# Helicopter Cube / Curvy Copter

The Helicopter Cube is a cube-shaped puzzle in which a normal move consists of twisting part of the cube around the midpoint of an edge by a half turn. The part of the cube that twists during a move is shaped like a triangular prism, with its cutting plane parallel to an edge and cutting through the midpoints of the two adjacent faces. All twelve edges can be twisted this way, so the cube is divided up into 8 corner pieces and 24 triangular centre pieces (4 on each face).

The Curvy Copter is a variant on the Helicopter Cube in which the cutting planes are curved. The effect of this is that each edge has an extra central piece. These extra pieces do not change position but their orientations do change, making this a more difficult puzzle. It was designed by Tom van der Zanden (TomZ).

A very interesting property of these puzzles is that they "jumble". Part way through a move, some of the adjacent edges suddenly become free to move. If you move these adjacent edges and mix it further in this way, then its shape changes and some axes can become blocked from moving. Even if you get it back into a cube shape, it is quite likely that it is not possible to solve it again without jumbling.

## The number of positions:

### Helicopter Cube, no jumbling

Lets assume that we have a Helicopter Cube which is only mixed using normal 180 degree moves, i.e. without jumbling. There are 8 corner pieces, with 3 orientations each. They can be arranged in at most 8!·38 ways. There are 24 centre pieces which can be arranged in at most 24! ways. Together that gives a maximum of 8!·38·24! positions. This limit is not reached because

• The orientation of 7 corners determines the orientation of the last corner (3)
• The centres actually occur in 4 distinct orbits, each containing all colours (24! / 6!4)
• The permutation of the face centres is even (2)
• The orientation of the puzzle as a whole does not matter (24)

This gives a total of 7!·36·6!4 / 2 = 493,694,233,804,800,000 positions, all of which can be reached.

### Curvy Copter, no jumbling

The difference with the Helicopter Cube is that the additional edge centres give the puzzle visible reference points, and that these edges have two orientations. So we have 8 corner pieces, arranged in at most 8!·38 ways, 24 centre pieces which can be arranged in at most 24! ways, and 12 edges which can be arranged in at most 212 orientations. Together that gives a maximum of 8!·38·24!·212 positions. This limit is not reached because

• The orientation of 7 corners determines the orientation of the last corner (3)
• The face centres actually occur in 4 distinct orbits, each containing all colours (24! / 6!4)
• The permutation parity of the face centres is even (2)
• The permutation parity of the corners matches the parity of the total number of edge flips (2)
• The permutation parity of each face centre orbit matches the parity of the number of edge flips along that orbit (24)

This gives a total of 8!·37·6!4·26 = 1,516,628,686,248,345,600,000 positions, all of which can be reached.

### Helicopter Cube, with jumbling

Lets assume that we have a Helicopter Cube which has been mixed using jumbling moves, but is back into a cube shape. There are 8 corner pieces, with 3 orientations each. They can be arranged in at most 8!·38 ways. There are 24 centre pieces which can be arranged in at most 24! ways. Together that gives a maximum of 8!·38·24! positions. This limit is not reached because

• The orientation of 7 corners determines the orientation of the last corner (3)
• The centres occur in 6 sets of indistinguishable pieces (4!6)
• The orientation of the puzzle as a whole does not matter (24)

This gives a total of 7!·36·24! / 4!6 = 11,928,787,020,628,077,600,000 cube-shaped positions, all of which can be reached.

To count non-cube positions, we need to count all the possible shapes (ignoring the colours). Counting those shapes is tricky, since sometimes moves are blocked purely due to the shape of the pieces rather than the underlying mechanism. Matt Galla has done a full analysis, and wrote up his results in this post on TwistyPuzzles Forum. I have reproduced and verified his results. He found 14,098 shapes, or 28,055 if mirror images are counted too. Some of these have symmetry however, and therefore occur in fewer than 24 (or 48) possible orientations. Here is a breakdown of those symmetries:

