From ccw@eql.caltech.edu  Sat Apr 18 23:17:20 1992
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From: ccw@eql.caltech.edu (Chris Worrell)
Message-Id: <920418201450.2b6009a5@EQL.Caltech.Edu>
Subject:  Some solutions to Rubik's Tangle have been found.
To: Cube-Lovers@ai.mit.edu

(No spoilers are included)

I now know two distinct solutions to Rubik's Tangle.
Each of these solutions can be done with each 5x5 set, 1-4.

I have not yet found any solutions to the 10x10 puzzle. I do know
that if the 10 by 10 is composed of four 5x5 solutions, than my solutions
of the 5x5 do not lead to a solution of the 10x10. I am now seeking a
solution where all 100 pieces can be used anywhere in the 10x10, not just in
a corner of the 10x10 for its own 5x5 set.

Unfortunately, these solutions were found by brute force, not by any
real calculation.  I have not been able to discover any "Science" about
the puzzle which contributes to any discovery of a solution.

These solutions were found by hand, not by a computer search. So there may be
other solutions to the basic puzzle which are still unknown.
My search path has approximately 750 cases in it, of which I have
tested about 300.  One solution was found by 'accident' in that I have not
yet looked at the case which actually yields that solution, but one of the
test cases had been 'close' to a solution, so I looked outside my search path.
The other solution was found in my search path, so it was not found by
accident. (except by luck that it was at case 300 not 750)

GENERAL THOUGHTS ON RUBIK'S TANGLE AS A PUZZLE
In short, it is not a good puzzle.  It will never be popular.
Solutions might only be findable by Computer, by Luck, and by Stubborness
(brute-force).
As far as I can tell the only real method of solution is by using a computer.
I did it by hand because I am stubborn, and I did get lucky.
My search path contained 750 cases, but I had already considered search paths
with an estimated 4000 and 8000 cases. These are so large that I had never
even completed the basic enumeration of cases.
A hand search of this magnitude is almost impossible.  There are too many
places for errors, and no real ways of checking.  The amount of time is also
absurd.  I worked on between 1 and 8 of my test cases at a time, with
about 250 of these groups in my search path. (later in
the search I would have, on ocassion, looked at 16 at once).  Sometimes a
group could be disposed of within 20 minutes, but sometimes it took several
hours.

Has anybody discovered more mathematical content than I have?

---Chris Worrell (ccw@eql.caltech.edu)