cycle {cubing}

Cycle and Twist Cubies

Description

Functions for cycling permuatations and altering orientations.

Usage

flipEdges(aCube, flip = 1:12)
twistCorners(aCube, clock = numeric(0), anti = numeric(0))
cycleEdges(aCube, cycle, right = TRUE, orient = TRUE)
cycleCorners(aCube, cycle, right = TRUE, orient = TRUE)

Arguments

aCube

A cubieCube object.

flip

An integer vector subset of 1:12 giving the set of edges to flip. By default, all edges are flipped.

clock

An integer vector subset of 1:8 giving the set of corners to twist clockwise. By default, none are twisted.

anti

An integer vector subset of 1:8 giving the set of corners to twist anti-clockwise. By default, none are twisted.

cycle

An integer vector representing the permutation cycle. See Details.

right

If FALSE, cycles to the left. See Details.

orient

Controls the orientation change for the permuation cycle. If TRUE each cubie is re-oriented according to the cubie it replaces, so the orientation vector does not change. If FALSE each cubie orientation is fixed so that the orientation vector also cycles.

Details

The cycle vector should be given according to mathematical cycle notation. For example, the vector c(5,3,7) means that the edge at position 5 moves to 3, 3 moves to 7 and 7 moves to 5. The vectors c(3,7,5) and c(7,5,3) are equivalent. The length of the vector is the length of the cycle, and so this example is a 3-cycle. If right is FALSE it cycles in the opposite direction, so 7 moves to 3, 3 moves to 5 and 5 moves to 7.

All of these functions can change the solvability of a cube. Solvability of a cube can be tested using the is.solvable function. For orientation solvability, the sum of the edge orientation vector must be even, and the sum of the corner orientation vector must be divisible by three. For the edge orientation to remain solvable, you must flip an even number of edges. For the corner orientation to remain solvable, the difference between the number of clockwise and anti-clockwise twists must be divisible by three. For example, the number of twists in each direction could be the same (a difference of zero), or you could have three clockwise twists and no anti-clockwise twists.

The sign of a permuation (even or odd) changes under a 2-cycle, which is just a swapping of two elements. In mathematical terminology this is called a transposition. A k-cycle can be written as k-1 transpositions, and therefore a k-cycle will change the sign of a permuatation if and only if k is even. So for a solvable cube to remain solvable, the length of cycle should be odd.

Two binary operators within the package that also impact solvability are %e% and %c%, which are composition operators for only edges and only corners respectively. If two cubes A and B are solvable, then A %e% B may be unsolvable because the edge and corner permuations may be of different sign; the orientations will always remain solvable. It is also possibe for A %e% B to be solvable but B %e% A to be unsolvable. The same reasoning applies to %c%.

In detail: if A and B have odd permutations, then both A %e% B and B %e% A become unsolvable. If A and B have even permutations, then both A %e% B and B %e% A remain solvable. If A has even and B has odd, then A %e% B is unsolvable but B %e% A is solvable.

Value

A cubieCube object.

See Also

%v%, is.solvable, is.solved, invCube

Examples

aCube <- getCubieCube("Superflip")
aCube <- flipEdges(aCube, flip = 1:12)
is.solved(aCube)
aCube <- twistCorners(aCube, clock = 3:6, anti = 2)
is.solvable(aCube)
aCube <- cycleEdges(aCube, c(2,10,5,6))
is.solvable(aCube)