Symmetry
mr4r3r2
Oh, m3m

mr3r2
D3d, 3m

r3r2
D3, 322

mfr2e
C2v, mm2

mer2e
C2h, 2/m

r2er2e
D2, 222

m4
S4, 4

me
Cs, m

r2e
C2, 2

r2f
C2, 2

mc
S2, 1

i
C1, 1
Total
Order 48 12 6 4 4 4 4 2 2 2 2 1
Index 1 4 8 12 12 12 12 24 24 24 24 48
Shapes 1 1 8 1 18 4 1 82 764 5 37 13,176 14,098
Shapes 1 1 16 1 18 8 1 82 1,528 10 37 26,352 28,055
Total 1 4 128 12 216 96 12 1,968 36,672 240 888 1,264,896 1,305,133

The row marked Order shows the sizes of the symmetry groups. The Index is the index of the symmetry group as a subgroup of the full cubic symmetry group, i.e. it is 48 divided by the order. The index is also the number of ways any particular shape with that symmetry can be oriented in space (including reflections). The first Shapes row lists the number of shapes that Matt found for each symmetry group but not counting mirror images, and the second Shapes row includes the mirror image shapes in its count. The row marked Total is the product of the index and the number of shapes.

Combining this with the previous result, we find that there are 1,305,133 * 7!·36·24! / 4!6 = 15,568,653,590,593,384,802,320,800,000 jumbled positions all together.

### Curvy Copter, with jumbling

Again, let's assume that the Curvy Copter has been mixed with jumbling moves, but that the puzzle is back into a cube shape. The difference with the Helicopter Cube is that the additional edge centres give the puzzle visible reference points, and that these edges have two orientations. So we have 8 corner pieces, arranged in at most 8!·38 ways, 24 centre pieces which can be arranged in at most 24! ways, and 12 edges which can be arranged in at most 212 orientations. Together that gives a maximum of 8!·38·24!·212 positions. This limit is not reached because

• The orientation of 7 corners determines the orientation of the last corner (3)
• The centres occur in 6 sets of indistinguishable pieces (4!6)
• The permutation parity of the corners matches the parity of the total edge orientations (2)

It is a little surprising that this last parity constraint still holds when the puzzle is jumbled, but it is due to the fact that a move will always involve two corner pieces on opposite sides of the edge centre.
This gives a total of 8!·37·24!·212 / (4!6·2) = 586,323,739,637,911,270,195,200,000 cube-shaped positions, all of which can be reached.

Calculating the number of positions of any shape is a little easier than with the normal Helicopter cube. We do not have to take symmetry into account because the edge pieces make the orientation of the puzzle visible. The previously mentioned work by Matt Galla shows there are 28,055 shapes, as there is no difference between the normal Helicopter Cube and the Curvy Copter as to what moves are blocked or available in each shape. Note however that there are some moves that are blocked but which can be done by forcing pieces past each other, but such moves are not taken into account here, and such moves may lead to otherwise unreachable shapes.

We can calculate the total number of jumpled positions by multiplying the number of shapes by the number of piece arrangements, as the latter is the same for all shapes including the cube shape. We therefore have 28,055 * 8!·37·24!·212 / (4!6·2) = 16,449,312,515,541,600,685,326,336,000,000.

## Notation:

Denote the 6 faces using the letters F, B, U, D, L, R for the Front, Back, Up, Down, Left, and Right faces. The edges can be uniquely labelled using the two letters of the faces that meet at that edge. For example FR is vertical edge shared between the Front and Right faces. The same two letter labels will also be used to denote 180 degree twists of the edges, so FR also means a half turn of the pieces around the front-right edge of the cube.
It is often useful to label the locations of the moving pieces. A corner location can be labelled using three letters, e.g. URF is the corner shared by the Up, Front and Right faces. The face centres will also be labelled with three letters, an upper case letter for the face it is in, and two lower case letters for which corner of that face. The four centre piece locations of the top face are therefore Ufr, Ufl, Ubl, and Ubr.

## Solution:

Phase 1: Return the puzzle to a cube shape.

