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Grip Theory

This whole page is under construction

Now that you understand the fundamentals of group theory, we can introduce grip theory. At its core, grip theory is very simple, but there are a lot of extensions that allow us to model more complicated puzzles.

Grips

  • A grip is a thing you can turn. Every puzzle has a grip set, denoted using \(\Gamma\) (capital gamma).
  • The grip group is a permutation group that acts on the grip set, denoted using \(G\).

Info

Note: The elements of the grip group are transformations, not grips. Each element of the grip group is a permutation of the grip set.

On a 3x3x3:

  • The grip set is \(\{ R, L, U, D, F, B \}\).
  • The grip group is the group of rotations of a cube.
  • The grip group acts on the set of grips the same way it acts on the faces of a cube.

This is already enough to construct cube rotations. For example, a y rotation applies the 4-cycle \((RFLB)\) to the grip set.

Exercise

Consider an x2 rotation on a 3x3x3.

  • What is the permutation applied to the grip set? Answer: \((UD)(FB)\)
  • What grips are fixed by the permutation? Answer: \(\{ R, L \}\)

Twists

A twist is defined using a grip \(g \in \Gamma\) and an element of the grip group \(\sigma \in G\) that keeps that grip fixed (i.e., \(\sigma[g] = g\)).

Confusing notation

  • \(\Gamma\) is the grip set. \(g\) is a grip.
  • \(G\) is the group group. \(\sigma\) is a group element.

I’m terribly sorry.

In the wider world of mathematics, \(G\) is often used for groups and \(\sigma\) is often used for permutations, so we really don’t have many letters left to choose for “grip” and “grip set.”

For example, on a 3x3x3 the R' twist is defined as \(R[(UFDB)]\).

On most puzzles we care about, all valid twists exist. In other words, any grip can be have any transformation applied to it as long as that grip stays fixed.

Exercises

  • What is the D2 twist? Answer: \(D[(RL)(FB)]\)
  • Is \(L[(URF)]\) a valid twist on 3x3x3? Answer: No, because \((URF)\) is not an element of the grip group.
  • Is \(L[(URF)(LDB)]\) a valid twist on 3x3x3? Answer: No, because \((URF)(LDB)\) doesn’t keep \(L\) fixed.

Pieces

  • A piece is a a thing that moves around. On a 33, there are 27 pieces. Every puzzle has a set of pieces.
  • A grip signature is a subset of a puzzle’s grips.
  • Each piece has a grip signature, which is a subset of the 6 grips that determines which twists affect the piece. For example, the grip signature of the RUF corner is \(\{ R, U, F \}\).

The grip signature tells us which twists affect which pieces.

Surprisingly, a grip signature is the only thing we need in order to describe how a piece behaves.

Applying twists

To apply a twist \(g[\sigma]\) to a puzzle:

  1. Take every piece that has the grip \(g\) in its grip signature.

The rest of this page is under construction

Grip group from twists

Complex puzzles

Anti moves

Indistinguishable pieces

TODO: attitude sets

Blocked grips

To model bandaging, …

TODO

Jumbling

TODO: ouch