An Alternate Universe of the Cube

I've been meaning to explore new variations on the 3x3x3 cube for a while and I think I've come up with something new.

If we consider the rotation of a 2x2x2 block as one move, say the UFR block, and call it the z-move for lack of a better name. Now all sorts of weird stuff becomes possible, e.g. (z, C_U2)^6 will generate a 5-spot pattern! The centres cycle (U,L,R,B,F) in befuddler notation.

So in this particular universe of the 3x3x3 cube we are considering 8 corner moves which move a 2x2x2 block rather than the usual 6 face moves. I won't spoil all the fun, but a 3-spot, 4-spot and 6-spot are possible and these spots patterns are quite different from what is possible in the 'normal' universe of the cube.

All this is merely a first step towards a program where the user can define their own moves on the 3x3x3 cube. Think of a flip-cube where the user could flip a face or 1x3x3 area of the cube 180 degrees. This would require a true 27-cube matrix instead of a normal Rubik's Cube so when you flip a face you still get colours all around, but I think it would be interesting enough as an alternate squares group.

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Triamid-like cube idea

The comment about flipping a face layer made me think of a puzzle I have called Rubik's Triamid. The puzzle is basically in the shape of a tetrahedron like the Pyraminx, but you can detach a smaller tetrahedron of pieces from any corner, and reorient that section one of 12 ways (11 if you don't count the original orientation) and reattach it to the puzzle. You could also just detach a single tip piece, and reattach it in a different orientation, but there was a rule "forbidding" that.

The same idea could be applied to the 3x3x3 (or Oh Cube) puzzle. You would be allowed to detach a 2x2x2 subcube from any corner and reattach that 2x2x2 subcube where it came from but in any of 24 orientations. If the 27 cubies can be distinguished as well as their 24 orientations, you could seemingly have 27!*(24^27) states (about 2.0*10^65) to the puzzle, or 27!*(24^26) if the orientation of the puzzle is not considered to matter. If my GAP model is correct, however, "only" about 2.49*10^56 of these states are reachable by detaching and reattaching the 2x2x2 subcubes. This should be divided by 4 if the orientation of the puzzle doesn't matter. It appears you can simulate a 180-degree flip of the entire puzzle by performing moves on it, but not other reorientations of the entire puzzle.

Corner Turning Cube Video

I like Bruce's idea very much and one definitely wants to hack up a simulation of this puzzle (actually making a real one seems rather difficult).

What I have done so far is hack up the corner turning Oh-cube and I made a short video showing the 5-spot pattern:

Corner Turning Cube

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Hi Bruce,

Cube orientations have a parity (given by the parity of the permutation of the 6 faces). The 27 cubelets locations split into 2 sets according to a checkerboard colouring. I think that the only way to move a cubelet from one set to the other is by a move with an odd reorientation parity. Therefore the orientation of any cubelet has even parity if it is in a location of the same checkerboard colour as its home location, and odd if it is in a location with a different colour. This alone reduces the number of positions the puzzle has by 227. Solving the puzzle in a position reoriented by a quarter turn is already ruled out by these constraints.

The parity of the cubelet permutation is even (a quarter turn of a 2x2x2 is two 4-cycles, an even permutation, and such quarter turns generate all possible moves). This gives another factor of 2.

The number of positions is therefore at most 27!*2427/228 = 7.4790e+56

The number you give is one third of that. If your Gap model is correct, this means that there is some kind of twist invariant as well. It's a tricky one.

Define the twist of a cubelet in one of the sets of locations as 0 if the U/D colours are in the U/D facelets, 1 if they are in the R/L facelets, 2 if they are in the F/B facelets. The twist of a cubelet in the other set of locations is similar but with 1 and 2 swapped.

A quarter turn of a 2x2x2 subcube consists of two 4-cycles. A turn around the U/D axis will leave all cubelets in the same orientation. A quarter turn around one of the other axes will increment the orientations in one cycle and decrement those on the other, and therefore still leave the total twist the same modulo 3.

This also shows that it is impossible to solve the puzzle in a reoriented position 120 degrees rotated about a corner. All the cubelets on one set will have to be twisted +1, and in the other set -1. Since there are 14 in one set and 13 in the other, the total twist would have to change.

