Also, this is equivalent to the "falling pictureframe" problem: Find a way to hang a pictureframe on the wall held up by n nails, such that if any one of the nails is removed, then the picture falls. I recall that some funster -- maybe Dan Asimov? -- sent this in to Marilyn vos Savant, many years ago, and she ran it in her column without giving him credit. --Michael On Fri, Oct 26, 2012 at 11:14 AM, Michael Kleber <michael.kleber@gmail.com>wrote:

Kim Whittlesey, a fellow grad student at Berkeley, needed to solve this problem for her Ph.D. dissertation. The answer is yes, they exist for arbitrary numbers of letters: just take commutators.

Define [a,b] = a b a’ b’. This is nontrivial in the free group, but becomes trivial if either a or b is.

Then [[[[[a,b],c], d], e], f], for example, is a nontrivial word in the free group, but becomes trivial if you mod out by any of a/b/c/d/e/f.

On Fri, Oct 26, 2012 at 10:39 AM, Adam P. Goucher <apgoucher@gmx.com>wrote:

...

Here's an interesting group-theoretic puzzle with connections to Borromean rings, Brunnian braids et cetera:

http://cp4space.wordpress.com/2012/10/26/borromean-strings/

I doubt the existence of Borromean strings, although I haven't been able to prove it.

Sincerely,

Adam P. Goucher

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