Face Turning Octahedron
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The Face Turning Octahedron is a three-layered face turning octahedral puzzle for which all three layers are the same width. The first working version was made by David Pitcher in the early 2000s and was later mass-produced by LanLan. In 2024, DianSheng released a magnetic version, which was followed by other speed versions by various companies including Dayan, Qiyi, and Shengshou. Before these versions were mass-produced, it was common to make one by attaching printed extentions to a Qiyi Rex Cube.
Basic Characteristics
The Face Turning Octahedron is a face-turning regular octahedral puzzle with a cut depth such that the cuts intersect at a point in the center of each face. It is the second shallowest possible planar cut depth of a face-turning octahedral puzzle (with the shallowest being the Dino Octahedron). Its pieces consist of 6 corners, 12 edges, and 24 X-centers. Since rotation about the faces of an octahedron result in two distinct sets of four faces each where a face from one set can never reach the starting position of any face from the other set, the stickers of an FTO are divided into two distinct orbits, where colors of adjacent faces can never end up on the same face. This means that the corners only have two possible orientations per location (when the puzzle is scrambled without cube rotations), the edges are always oriented, and there are two sets of X-centers that never mix. Interestingly, this also means than an FTO stickered with only four colors, with opposite sides having the same color, solves exactly the same way as an FTO stickered with all eight colors.
The corners turn out to be equivalent to pyraminx edges, which means that only even permutations are possible and that a single corner "flip" is impossible. Additionally, since each face turn does a 3-cycle of edges, only even permutations are possible for edges as well.
As calculated by Jaap, the FTO has a total of 31,408,133,379,194,880,000,000 visually distinct positions [1].
If the X-centers are to be each considered unique, then the number becomes [(6!/2)(2^6/2)(12!/2)(12!/2)^2]/12 = 13,188,400,838,457,446,891,520,000,000 total positions.
Related Puzzles
Skewb Diamond
| Main Article : Skewb Diamond |
The Skewb Diamond, also known as the 2x2 FTO, can be seen as either a face-turning octahedron with deep cuts or a 2-layer version of the standard FTO. It is also an octahedral shape transformation of a Skewb where the Skewb's centers correspond to the Skewb Diamond's corners and the Skewb's corners correspond to the Skewb Diamond's centers, which means that orientation matters on the opposite pieces as compared to the standard Skewb.
Master FTO
The Master FTO, or 4x4 FTO, is a four-layered face-turning octahedron with two different cut depths (one deep and one shallow) such that the faces appear like those of a Master Pyraminx.
Other Related Puzzles
There are four distinct face-turning octahedral puzzles that each have a single planar cut depth. From shallowest to deepest, they are the Dino Octahedron, the standard FTO (the main subject of this article), the Master Skewb Diamond, and the Skewb Diamond (discussed above). The Dino Octahedron, briefly produced by MF8, is the shallowest possible face-turning octahedron with one cut per side. It is equivalent to a standard FTO with the addition of fixed centers. The Master Skewb Diamond is a face-turning octahedral puzzle with a cut depth between the standard FTO and the Skewb Diamond, and can be seen as a shape transformation of the Master Skewb. Although it has not yet been mass-produced in its pure form, MF8 has made a "crazy" version with added circular cuts on the faces. The only functional difference between the Dino Octahedron and the Master Skewb Diamond is that the centers are fixed on the Dino Octahedron, whereas they turn with the faces on the Master Skewb Diamond.
Methods
There are many methods used to solve the Face Turning Octahedron, the most popular of which are described below.
Bencisco
The Bencisco method is a method based on ending in a step called Last Three Triples and was created by Ben Streeter in 2020.
- First Block: Solve a block on the BL layer consisting of a large hexagon center and two triples. A triple is a corner and two opposite small triangle centers.
- Centers: Solve the remaining three hexagon centers in the layers to the right of the block built in the first step.
- Triple: Rotate to have the upper left corner be at the bottom and solve the unsolved triple at that position.
- Last Three Triples: Middle layer center triangles, last layer center triangles, and last layer corners remain to be solved. Finish the solve by forming three triples then solving them to their correct positions, or solve everything in a single step using 179 algorithms.
Nautilus
The Nautilus method is an F2L and Last Layer method that was developed by a team consisting of James Straughan, Liam Highducheck, Vincent Trang, and Malte Ihlefeld, with additional input by Ryan Hudgens and mrmangohands.
- First Block: Solve a block on the BL layer consisting of a large hexagon center and two triples.
- Centers: Solve the remaining three hexagon centers.
- Triple: Solve the last triple at the bottom right, completing the first two layers.
