What Makes a Position Legal?
A "legal" position is any arrangement that can actually be reached from the solved cube by turning its faces. Arrangements that can only be made by disassembling the cube and reassembling it, or by peeling off stickers, are called "illegal".
The 3 conditions for a legal 3x3x3
It is a well-known fact from group theory that a 3x3x3 arrangement is actually reachable (legal) only if all three of the following hold:
(1) The corner permutation and the edge permutation must both be even (one cannot be odd while the other is even).
(2) The sum of the corner orientations must be a multiple of 3.
(3) The sum of the edge orientations (the number of flipped edges) must be a multiple of 2.
Conversely, every arrangement satisfying these three conditions is known to be reachable.
How many arrangements are there?
The 3x3x3 cube has 43,252,003,274,489,856,000 distinct arrangements (about 43 quintillion), and the 2x2x2 cube has 3,674,160. These numbers are widely documented.
The 2x2x2 case
The 2x2x2 cube has no edges, so conditions (1) and (3) above drop out. The only remaining condition is that the sum of the corner orientations be a multiple of 3 (the permutation itself is unrestricted).
Parity on 4x4x4 and 5x5x5
On 4x4x4 and 5x5x5, same-colored centers and edge pairs are indistinguishable, so reducing to 3x3x3-style solving can leave what looks like a single swapped pair (commonly called "OLL parity" and "PLL parity"). This is not an illegal state — it's a well-known phenomenon specific to even-layered cubes.
What cubepic lets you do
cubepic instantly renders a photorealistic cube image or mirror-view image straight from a net (unfolded) form input, whether the arrangement is legal or not — and tells you on the spot whether it's legal.
→ Make a cube image on the top page© アサラボ