A friend on mine applied too much force in an odd direction and
produced a tangled mess. I was nominated to fix it. My approach
was to take it apart completely and then put it back together.

Tools required:
Well, I used a paper clip.

Components:
A Magic is made out of 16 loops of string (actually nylon fishline)
and 8 panels. The loops of string are all the same length. The
loops of string are not tangled in any way. The panels each
decompose into two clear plastic covers and a piece of paper
(actually plastic). The panels are held together by the string.

Construction:

Each loop of string is twisted around three panels. It follows a
path like these:

 ---- ---- ----           ---- ---- ----
|/\  |  /\|/\  |         |  /\|/\  |  /\|
|\ \ | / /|\ \ |   or    | / /|\ \ | / /|
| \ \|/ / | \ \|         |/ / | \ \|/ / |
|  \/|\/  |  \/|         |\/  |  \/|\/  |
 ---- ---- ----           ---- ---- ----

For each string, there is another string that lies in the same
channels. When stringing a loop through the channels, there is a
choice at the points where the string passes from one panel the
next: Which string is closer to the center of the panel? That
question is answered differently for the two strings running throught
the same channels.

[I believe that this is done so that the net force trying to twist
the toy at that point is near zero. The redundancy also probably
strengthens the toy.]

The panels can be divided into two sets of four: The panels that are
centers in these triples and the panels that are ends. Each "end panel"
is an end for two different triples.

Strings don't lie in crossing channels on the same side of a panel.
(That describes how the two pairs of loops go on the same triple and
how two triples interact on their common end panel.)

-jim boyce