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Author Topic: A knot-theoretic argument for the impossibility of Figure Eight - puzzle
Mikko Kiviharju
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Member # 1521

posted February 12, 2009 05:26 AM     Profile for Mikko Kiviharju   Author's Homepage   Email Mikko Kiviharju     Send New Private Message   Edit/Delete Post   Reply With Quote
Hello all,

In the "Solution Help" I again saw questions of the "Figure Eight"-puzzle. It seems to be an open question, whether the puzzle is at all solvable. Many experts think so, but I haven't so far come across any proof yet. (Please let me know, if anyone has)

So I spent a few hours thinking through a vein of thought:

- If the wire-part were flexible, and would not even by itself come off as a straight cord with two loops at ends, it should be then topologically impossible to remove a closed string from around it.
- Structures like this can be investigated with knot theory, but unfortunately it does not apply for strands, which split in two.
- However, one could imagine that the connecting wire between the two loops was split in half in the middle, forming one single closed loop.
- An equivalent question would then seem to be, if the resulting knot can be "untangled" to the unknot.
- However, by manipulating the resulting knot, it can be seen that it is actually a knot sum of two trefoils, also known as the square knot. Square knot's unknotting number is 2, so it cannot be untangled to the unknot.
- By this reasoning, the "Figure Eight" form can neither be untangled to a straight line, making the puzzle topologically impossible.

If there are any professional mathematicians or topologists amongst you, I'd appreciate comments and pointing out possible faults within this reasoning.

br: Mikko K


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Mikko Kiviharju
Multiple
Member # 1521

posted May 14, 2009 06:59 AM     Profile for Mikko Kiviharju   Author's Homepage   Email Mikko Kiviharju     Send New Private Message   Edit/Delete Post   Reply With Quote
After researching the Knot Atlas, I came across an equivalent link for the modified "Figure Eight" puzzle. As this is a separate link not isotopic to the unlink, it follows that "Figure eight" is also topologically unsolvable.

Formally, one would still need to show that the "zipped"-open trunk is in this case equivalent to the original trunk. However, this seems so intuitive that I'm inclined to leave it as an exercise.

br: Mikko K


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Ron Zezima
Multiple
Member # 215

posted June 22, 2009 12:10 AM     Profile for Ron Zezima   Author's Homepage   Email Ron Zezima     Send New Private Message   Edit/Delete Post   Reply With Quote
Hey Mikko:

There are no experts on this subject. I removed the rope in December of 1998 and replaced it 20 mins later. I've never been able to repeat this feat, nor to document the solution. Others have also tried and failed. Many have tried to reason that there's no solution, a closed loop, etc. But, when you've seen the solution with your own eyes, their words really don't matter: because they are incorrect. Hopefully someone will one day document this truth. Personally, I find it facinating, frustrating, and puzzling. Best Regards, Ron Zezima


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Mikko Kiviharju
Multiple
Member # 1521

posted August 30, 2009 10:25 AM     Profile for Mikko Kiviharju   Author's Homepage   Email Mikko Kiviharju     Send New Private Message   Edit/Delete Post   Reply With Quote
Hello Ron,

Do you have a picture of your puzzle? Love to see it.

br: Mikko K


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Stephen Noltie
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Member # 1644

posted September 23, 2009 09:06 PM     Profile for Stephen Noltie   Author's Homepage   Email Stephen Noltie     Edit/Delete Post   Reply With Quote
Hi Mikko,

The “figure eight” puzzle is unsolvable. A proof is given in a note entitled “A Topological Puzzle”, authored by the Swiss mathematician Inta Bertuccioni, which appeared in the December 2003 issue of the The American Mathematical Monthly.

Here is a link to that paper: http://gergo73.tar.hu/3647964.pdf
Be sure to scroll down to see the complete three page paper. Understanding the proof requires some background in the fundamental group from algebraic topology.

Some three years later another article on the same topic appeared in the April 2006 issue of the same journal. It is entitled “A Topological Menagerie” by Paul Melvin. I haven’t seen Melvin’s article but believe he has a put forth a more intuitive, less technical argument. This article is also available from JSTOR, but you need access, likely through a university. If you are interested, I can obtain a copy of Melvin’s article sometime, scan it and e-mail it to you.

I tried the puzzle after reading about it in “Creative Puzzles of the World” by Delft and Botermans. I constructed a model myself, but soon believed it impossible. I stumbled onto the above article when browsing some math journals and was thrilled to find someone had actually worked up a proof. A history of the puzzle by the designer, Stewart Coffin, is given in this second link: http://g4gardner.pbworks.com/f/mm-coffin2.pdf

Steve Noltie


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Mikko Kiviharju
Multiple
Member # 1521

posted December 10, 2009 02:27 AM     Profile for Mikko Kiviharju   Author's Homepage   Email Mikko Kiviharju     Send New Private Message   Edit/Delete Post   Reply With Quote
Hi Steve, and thanks for the links.

I did some independent research and bumped into the same articles. I since focused on a more "pictorial" proof, resulting in a two-part article now appearing in the CFF magazine.

The backgound part is published in CFF 80, the actual proof will appear in a later issue, when they have room for it. I felt obligated to write a background, since the Coffin's article appeared already in the 90's, and all the new developments are not covered.

I was surprised, too, to see that the wire-frame is actually the same as the square knot.

br: Mikko K


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