- Last Layer: Solve the last layer. This is typically accomplished by first solving the center triangles then the corners. 1LLL has also been developed.
There are some alternative steps to solving the first two layers that users may prefer:
- First Block: Solve a block on the BL layer consisting of a large hexagon center and two triples.
- Stripe: Solve the F2L stripe to the right of the first block. This leaves just the U and R layers unsolved.
- F2L: Solve the remainder of the F2L using just U and R turns.
- Last Layer: Solve the last layer.
Vertigo
By Ryan Hudgens and Ben Streeter.
- Vertex: Solve a corner and its surrounding four triangles and four edges.
- Second Vertex: Solve another vertex adjacent to the first. Keep these two at the bottom left of the first two layers.
- Edges and Triangles: Solve the last four edges and the two triangles at either the intersection of DR or UL.
- Last Six Triangles: If in the last step the two triangles at DR were solved, solve the three remaining U face triangles and the three that belong in the middle layer beneath the U layer, equivalent to the E layer on 3x3. Do the opposite of this if in the last step the two triangles at UL were solved.
- Last Three Triples: Finish the solve by forming the last three triples then solving them to their correct positions, or solve everything in a single step using 179 algorithms.
CFL
CFL is a method similar to CFOP and was created by James Straughan.
- Centers: Solve the three large hexagon centers around the sides of the first two layers.
- F2L: Solve the three triples of the first two layers.
- Last Layer: Solve the last layer. This is typically accomplished by first solving the triangle centers then the corners, but other systems also exist such as OLL then PLL or 1LLL.
Many CFOP techniques can be carried over such as pseudo slotting, edge control (corner control or triangle control in CFL), ZBLS (orienting corners or positioning triangles during the last slot), and XCross (arrow block + centers in CFL).
L4T Redux
A version of reduction that combines modern techniques with reduction to Pyraminx steps. All of the various Pyraminx steps can be incorporated into the method. The method was developed by Vincent Trang, with some algorithm development and an EIF version proposal by James Straughan.
- First Block: Solve a block on the BL layer consisting of a large hexagon center and two triples. The two triples can be any triple and not just the ones that belong in those positions. This allows for more freedom in the reduction step.
- Centers: Solve the remaining three hexagon centers.
- Reduction: Form the last four triples using big cube style slicing techniques. This reduces the FTO to a Pyraminx stage.
- Pyra Stage: Solve as a Pyraminx.
It is also possible to perform reduction in an edge in front grip. The R layer center and a triangular first block on BL would be solved first, then the remainder of the middle slice of the first two layers leaving out the bottom triangle. This allows for a U R r pairing move group that doesn't break the R layer center. Reduction would be performed using r turns to attach triangles, then pairs and triples are easily moved around using U and R turns.
Julian
- Center: Solve a large hexagon center.
- Arrow Blocks: Add two adjacent arrow blocks to the hexagon center.
- Third arrow: Solve the last arrow connected to the hexagon center, but leave out the two triangles that would go around the sides of the layer where the hexagon center is.
- Corners: Solve the last three corners.
- Triangles: Use commutators to solve all remaining small triangle centers.
Other F2L Methods
Two of the mentioned methods, Nautilus and CFL, involve solving F2L then the last layer. Various 3x3 methods can be adapted to FTO to solve the first two layers [2]. Some of these involve solving F2L from an edge in front perspective, which involves solving a right layer center and left layer block, similar to many big cube methods. This perspective allows for U R r turning that doesn't disturb the built right layer center.
LEOR (CIF)
- First Block: Solve a block on the BL layer consisting of a large hexagon center and two triples.
- Stripe: Solve the F2L stripe to the right of the first block. This leaves just the U and R layers unsolved.
- F2L: Solve the remainder of the F2L using just U and R turns.
- Last Layer: Solve the last layer.
LEOR (EIF)
- First Block + Center: Solve the R layer center and a block on the BL layer consisting of a half center and its two outer triangles and a triple on the bottom. This leaves the U layer, R layer triples, and the slice layer between the R and BL layers unsolved.
- Stripe: Solve the F2L stripe between the first block and R layer center.
- F2L: Solve the two remaining F2L triples.
- Last Layer: Solve the last layer.
Nautilus (EIF)
- First Block + Center: Solve the R layer center and a block on the BL layer consisting of a half center and its two outer triangles and a triple on the bottom. This leaves the U layer, R layer triples, and the slice layer between the R and BL layers unsolved.
- Pyramid: Build a 2x2x2 pyramid on the bottom in the back.
- Stripe: Finish the stripe between the first block and R layer center.
- Last Slot: Solve the last slot, which is the triple on the bottom in the front.
- Last Layer: Solve the last layer.
