ru-rd, bu-bd (no Side Effects) Minimalist (but complicated, based on c): c R B R- B- 4 urf:bru:rfu drb:ldb:bdr swap 2 corner pairs (3E) c2 8 ur:R db:L twist 2 corner pairs (3E) c3 12 ur & db swap 2 corner p. (no SE) cs2 U c U- B 7 bur-bdl, twist bul:R swap 2 cor, twist 1 (3E) ct2 c2 B L- c2 L B- 20 fur:L bdl:R twist 2 corners (3E) ct3l c F2D2[R-] c B2D2[R] 12 bur,bdr,bdl:L twist 3 corners left (4E) Minimal Efficient (few turns or side effects): mcs R U- R2 B- U R- U- B R2 B U 11 urf-urb 2 corner swap (7E) mct R B R2 U2 B- R- B U- R B- 10 urf:L urb:R 2 corner twist (6E) mctl R2 B R2 B- R- B R- B- 8 ruf,rub,rdb:L 3 corner twist L (3E) mctr R2 B- R2 B R B- R B 8 rub,rdb,rdf:R 3 corner twist R (3E) No Side-Effects: nct (U2 R- B D2 B- R) 2 12 blu:R fur:L 2 corner twist nctl U R- U L U- D- F2 D R2 U2 L- U R- 13 urf,urb,ulb:L 3 corner twist left nctr U- B U- F U2 B2 D- R2 U D F- U- B 13 urf,urb,ulb:R 3 corner twist right ne3 R2 U R U R- U- R- U- R- U R- 11 uf:ul:ur 3 edge cycle nef L F R- F- L- U2 R U R U- R2 U2 R 13 fu ru 2 edge flip cses F-UBU-FU2B-UBU2B- 11 2 corner, 2 edge swap While cs has more turns than mcs, it is simpler, easier, less error-prone, and scrambles fewer edges. While ct may take 36 turns, these turns are repetitive, robust, and fast. (The 36 turns can be reduced to 24 by using the Left version of ct.) The Minimalist Transforms (2 edge pair swap and 2 corner pair swap) match beautifully. Holding the cube near the navel, R&B turns are easy. Near the nose, R&U turns are easy. Observations on solving the corner cube (2x2x2) (aka Pocket Cube). 11 face turns (14 quarter turns) suffice to solve the 2x2x2 cube, with 3674160 possible states. (3674160 is remarkably close to the cube root of 43252003274489856000.) I like to do the white layer first. Once we pick a white cubie, there are 1889 ways of scrambling the other 3 white cubies. Having done the white layer, once we place one yellow cubie, there are 161 ways of scrambling the yellow cubies. Using Jaap's Javascript 2x2x2 Rubik's Cube I showed that the 2nd layer may need 11 face turns (to undo cs), even though there are only 161 scrambled states. What is the maximum number of face-turns needed for the first (white) layer, with 1889 scrambled states? Is there a state that requires 11 turns? Flip and Twist Parity (Anne Scott). (Fixing colour positions, e.g.: R=Red, B=Blue:) Chief Face of Cubicle = U if in U-layer, D if in D-layer, R or L if a middle layer edge. Chief Facelet of Cubie = Facelet that matches home cubicle's Chief Face. If Cubie's Chief Facelet is in Cubicle's Chief Face, it is Sane, else Flipped or Twisted. Thus, U,D change no sanity. F,B flip no edges, R,L flip 4 edges. F,B,R,L twist 4 corners. Flip (mod 2) and Twist (mod 3) total to 0. I invite comments and questions. I would like to learn about other elegant solutions. |
[ peter [at] dancing [dot] org ]