Does anyone have any hints on how to solve the Square-1 puzzle?

                                SHAPES

1. There are 29 different shapes for a side, counting reflections:
    1 with 6 corners, 0 edges
    3 with 5 corners, 2 edges
   10 with 4 corners, 4 edges
   10 with 3 corners, 6 edges
    5 with 2 corners, 8 edges

2. Naturally, a surplus of corners on one side must be compensated
   by a deficit of corners on the other side.  Thus there are 1*5 +
   3*10 + C(10,2) = 5 + 30 + 55 = 90 distinct combinations of shapes,
   not counting the middle layer.

3. You can reach two squares from any other shape in at most 7
   transforms, where a transform consists of (1) optionally twisting
   the top, (2) optionally twisting the bottom, and (3) flipping.

4. Each transform toggles the middle layer between Square and Kite,
   so you may need 8 transforms to reach a perfect cube.

5. The shapes with 4 corners and 4 edges on each side fall into four
   mutually separated classes.  Side shapes can be assigned values:
   0: Square, Mushroom, and Shield; 1: Left Fist and Left Paw; 2:
   Scallop, Kite, and Barrel; 3. Right Fist and Right Paw.  The top
   and bottom's sum or difference, depending on how you look at them,
   is a constant.  Notice that the side shapes with bilateral symmetry
   are those with even values.

6. To change this constant, and in particular to make it zero, you
   must attain a position that does not have 4 corners and 4 edges on
   each side.  Almost any such position will do, but returning to 4
   corners and 4 edges with the right constant is left to your
   ingenuity.

7. If the top and bottom are Squares but the middle is a Kite, just
   flip with the top and bottom 30deg out of phase and you will get a
   cube.

                                COLORS

1. I do not know the most efficient way to restore the colors.  What
   follows is my own suboptimal method.  All flips keep the yellow
   stripe steady and flip the blue stripe.

2. You can permute the corners without changing the edges, so first
   get the edges right, then the corners.

3. This transformation sends the right top edge to the bottom
   and the left bottom edge to the top, leaving the other edges
   on the same side as they started:  Twist top 30deg cl, flip,
   twist top 30deg ccl, twist bottom 150deg cl, flip, twist bottom
   30deg cl, twist top 120deg cl, flip, twist top 30deg ccl, twist
   bottom 150deg cl, flip, twist bottom 30deg cl.  Cl and ccl are
   defined looking directly at the face.  With this transformation
   you can eventually get all the white edges on top.

4. Check the parity of the edge sequence on each side.  If either is
   wrong, you need to fix it.  Sorry -- I don't know how!  (See any
   standard reference on combinatorics for an explanation of parity.)

5. The following transformation cyclically permutes ccl all the top
   edges but the right one and cl all the bottom edges but the left
   one.  Apply the transformation in 3., and turn the whole cube
   180deg.  Repeat.  This is a useful transformation, though not a
   cure-all.

6. Varying the transformation in 3. with other twists will produce
   other results.

7. The following transformation changes a cube into a Comet and Star:
   Flip to get Kite and Kite.  Twist top and bottom cl 90deg and flip
   to get Barrel and Barrel.  Twist top cl 30 and bottom cl 60 and flip
   to get Scallop and Scallop.  Twist top cl 60 and bottom cl 120 and
   flip to get Comet and Star.  The virtue of the Star is that it
   contains only corners, so that you can permute the corners without
   altering the edges.

8. To reach a Lemon and Star instead, replace the final bottom cl 120
   with a bottom cl 60.  In both these transformation the Star is on
   the bottom.

9. The following transformation cyclically permutes all but the bottom
   left rear.  It sends the top left front to the bottom, and the
   bottom left front to the top.  Go to Comet and Star.  Twist star cl
   60.  Go to Lemon and Star -- you need not return all the way to the
   cube, but do it if you're unsure of yourself by following 7
   backwards.  Twist star cl 60.  Return to cube by following 8
   backwards.  With this transformation you should be able to get all
   the white corners on top.

10. Check the parity of the corner sequences on both sides.  If the
   bottom parity is wrong, here's how to fix it:  Go to Lemon and
   Star.  The colors on the Star will run WWGWWG.  Twist it 180 and
   return to cube.

11. If the top parity is wrong, do the same thing, except that when you
   go from Scallop and Scallop to Lemon and Star, twist the top and
   bottom ccl instead of cl.  The colors on the Star should now run
   GGWGGW.

