Algorithm transformation is the process of adjusting or altering an existing sequence of cube moves. It includes manipulations such as reverse, inversion, permutation, and symmetries.
While these concepts may sound challenging, they are fundamental to understanding algorithms. They're part of the broader concept of cube group theory^{1}, which provides a mathematical foundation for studying cube symmetries and transformations.
The goals of algorithm transformations and symmetries include, among others:
 Finding new algorithms
 Reducing cube solving times
 Optimizing finger tricks
 Enhancing solving methods
 Customizing existing algorithms to suit solvers' preferences
 Identifying better memorization patterns
All these algorithm transformations can be applied within the CubePal app.
You can download the CubePal app for free on the App Store.
Reverse
This transformation involves reversing all moves by flipping the entire sequence and inverting each move. In fact, it is identical to undoing the initial sequence of moves.
For example, if your original sequence is R' B' U
, its reverse would be U' B R
.
By applying the reverse operation to certain algorithms, we can obtain a new algorithm that performs the same transformation as demonstrated in the following PLL Headlights algorithm:
To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Inversion
Inverting each move in a sequence means replacing each move with its opposite. It may sound similar to reversing, but its concept differs from the reverse transformation.
For example, if you have a sequence of moves like R' B' U
, its inversion would be R B U'
.
By applying the inversion operation to certain algorithms, we can obtain a new algorithm that performs the same transformation as demonstrated in the following 2 Checkers 4H pattern:
To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Permutations
Permutation involves manipulating the first and last moves in a sequence while keeping the rest of the algorithm intact. In this case, the sequence of moves can be shifted left or right, or the first and last moves can be inverted or swapped.
Sequence shifting
Shifting operation involves permuting all moves of a sequence to the left or right and rearranging the position of
the first or last moves. Please note that half turns, like R2
, are initially split into two 90degree
turns (R
and R
) before being rearranged to the beginning or end of the sequence.
Shift Left
This permutation means moving each move in the sequence one position to the left, and moving the first move to the end of the sequence.
For example, if your original sequence is R' B' U
, its shifting to left permutation results in
B' U R'
.
By applying the Shift Left operation to certain algorithms, we can obtain a new algorithm that creates a new patten as demonstrated in the following 4 Crosses Pattern that after Shift Left operation transforms into a 6 Crosses Pattern:
To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Shift Right
This permutation means moving each move in the sequence one position to the right, and moving the last move to the beginning of the sequence.
For example, if your original sequence is R' B' U
, its shifting to right permutation results in
U R' B'
.
By applying the Shift Right operation to certain algorithms, we can obtain a new algorithm that performs the same transformation as demonstrated in the following 4 H Pattern::
To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Flipping (inverting) the first and last moves
Flipping or inverting the similar first and last moves results in another permutation. This permutation is based on
the fact that
two similar moves, inverted or not, will result in a halfturn move: R
+ R
and
R'
+ R'
are both equivalent to R2
half turn.
An example of this would be flipping first and last moves in the sequence R B U R
which will result in
R' B U R'
.
By applying the flip operation to certain algorithms, we can obtain a new algorithm that performs the same transformation as demonstrated in the following Superflip pattern:
Flip
↓R' L' U R F' D B' L2 D R' F' R' B' L D' B L F' L D' F2 U' R' F B R'
To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Swapping first and last moves
This permutation only deals with cases where the first and last moves are along the same axis of a cube. This permutation swaps the first and last moves.
For example R B U L
will result in L B U R
.
By applying the swap operation to certain algorithms, we can obtain a new algorithm that performs the same transformation as demonstrated in the following 4 Crosses pattern:
To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Notes:

Cube group theory is a branch of mathematics that explores the symmetries and transformations inherent in a cube. It provides a formal framework for understanding how sequences of cube moves can be analyzed mathematically, aiding in the development of solving algorithms and methods. ↩