Face Turning Octahedron

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,{{Cite web |last=Hewlett Jr. |first=Clarence |date=May 29, 1984 |title=Magic Octahedron |url=https://patents.google.com/patent/US4451039A/ |access-date=April 1, 2025 |website=Google Patents}} and just two weeks later, on February 24, 1982, Karl Rohrbach filed a similar patent.{{Cite web |last=Rohrbach |first=Karl |date=February 24, 1982 |title=Logisches Stereosspielzeug |url=https://depatisnet.dpma.de/DepatisNet/depatisnet?window=1&space=menu&content=treffer&action=bibdat&docid=DE000003206560A1 |access-date=April 1, 2025 |website=Deutsches Patent- und Markenamt}} 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.{{Cite book |last1=Rubik |first1=Ernő |title=Rubik's Cubic Compendium (Recreations in Mathematics) |last2=Varga |first2=Tamás |last3=Kéri |first3=Gerzson |last4=Marx |first4=György |last5=Vekerdy |first5=Tamás |publisher=New York: Oxford University Press |isbn=9780198532026 |publication-date=April 21, 1988 |pages=15 |language=en}} Rubik envisioned a version of the puzzle that incorporated only corners and centers, and a patent was filed on February 9, 1981.{{Cite web |last=Rubik |first=Ernő |date=April 15, 1982 |title=Three-dimensional toy |url=https://www.search-for-intellectual-property.service.gov.uk/GB2084471 |access-date=April 1, 2025 |website=Search for intellectual property - GOV.UK}}

On September 15, 1997, Xie Zongliang (謝宗良) from Taiwan applied for a patent for the FTO.{{Cite web |last=Xie |first=Zongliang |title=鑽石型魔術方塊 Diamond-like magic block |url=https://tiponet.tipo.gov.tw/ |access-date=April 1, 2025 |website=Taiwan Intellectual Property Office}} 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.{{Cite web |title=Re: [方塊] 八面體方塊 |url=https://www.ptt.cc/bbs/Rubiks/M.1226659344.A.311.html |access-date=2025-04-01 |website=Ptt 批踢踢實業坊}}

On July 9, 2003, David Pitcher filed a patent for an FTO.{{Cite web |last=Pitcher |first=David |date=July 9, 2003 |title=Octahedral puzzle apparatus |url=https://patents.google.com/patent/US20050006842A1/ |access-date=April 1, 2025 |website=Google Patents}} 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.{{Cite web |title=TwistyPuzzles.com > Museum > Show Museum Item |url=https://twistypuzzles.com/app/museum/museum_showitem.php?pkey=1663 |access-date=2025-04-01 |website=twistypuzzles.com}}{{Multiple images

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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.

Consider these constraints for calculating the total number of unique positions:{{Cite web |title=Face-turning Octahedron |url=https://www.jaapsch.net/puzzles/octaface.htm |access-date=2025-04-01 |website=www.jaapsch.net}}

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:{{Cite web |title=The Complexity Dynamics of Magic Cubes and Twisty Puzzles |url=https://dhushara.com/cubes/ |access-date=2025-04-01 |website=dhushara.com}}{{Cite web |title=Rob's Puzzle Page - Rearrangement |url=https://www.robspuzzlepage.com/rearrangement.htm |access-date=2025-04-01 |website=www.robspuzzlepage.com}}

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

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.{{Cite web |title=Will FTO become an official WCA event? |url=https://speedcubing.org/blogs/news/will-fto-become-an-official-wca-event |access-date=2025-04-01 |website=speedcubing.org |language=en}}

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

Category:Mechanical puzzles

Category:Puzzles

Category:Combination puzzles