Jaap's Puzzle Page:

GAP code

Thanks, Jaap, for analyzing the invariants of the puzzle.

If it's useful to anyone, my GAP definitions are found below.

I used 6 consecutive numbers for each cubie. The order of the facelets is U,D,L,R,F,B. The first cubie is ULB. The rest of the U layer cubies follow in the order UB, UBR, UL, U, UR, UFL, UF, and URF. This is followed by the middle (horizontal) layer, and finally the bottom layer.

To simplify the definitions, I defined functions that generate "moves" for 3-cycling whole layers. I then only needed to explicitly define 3 per permutations of the puzzle facelets (raw numeric definitions). These "moves" were "U1" (equivalent to performing U on a Rubik's Cube) and "L1" (equivalent to performing L on a Rubik's cube), and "cyu" (similar to U1, except only cycles four cubies on the U layer). From "U1" and "L1" and the layer 3-cycles I could make definitions for whole cube rotations. Conjugating "cyu" by a layer shifter allowed defining "cyd" which cycles the 4 cubies below the "cyu" cubies. Combining "cyu" and "cyd" then gives one of the basic moves of the puzzle. Then conjugating by whole cube rotations allows defining all the other basic moves of the puzzle.

The actual code is:
U1a := (1,13,49,37)(2,14,50,38)(7,31,43,19)(8,32,44,20);
U1b := (9,36,46,23)(10,35,45,24)(27,30,28,29);
U1c := (3,18,52,41)(12,34,47,21)(6,16,53,39);
U1d := (4,17,51,42)(11,33,48,22)(5,15,54,40);
U1 := U1a*U1b*U1c*U1d;

MakeV1 := function ()
  local i, g;
  g := ();
  i := 1;
  while i <= 54 do
    g := g*(i, i+108, i+54);
    i := i + 1;
  return g;
V1 := MakeV1 ();

E1 := V1*U1*(V1^-1);
D1 := (V1^-1)*U1*V1;
y := U1*E1*D1;
y2 := y*y;
y3 := y2*y;

L1a := (3,39,147,111)(4,40,148,112)(21,93,129,57)(22,94,130,58);
L1b := (1,41,146,114)(19,95,128,60)(6,37,149,110);
L1c := (5,38,150,109)(20,96,127,59)(2,42,145,113);
L1d := (23,92,132,55)(73,77,74,78)(24,91,131,56);
L1 := L1a*L1b*L1c*L1d;

MakeH1 := function ()
  local i, j, g;
  g := ();
  i := 0;
  while i <= 144 do
    j := 1;
    while j <= 6 do
      g := g*(i+j, i+j+12, i+j+6);
      j := j + 1;
    i := i + 18;
  return g;
H1 := MakeH1 ();

M1 := H1*L1*(H1^-1);
R1 := (H1^-1)*L1*H1;
x3 := L1*M1*R1;
x2 := x3*x3;
x := x2*x3;

cyu := (25,31,49,43)(26,32,50,44)(27,36,52,47)(28,35,51,48)(29,33,54,46)(30,34,53,45);
cyd := V1*cyu*(V1^-1);
cy := cyu*cyd;
cx := y3*x3*cy*x*y;

URFcy := cy;
UFLcy := y3*cy*y;
ULBcy := y2*cy*y2;
UBRcy := y*cy*y3;
DRBcy := x2*cy*x2;
DBLcy := x2*y3*cy*y*x2;
DLFcy := x2*y2*cy*y2*x2;
DFRcy := x2*y*cy*y3*x2;

URFcx := cx;
UFLcx := y3*cx*y;
ULBcx := y2*cx*y2;
UBRcx := y*cx*y3;
DRBcx := x2*cx*x2;
DBLcx := x2*y3*cx*y*x2;
DLFcx := x2*y2*cx*y2*x2;
DFRcx := x2*y*cx*y3*x2;

G := Group (URFcy,UFLcy,ULBcy,UBRcy,DRBcy,DBLcy,DLFcy,DFRcy,

Oh Cube

The first puzzle you describe exists for real: Oh Cube by Oskar van Deventer.

Jaap's Puzzle Page:

Oh Cube is a surprise

This took me by surprise. I never would have thought there would be enough clearance to rotate the 2x2x2 block on a real puzzle. Amazing work! I must post some patterns later.

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