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← Thinking in Turns

Thinking in Turns

Three short lessons on the puzzle cube, for anyone who never got past one face: what a turn is and how to undo it, why four turns bring you home, and why the order of moves matters.

Spark & Whetstone · sparkwhetstone.org/learn/thinking-in-turns · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. A turn and its undo

Cubers name the six faces by where they sit: U for up, D for down, R right, L left, F front, B back. "R" means turn the right face a quarter turn clockwise, as if you were looking straight at that face. "R′" (said "R prime") is the same face turned back the other way.

Every move has an undo. To undo a sequence, undo the last move first and work backwards: the undo of "R U" is "U′ R′". It is like taking off shoes and socks in the opposite order you put them on.

Question. What undoes the sequence "F R U"?

  1. F′ R′ U′
  2. U′ R′ F′
  3. F R U

Say it back. In your own words: why do you undo the last move first?

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Lesson 2. Four turns home

Turn one face a quarter turn four times and the cube is back where it started. That is the simplest example of a pattern that runs through the whole puzzle: repeat any sequence enough times and it returns home.

The famous four-move sequence R U R′ U′, which cubers nickname the "sexy move", comes home after six repeats. The two-move sequence R U takes 105 repeats. You can check both on the cube page.

Question. You turn the top face U three times. What single move brings the cube back?

  1. One more U
  2. U′, a turn the other way
  3. D

Say it back. Say it back: what happens if you repeat any sequence enough times?

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Lesson 3. The order matters

In ordinary arithmetic, 2 + 3 is the same as 3 + 2. Cube moves are not like that: R then U leaves the cube differently from U then R. Mathematicians say the moves do not commute.

This is why cube methods are written as exact sequences, and why solvers work face by face, using sequences that change a few pieces and leave the rest alone.

Question. Is "R then U" the same as "U then R" on a cube?

  1. Yes, always
  2. No, they leave the cube differently
  3. Only on a solved cube

Say it back. In your own words: what does it mean that two moves do not commute?

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At the table: Read me the moves

Players: two people and one real cube. You'll need: a puzzle cube, or the cube page on one device.

  1. One person reads a short scramble aloud from the cube page; the other turns the cube.
  2. Swap: the turner now reads the undo, last move first, and the reader turns it home.
  3. Try R U R′ U′ six times together, counting aloud.
  4. Stop when it stops being fun. Nobody times anything.

Seat sheets and a run-through: sparkwhetstone.org/cube

Answers

  1. U′ R′ F′ — Reverse the order and turn each face back: U′, then R′, then F′.
  2. One more U — Four quarter turns make a full turn, so one more U brings the face home.
  3. No, they leave the cube differently — They give different cubes: the two moves share pieces, so their order matters.