Face Turning Octahedron

The Face Turning Octahedron in its solved state shown from the top

A DianSheng Face Turning Octahedron, an example of newer hardware

The Face Turning Octahedron (often abbreviated as FTO) is a combination and mechanical puzzle. Unlike cubic puzzles, the FTO is based on an octahedral geometry with eight triangular faces that rotate independently. Its deep-cut mechanism and interplay of the various piece types give the puzzle a distinctive solving approach compared to other cubic puzzles. ^[1] The FTO is notable for being the first octahedral twisty puzzle to feature straight cuts, setting it apart from earlier octahedral designs.

History

The idea for the FTO was initially developed through a series of early patent filings. On February 9, 1982, Clarence W. Hewlett Jr. filed the first patent for a face-turning octahedron, ^[2] and just two weeks later, on February 24, 1982, Karl Rohrbach filed a similar patent. ^[3] However, neither patent led to a commercial product which left the concept theoretical for years.

Ernő Rubik, the creator of the Rubik's Cube, expressed interest in the development of an FTO. ^[4] Rubik envisioned a version of the puzzle that incorporated only corners and centers, and a patent was filed on February 9, 1981. ^[5]

On September 15, 1997, Xie Zongliang (謝宗良) from Taiwan applied for a patent for the FTO. ^[6] According to a report, approximately 1,000 units were produced by Xie in 2008, and there is some indication that the puzzle may have been constructed as early as a decade before that production run. ^[7]

On July 9, 2003, David Pitcher filed a patent for an FTO. ^[8] However, the patent was never formalized due to non-payment of issuance fees, allowing the invention to enter the public domain. Between 2001 and 2003, Pitcher developed a working mechanism for the puzzle and later claimed that his design was the first functional prototype of an FTO. However, Pitcher's prototype did not enter mass production, leaving uncertainty on whether Pitcher or Xie created the first working prototype. ^[9]

Mechanism

The FTO consists of three distinct piece types, totaling 42 external elements:

• Corner pieces: There are 6 corners, each occupying a vertex of the octahedron

• Edge pieces: There are 12 edges that are located on the intersections of the turning planes
• Triangle pieces: In addition to the corners and edges, there are 24 triangle pieces that fill the remaining gaps

The number of internal components varies depending on the manufacturer.

Number of unique positions

Consider these constraints for calculating the total number of unique positions: ^[10]

Permutations and orientations:

• 6 vertices (corners) can be arranged in 6! ways, with 2 orientations each
• 12 edges can be arranged in 12! ways
• Two sets of 12 centers (triangle pieces) can be arranged in (12!)² ways

Restrictions:

• Only an even number of vertex pieces can be flipped (division by 2)
• Vertex and edge permutations must be even (division by 2)
• Centers are grouped in identical triplets (division by 3!⁸)
• The puzzle's orientation is fixed by one unique piece, offering 12 possible (division by 12)

Combining these factors, the total number of unique positions is: ^{[11] [12]}

${\displaystyle {\frac {6!\times 2^{3}\times 11!\times (12!)^{2}}{(3!)^{8}}}=31,408,133,379,194,880,000,000}$

Records

Although the FTO is not an official World Cube Association event, it has an active speedsolving community, largely due to the resurgence of newer hardware in recent years. As one of the most frequently featured unofficial events at official competitions, there is growing advocacy for the FTO to gain official recognition by the WCA. ^[13]

Top 5 solvers by single solve

Number [14]	Name	Fastest solve	Competition
1.	Aedan Bryant	12.15s	Skowhegan Cubikon 2025
2.	Chris Choi	13.77s	Pittsburgh Winter 2025
3.	Dan Pastushkov	14.31s	Bay Area Speedcubin' 64 LIVE - SSF 2024
4.	Chandler Pike	14.48s	FMC and More Maine 2025
5.	Michael Larsen	15.52s	Davis Fall 2024

Top 5 solvers by Olympic average of 5 solves

Number [15]	Name	Fastest average	Competition	Times
1.	Aeden Bryant	14.29s	FMC and More Maine 2025	(12.57), 13.56, 14.38, (14.93), 15.26
2.	Michael Larsen	17.10s	Cubing with Dinosaurs Lehi 2025	(16.24), 20.71, 16.11, 17.04, (18.03)
3.	Chris Choi	17.11s	Pittsburgh Winter 2025	(17.24), 17.45, (22.52), 16.63, 13.77
4.	Chandler Pike	17.40s	Orono Open 2025	(17.07), 15.77, (21.44), 17.17, 17.96
5.	Dan Pastushkov	18.12s	Bay Area Side Events Day 2025	(18.97), 20.76, 17.69, 17.69, (15.72)

See also

• Skewb Diamond, an octahedron puzzle that would result if the middle layers from the FTO were removed
• Rubik's Cube
• Octahedron

References

1. "Face-turning Octahedron". www.jaapsch.net. Retrieved 2025-04-01.
2. Hewlett Jr., Clarence (May 29, 1984). "Magic Octahedron". Google Patents. Retrieved April 1, 2025.
3. Rohrbach, Karl (February 24, 1982). "Logisches Stereosspielzeug". Deutsches Patent- und Markenamt. Retrieved April 1, 2025.
4. Rubik, Ernő; Varga, Tamás; Kéri, Gerzson; Marx, György; Vekerdy, Tamás (April 21, 1988). Rubik's Cubic Compendium (Recreations in Mathematics). New York: Oxford University Press. p. 15. ISBN   9780198532026.
5. Rubik, Ernő (April 15, 1982). "Three-dimensional toy". Search for intellectual property - GOV.UK. Retrieved April 1, 2025.
6. Xie, Zongliang. "鑽石型魔術方塊 Diamond-like magic block". Taiwan Intellectual Property Office. Retrieved April 1, 2025.
7. "Re: [方塊] 八面體方塊". Ptt 批踢踢實業坊. Retrieved 2025-04-01.
8. Pitcher, David (July 9, 2003). "Octahedral puzzle apparatus". Google Patents. Retrieved April 1, 2025.
9. "TwistyPuzzles.com > Museum > Show Museum Item". twistypuzzles.com. Retrieved 2025-04-01.
10. "Face-turning Octahedron". www.jaapsch.net. Retrieved 2025-04-01.
11. "The Complexity Dynamics of Magic Cubes and Twisty Puzzles". dhushara.com. Retrieved 2025-04-01.
12. "Rob's Puzzle Page - Rearrangement". www.robspuzzlepage.com. Retrieved 2025-04-01.
13. "Will FTO become an official WCA event?". speedcubing.org. Retrieved 2025-04-01.
14. "Rankings | Cubing Contests". cubingcontests.com. Retrieved 2025-04-01.
15. "Rankings | Cubing Contests". cubingcontests.com. Retrieved 2025-04-01.