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I devised a Rubik's Cube variant, called the Delta Cube. I am interested in calculating its God's Number using the quarter turn metric. Below is a solved Delta Cube.

To clarify exactly how moves work here:

We take the Delta cube below (here the surfaces are not colored but the edges are colored) this is to highlight the edges. Assume the Delta cube is centered at $(0,0,0)\in \Bbb R^3$.

An example of a Delta move is: Slice by a $z=0$ plane which slices and disconnects the object, into equal halves, through the blue (rubber band) arc shown. Then rotate the top half by a quarter turn or 90 deg. and re-attach.

enter image description here

The Delta Cube is functionally equivalent to a Pocket Cube (2×2×2 Rubik’s Cube),
except that pairs of antidiagonal corner cubes are assigned the same color. In addition, we define edges as paths between adjacent corner cubes.
These edges are partitioned into three equivalence classes, each forming a closed loop around the cube. We assign each equivalence class a distinct color, resulting in three edge colors total. Thus, when the cube is rotated (by standard Pocket Cube moves), both the colored corner pieces and the colored edge classes move accordingly.
The underlying move structure is identical to the Pocket Cube — the Delta Cube simply emphasizes the antidiagonal color pairing and colored edge loops instead of face stickers.

If the Delta cube is centered at the origin in $\mathbf R^3$ then the $z=0$ plane partitions the cube into "top" and "bottom." Then a Rubik's "move" otherwise called a Delta move would rotate the upper half by $90$ degrees. This sends the Yellow to the Green, and the Green to Red and the Red to Purple and the purple to Yellow. This is an example of a "quarter turn." Including $x,y=0$ planes into this, we get the full set of moves or Delta moves, which again is functionally the same as a pocket cube's moves.

A functional diagram of a Delta move wrt. to the $z=0$ plane.

enter image description here

Solved Delta Cube

Delta Cube Solved Configuration

  • As you can see, the cube consists of $8$ corner pieces (located at cone points) and $12$ edge pieces (represented by colored arcs between corners).
  • The $8$ corner pieces are colored using $4$ colors in total.
  • The $12$ edge pieces are colored using $3$ colors in total.
  • Rotations are permitted along $1$-dimensional arcs analogous to standard Rubik's cube face turns.

The structure imposes the following constraints:

  • Only even permutations of corners and edges combined are allowed (parity constraint).
  • The sum of the corner orientations must be congruent to $0 \mod 3$.
  • The number of flipped edges must be even.

After modding out symmetries due to color indistinguishability, the total number of distinguishable states is approximately

$$ 1.955 \times 10^{14}. $$

What is God's Number for the Delta Cube (using the quarter turn metric)? That is, what is the minimum number $N$ such that any scrambled state of the cube can be solved in at most $N$ moves?

It should be around $23$ or $24$ based on comparing the known God's numbers of the usual $2\times2\times2$ Rubiks cube and the $3\times3\times3$ Rubiks cube respectively. Therefore, the Delta Cube is technically easier to solve than the $3\times3\times3$ due to less states in the state space, but would likely take some time for a professional or amateur cuber to get used to.

J. Zimmerman
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    It's not clear from the picture how the puzzle works. (I can't even tell where the edge pieces are.) Can you add a picture of what it looks like after performing one move from the solved state? Alternatively, can you say in words what axes you twist, which exact pieces move when you do so, and which pieces/stickers are indistinguishable? – Ravi Fernando Apr 27 '25 at 16:05
  • I assume that the number of positions is $195547613184000 = 2^{14} \cdot 3^{11} \cdot 5^3 \cdot 7^2 \cdot 11$ (which seems to be the only divisor of the order of the Rubik's cube group near the number you mention). – Ravi Fernando Apr 27 '25 at 16:12
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    I am also not clear how the puzzle works. maybe label every color face, with its own unique number, then tell us how each basic rotation moves them using cycle notation. so something like this https://ruwix.com/pics/rubiks-cube/mathematics-permutation-group.jpg using this https://en.wikipedia.org/wiki/Permutation#Cycle_notation. it might be a little overkill but then at least there will be no ambiguity. – timidpueo Apr 28 '25 at 02:53
  • @RaviFernando It's basically a 2×2×2 cube where antidiagonal corner pairs share colors, and the edges are grouped into three colored loops. Rotations are identical to Pocket Cube moves — just acting on corners and colored loops instead of faces. – J. Zimmerman Apr 28 '25 at 17:59
  • @timidpueo I've tried to do that - let me know if my update is on the right track. – J. Zimmerman Apr 28 '25 at 18:12
  • A couple of things I'm still unclear on: (1) it looks like four of the edges are on the equator. What happens to these when you turn the top half of the puzzle: do they move with the top or stay put with the bottom? (2) From your picture, it doesn't look like the orientations of any of the pieces would be visible. Is that your intention? – Ravi Fernando Apr 29 '25 at 15:42
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    @J.Zimmerman, I don't see any cycle notation in your edit? here is a more explicit example, for the rubiks cube: https://math.stackexchange.com/a/1641564/54230. don't worry about what GAP is, just look at how he defines each piece with a number and where each type of rotation moves that number. – timidpueo Apr 29 '25 at 23:29
  • @RaviFernando the four edges on the equator just get sliced in half or copied to the top and bottom halves then you do the turn and reattach them. – J. Zimmerman Jun 17 '25 at 16:37
  • If a piece gets sliced in half by a move, then it's not one piece, it's two. In particular, if you do (say) R U2 to your picture of the 3d printed puzzle, each of the four edges around the equator will be half blue and half red. – Ravi Fernando Jun 18 '25 at 14:26
  • In fact, if I'm understanding your description correctly, you could simulate this puzzle with an ordinary 2x2 cube by painting the corners in your four corner colors, and then painting a border around each corner to represent your edge pieces. (This is somewhat similar to how 4x4 and 5x5 supercube stickers usually have little colored fringes to disambiguate between centers of the same color.) – Ravi Fernando Jun 18 '25 at 14:32
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    If I'm understanding correctly, then I claim the Delta Cube is literally equivalent to a standard 2x2 cube. Reason: the moves are the same, so the only question is which positions are indistinguishable. But I claim that the corner colors and edge colors combined disambiguate everything... – Ravi Fernando Jun 18 '25 at 14:39
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    For example, consider the corner facing the camera in your 3d printed picture. In the actual puzzle, it would have a corner color as well--let's say yellow. The only other yellow corner is the opposite one, whose surrounding edges are blue, red, purple in clockwise order instead of blue, purple, red. So the edge colors distinguish opposite corners from each other. They also distinguish each corner from twisted versions of itself, of course. So no two (corner, orientation) pairs have the same appearance; i.e., the colors disambiguate everything. – Ravi Fernando Jun 18 '25 at 14:41
  • @RaviFernando Okay that makes sense - thanks for helping me see that the Delta Cube is equivalent to the 2x2x2 cube. – J. Zimmerman Jun 18 '25 at 18:03
  • I agree that it is equivalent and so has the same number of states as the normal 2x2x2 cube. It does seem to me however that whereas the 2x2x2 has only one solved state, this puzzle has more than one (I think 6; once one pair of opposing corners is fixed, the other three pairs can be solved with any permutation of the the tip colours). So it will be a slightly easier puzzle to solve. – Jaap Scherphuis Jun 20 '25 at 09:48

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