How do you solve the Pyraminx?

                                The Pyraminx

Combinations: 75,582,720
Mechanism: axles (mini corners), 4 armed spider

Again, don't let the number of combinations intimidate you.  This puzzle
is really quite simple, probably more so than the Skewb.  The pyraminx
is a pyramid shape made up of 4 equilateral triangles, each chopped into
9 segments, like this:

                             /\
                            /__\
                           /\  /\
                          /__\/__\
                         /\  /\  /\
                        /__\/__\/__\

It can turn along any of these lines.  Each turn affects 3 of the 4
sides.

NOTATION

For the most part, the puzzle will be solved with the pyramid upside
down.  The 3 major corners (ie. mini corners which can twist freely,
plus the next slice over) facing you will be referred to as L(eft),
R(ight), and D(own).  For the L and R corners, a - means move the corner
towards you (clockwise for L, counter clockwise for R), and a + means
move it away from you.  For the D corner, D- means move it left, and D+
means move it right.

THE SOLUTION

The first step is to get all the corners oriented properly.  First,
twist the mini corners to align them with their centres.  Next, align
the centres with each other.  To do this, pick a mini corner.  Whatever
colour is NOT on that corner (eg. red) belongs on the opposite side.
Rotate these major corners so that the appropriate coloured centres (in
this case, red) are on the same side, and then rotate the remaining
corner to match the other centres.  Congratulations.  Thanks to these
few simple moves (8 at most), you've slashed the number of combinations
down to 11,520.  It doesn't seem quite so intimidating now, does it?
Now, pick a colour to use as the top layer, and we'll get to work on the
edge pieces.

                             THE TOP LAYER

There are 3 basic patterns necessary to complete the top layer.  1) move
the left edge into the top layer, 2) move the right edge into the top
layer, or 3) move the top edge somewhere else in the top layer, or flip
the top edge.  First of all, if you're moving a bottom layer edge piece
into the top layer, rotate the bottom layer so that the edge piece in
question shows the same colour on the front as the 2 top corners.  Then
perform:

1) R- D+ R+
2) L- D- L+

If an edge piece with your working colour is in the top layer, but in
the wrong place or is oriented wrong, then:

3) perform 1 or 2 (your choice) to move the errant edge piece to the
   bottom layer.  Then perform 1 or 2 as necessary.

Once the top layer is done, there's only 1 of 12 possible positions
left.  We're almost there...

                           THE BOTTOM LAYER

All that's left to do now, is to rotate the 3 bottom edges and orient
them properly.  An X in these patterns means use either the left OR the
right corner.  Your choice.  Just keep it consistent all the way through
the pattern.

1) Rotate bottom edges left:  X- D+ X+ D+ X- D+ X+
2) Rotate bottom edges right: X- D- X+ D- X- D- X+
3) Flip T and L edges: R- D+ R+ D- L- D- L+ D+
4) Flip T and R edges: L- D- L+ D+ R- D+ R+ D-

                   C O N G R A T U L A T I O N S!

You've just managed to conquer the Pyraminx!  Give yourself a pat on the
back.  Before I go, here's a pattern for you to try out.  After each
repetition of (X), rotate the whole Pyraminx in the same direction as
the D turn.  Choose one of (R- D+ R+)^5 D+ OR (L- D- L+)^5 D-.  This
pattern can be combined with the edge piece and mini corner rotations to
create many amazing patterns.  Now that you're a master of the Pyraminx,
get creative and develop your own.  Most important of all, HAVE FUN!

-- "The Simplest Solutions" web page,
   http://www.victoria.tc.ca/~ue451/solves.html (author not named)