page_title: Rubik's family cubes of all sizes
section_title: Final layer problems
hierarchical_section_title: Rubik's family cubes of all sizes / Cube solving / Final layer problems
caption_attribution_description: English: Orientation of last layer after correction for 9x9x9 Rubik’s family cube.
context_page_description: The original Rubik's cube was a mechanical 3×3×3 cube puzzle invented in 1974 by the Hungarian sculptor and professor of architecture Ernő Rubik. Extensions of the Rubik's cube have been around for a long time and come in both hardware and software forms. The major extension have been the availability of cubes of larger size and the availability of the more complex cubes with marked centres. The properties of Rubik’s family cubes of any size together with some special attention to software cubes is the main focus of this article. Many properties are mathematical in nature and are functions of the cube size variable.
context_section_description: A "final layer problem" is defined here to mean a need for a rearrangement of final layer edge cubies that cannot be achieved using standard size 3 cube moves. These are often referred to as parity problems or errors, but such terminology may be misleading. If moves were limited to those available to the size 3 cube, such states would be unreachable (break parity rules). There are numerous variations in the way the final layer problems are presented and the algorithms to resolve them, but the correction requirement will be similar to that described below. The problems considered here apply equally to standard cubes and to those with marked centres but in the latter case additional final layer issues arise for aligning centre cubies. The problems for larger cubes can be considered as straightforward extensions of those that apply to the size 4 cube. Basically, two types of problem can arise:
  There is a need to flip a complementary pair or a complete set of edge cubies in the final edge set. This condition will be referred to as an OLL (orientation of last layer) requirement.
  There is a need to swap the positions of two edge cubie sets in the final layer. This condition will be referred to as a PLL (permutation of last layer) requirement.
  OLL and PLL as used here can be considered to be sub-sets of the usual definitions of these terms. There are many references to moves that can be used to resolve these problems. Fewer references demonstrate how these moves satisfy parity rules. From a parity perspective, there is a need to consider the rearrangement of centre cubies which is not readily observable in cubes with unmarked centres. Only OLL parity compliance will be illustrated here.
  A typical OLL correction for a size 9 cube is shown. The cubies shown in colour are the only ones in the cube that change positions.
  For the OLL correction there are  centre cubie swaps and overall there are  swaps when the edge pair is included. For odd size cubes  is always even (and conforms with the universal even parity requirement for odd size cubes). For even size cubes  is always odd which means in this case a parity reversal always occurs, an allowable parity condition for even size cubes.
  For the complete edge set flip (a requirement that can arise only for cubes of even size), the number of swaps will be . The overall number of swaps will be even if  is even (i.e.  is odd). The overall number of swaps will be odd if  is even. Hence overall parity will be even if  is odd and odd if  is even.
  The parity of a given algorithm can, of course, also be deduced from its content using the rules detailed in the Rules for Rubik’s family cubes section.
  For standard cubes the rearrangement of centre cubies to resolve the OLL and PLL problems is unimportant. For cubes with marked centre cubies the effect of this rearrangement of these cubies is a serious drawback. For cubes with marked centres it is not possible (except for the size 4 cube) to align all final layer centre cubies until all edge cubies have been placed in their final positions.
caption: OLL after correction for size 9 cube
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