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.
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
This gives a total of 7!·36·6!4 / 2 = 493,694,233,804,800,000 positions, all of which can be reached.
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
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.
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
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:
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 654,117 * 7!·36·24! / 4!6 = 7,802,822,379,572,176,235,479,200,000 jumbled positions all together.
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
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 jumbled 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.
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
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.
Phase 1: Return the puzzle to a cube shape.
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.
Phase 3: Solve the bottom face centres (and edges)
Phase 4: Solve the bottom corners
Phase 5: Solve the bottom centres (and middle edges) on the side faces
Phase 6: Solve the top corners
Phase 7: Solve the face centres (and edges) of the top half