1. Do any moves you can to get the shape closer to being a cube.
2. Examine each edge, and twist it to its normal orientation if possible. This is easier on the Curvy Copter because it has visible edge pieces, but even on the Helicopter Cube these is a hidden edge piece at the top of each axis of the mechanism. Try to figure out how to twist each axis to align this hidden piece.
3. If you have done the previous step correctly, then the pieces that stick out come in pairs - always a corner and a centre piece together. Furthermore, there will be an even number of such pairs. Rotate the cube or do any moves necessary such that one pair lies at UFL, and another at UFR.
4. Depending on which direction the two corners are sticking out, do one of the following:
Luf, Fur: UL- UF UL+
Luf, Ufr: FL+ UF FL-
Ful, Rfu: UR+ UF UR-
Ufl, Rfu: FR- UF FR+
Luf, Rfu: UR UB, and try steps c,d again.
Ful, Fur, or Ufl, Ufr: Impossible as the sticking-out corners interfere
Ufl, Fur, or Ful, Ufr: I haven't found a good way to resolve this case. Let me know if you have.

The following phase needs some further explanation. Consider any face centre piece, and imagine where it can move to using normal move. There are always only two moves that affect the piece, so after the first move it can only move in one direction to a new location. This results in a loop of 6 locations around the cube that the piece can move to. For example, the piece at Ufr can move to Flu, Ldf, Dbl, Brd, Rub, and then back to Ufr again. This is a ring of six locations, one on each face. There are four of these rings, each with one location on each face.
The six pieces in such a ring can be mixed, but can never leave their ring if you do only normal moves. If a cube has not been jumbled, then each ring must contain one piece of each face colour. After jumbling this is no longer true, so this will be resolved in the next phase.

Phase 2: Make the face centre rings contain all six colours.

1. Find any ring that contains (at least) two pieces of the same colour. Remember that colour.
2. Find any other ring that does not contain a piece of that colour.
3. Hold the cube such that the Ufr location is part of the first ring, and Dfr is part of the second ring.
4. Using only normal moves, bring one of the two pieces of the same colour to the Ufr location.
5. Examine the second ring, find out which colour occurs more than once, and then use normal moves to bring a pieces of that colour to the Dfr location. Note that you need not disturb the Ufr piece for this.
6. Swap the pieces using the following move sequence: UF+ DR+ FR DR- UF-

Phase 3: Solve the bottom face centres (and edges)

1. Choose a colour for the bottom face.
2. If you have a Curvy Copter, then find the four edges with the chosen colour. Turn those edges so that they have the chosen colour in the same face, and then hold the cube with that face on the bottom. On a normal Helicopter Cube it does not matter which face is held at the bottom.
3. Find any centre piece of your chosen colour that does not already lie in the bottom face.
4. Using normal moves, and without disturbing the bottom face, bring the piece to the lower half of the cube.
5. Hold the puzzle so that the piece lies in (the bottom half of) the front face.
6. Do one of the following move sequences to insert the piece into the bottom face:
Fdl: FL DL FL DL
Fdr: FR DR FR DR
7. Repeat steps c-f until the bottom face centres are all of the chosen colour.

Phase 4: Solve the bottom corners

1. Find any unsolved corner piece that has the bottom face colour on it.
2. Determine the location where the corner piece belongs. On a Curvy Copter, you must look at the side colours of the bottom edges to determine where the corner should be placed to match those colours. If you are using a normal Helicopter Cube and this is the first corner to be solved, then you can choose any of the four bottom locations. For later corners you must use the first corner as your guide for where to place them.
3. Without disturbing the bottom face centres, use normal turns to bring the corner to the location in the top face directly above the location where it belongs.
4. Hold the puzzle so that the corner piece is at URF, and the location it belongs below it at DRF.
5. The corner's side with the bottom-face-colour needs to be on top. If this is not so, you can twist the corner as follows:
Clockwise: UF UL UB UR
Anti-clockwise: UR UB UL UF
6. Repeat steps a-e until the bottom face corners are all solved.

