Problem

[William Rex Marshall]

----- Original Message ----- 
From: "Roberto Tauraso" <tauraso at mat.uniroma2.it>
To: "Richard Guy" <rkg at cpsc.ucalgary.ca>
Cc: "Roberto Tauraso" <tauraso at mat.uniroma2.it>; "berend daniel" <berend at cs.bgu.ac.il>; <seqfan at ext.jussieu.fr>; "Elwyn Berlekamp" <berlekassistant at yahoo.com>; <berlek at gmail.com>; <berlek at math.berkeley.edu>
Sent: Monday, January 23, 2006 9:56 PM
Subject: Re: Problem


> On Sun, 22 Jan 2006, Richard Guy wrote:
> 
> > The square bracelet conjecture is answered
> > in a preprint of Berlekamp & Guy.  We should
> > make this more available.
> 
> Really! I'm very happy of that. Is it possible to receive
> an electronic copy of this work?
> In these days I searched in the web but I did not find it.
> Just yesterday I submitted the sequence A115418
> about the minimum lenght of k-th-power loop (or bracelet).
> The sequence is not very long (it starts with index 1):
> 2, 32, 473, 9641
> 32 is well known and 473 was found also by Resta in
> http://www.primepuzzles.net/puzzles/puzz_311.htm
> It would be nice to extend it but for k=5. The first number
> such that all the numbers have degree at least 2 is 246877
> and to find the cycle for such a huge graph is not easy
> (at least for me).
> I wonder if the Berlekamp & Guy result can be extended also
> for cube loops: do cube loops exist from 473 on?
> 
> Bye,
> Roberto Tauraso


I had clean forgotten that I had included the problem for cubes when I originally submitted A071983 and A071984 (the comment for A071984 still needs updating, however).

Examples of cube loops of length 473 were discovered in 2002 by Robert Israel. His first example appears in the following Usenet post to rec.puzzles and alt.brain.teasers:

http://groups.google.com/group/rec.puzzles/msg/6c5f7b70f35239fe?dmode=source&hl=en

An early reference to square loops (which includes the square loop of length 32 in the solution) was given as a problem in JRM:

Ruemmler, Ronald E., "Square Loops," Journal of Recreational Mathematics 14:2 (1981-82), page 141; Solution by Chris Crandell and Lance Gay, JRM 15:2 (1982-83), page 155.

Did the square loop of order 32 ever appear in any earlier publication of the mathematical literature? Are there any published examples of cube loops of length 473 earlier than Robert Israel's Usenet post?