Phase 5: Solve the bottom centres (and middle edges) on the side faces

1. Hold the cube with the solved face still at the bottom, and so that the bottom-right centre of the front face (Frd) is unsolved. If all the side faces have their bottom-right centre solved, then skip ahead to step g.
2. Find the centre that belongs at Frd. It must lie at one of the locations Ruf, Ubr, Blu, Ldb.
3. If it lies at Ldb, then bring it to the top with the moves BL UL BL UL BL
4. Use top layer moves to bring the centre to the Ruf location.
5. Insert the centre piece with the moves FR UF FR UF FR.
6. Repeat steps a-e until the bottom-right centres of all four sides are solved.
7. Hold the cube with the solved face still at the bottom, and so that the bottom-left centre of the front face (Fld) is unsolved. If all the side faces have their bottom-left centre solved, then skip ahead to step l.
8. Find the centre that belongs at Fld. It must lie at one of the locations Luf, Ubl, Bru.
9. Use top layer moves to bring the centre to the Luf location.
10. Insert the centre piece with the moves FL UF FL UF FL.
11. Repeat steps g-j until the bottom-right centres of all four sides are solved.
12. On the Helicopter Cube this phase is now finished, and you can skip to the next phase. On the Curvy Copter it may be that some the middle edges are flipped.
13. Hold the Curvy Copter cube with the solved face still at the bottom, and so that the front-right edge is flipped.
14. Do the move sequence FR UR UF FR UF UR FR to flip the edge.
15. Repeat steps l-n until all middle edges are correct.

Phase 6: Solve the top corners

1. Find the corner that belongs at URF, which is the corner piece that has the same colours as the front and right faces.
2. Use top layer moves to bring that corner piece to the URF location where it belongs.
3. If the corner is not correctly oriented, you can twist the corner as follows:
Clockwise: UF UL UB UR
Anti-clockwise: UR UB UL UF
4. Find the corner that belongs at ULF, which is the corner piece that has the same colours as the front and left faces.
5. Use UB and UL moves to bring that corner piece to the ULF location where it belongs.
6. If the corner is not correctly oriented, you can twist the corner as follows:
Clockwise: UL UB UR UF UR
Anti-clockwise: UF UR UF UB UL
7. If the UBR and UBL corners need to be swapped to be in their correct positions, do a UB move.
8. If the UBL and UBR corners are not correctly oriented, you can twist them as follows:
UBL clockwise, UBR anti-clockwise: UB UR UF UL UF UR
UBR clockwise, UBL anti-clockwise: UR UF UL UF UR UB

Phase 7: Solve the face centres (and edges) of the top half

1. If the Fur location is not solved, then find out where the centre piece is that belongs there (it will be at Ulf or Lbu) and do one of the following to solve it:
Urf: (UF FL UL FL)2
Lbu: (UL FL UF FL)2
2. If the Ruf location is not solved, then find out where the centre piece is that belongs there (it will be at Ubr or Blu) and do one of the following to solve it:
Ubr: (UB BR UR BR)2
Blu: (UR BR UB BR)2
3. Turn the whole cube to the left, so that the right face becomes the front.
4. Repeat steps a-b, so that you will have solved the sides of the front half of the cube.
5. Turn the whole cube around, so that the back face becomes the front.
6. If the Ubr centre location is not yet solved, do (UR UF)3 to solve it.
7. If the Ubl centre location is not yet solved, do (UL UF)3 to solve it.
8. If the Ful and Ufr locations are not yet solved, then do UR FR UF (UR FR)3 UF FR UR.
9. If the cube was jumbled, then it is possible that the final two face centres, Fur and Ufl, need to be swapped. This is done by the move sequence: FR FD FL FR (FR- UL- UF FR+ UL+ UF) FR FL FD FR
10. A normal helicopter cube should now be in the solved state. On the curvy copter, there may be some edges that need to be flipped. To fix them, you can use one of the following sequences, of which only the first is possible without jumbling:
1. To flip all U edges: (UF UR)3 UL UF UB UR (UB UL)3 UR UB UF UL
2. To flip the UF and UR adjacent edges: UR UF FR (UR UF)3 FR UF UR, UL+ FR+ UF FR- UL-, UR- FL- UF FL+ UR+.