12. Once the parity is right on both sides, the basic method is to
   go to Comet and Star, twist the star 120 cl (it will be WGWGWG),
   return to cube, twist one or both sides, go to Comet and Star,
   undo the star twist, return to cube, undo the side twists.
   With no side twists, this does nothing.  If you twist the top,
   you will permute the top corners.  If you twist the bottom,
   you will permute the bottom corners.  Eventually you will get
   both the top and the bottom right.  Don't forget to undo the
   side twists -- you need to have the edges in the right places.

Happy twisting....
-- 
Col. G. L. Sicherman
gls@windmill.att.COM

I did once post my solution to Square one here, and I'll repost it
below.  It uses a neat shorthand notation that I think is much easier
than the one used at the S1 site.

Here is my solution to the Square-1 puzzle.  I assume you can get the
cube shape and that by giving it suitable twists (leaving it in cube
shape) that you can get three of the corner pieces of the top layer
correct.

To put the last corner piece from the bottom right position to the top
right position without disturbing the other three top corners, do
(1,0)/(0,-3)/(0,3)/(-1,0)
where each '/' means twisting the right half of the puzzle 180 degrees,
and each pair of coordinates denotes how much to turn the top and
bottom layers respectively.  3 means clockwise 90 degrees, 1 means 30
degrees (ie one edge along) and negative numbers mean anticlockwise
turns.

The letters F, B, L, R denote the Front Back Left and Right sides of
the puzzle, and U and D denote the top and bottom layer resp.
(Up/Down).  So the sequence above moves DFR to UFR without disturbing
UFL, UBL, and UBR.

Next the bottom layer corners:
Swap the FR and BR corners of the D layer by
/(0,-3)/(-3,6)/(0,-3)/(-3,3)/(0,-3)/(-3,6)
or the DFL and DBR corners by
/(3,3)/(0,3)/(3,3)/(0,3)/(3,3)/(0,-3)

Now put the edges in the correct layer by the following sequences:
Swap FD and FU, BD and BU:
(1,6)/(-1,-1)/(0,-5)
Swap RD and RU, BD and BU:
/(3,0)/(-3,-1)/(-5,-5)/(0,2)/(-3,6)/
Swap RD and BU, RU and BD:
(6,6)/(0,-3)/(-3,5)/(-5,-5)/(0,-4)/(-3,6)/
Swap RD and BU, FU and BU:
(1,0)/(3,0)/(0,3)/(3,-3)/(-3,3)/(-1,0)/(3,-3)/(-3,3)/(1,-3)/(-3,0)/(-1,0)
Cycle the three edges FU to BD to RU:
/(0,3)/(1,-3)/(0,-1)/(0,1)/(-1,0)/(0,3)/(0,-3)/
Cycle the three edges LU to BD to RU:
/(0,6)/(3,6)/(3,-4)/(3,-2)/(-3,0)/(0,3)/(-4,3)/(-3,4)/(0,-3)/

Now put the pieces in each layer in their correct position:
Top layer only:
Swap FU and BU, LU and RU:
/(3,-3)/(-3,3)/(1,0)/(3,-3)/(-1,0)
Swap FU and LU, BU and RU:
/(3,3)/(0,3)/(1,1)/(-4,5)/(3,3)/(-3,-3)
Cycle the three edges FU to RU to BU:
/(3,6)/(3,-2)/(2,-3)/(0,-3)/(0,3)/(-2,3)/(-3,2)/(0,3)/(-3,3)/
Swaps in both layers:
Swap FU and BU, FD and BD:
(1,0)/(-1,-1)/(-5,1)/(-1,1)/(6,1)
Swap FU and RU, FD and RD:
(1,-3)/(0,3)/(-1,-1)/(1,-2)/(-1,3)

All of these edge sequences were even permutations.  The nicest odd
edge permutation I know is this:
Swap FU and BU, LU and RU, FD and BD:
/(3,3)/(1,2)/(2,-4)/(-2,4)/(-1,1)/(3,3)/(0,3)/(3,3)/(0,3)/(6,0)/(6,0)/

This last one may need some explaining.  You may arrive at a position
where it is all solved except that 2 edges need swapping.  If you only
do the move sequences which do an even number of edge swaps, this
cannot be solved.  You will have to do this last sequence that performs
3 swaps, and then use the other sequences again to finally solve it.

-- Jaap Scherphuis <jaap@ermine.ox.ac.uk>, May 1997

[ Editor: the following was mailed to the archive by Carl Witthoft
<cgw@world.std.com> in October 1997, as a reply he had received to his
question "Does anyone know whether it's possible to get all of the
small wedges aligned next to each other?" ]

Yes.  Here's a way, starting from a cube:
(1) turn the right half of the puzzle 180 degrees -- two kitelike
    shapes result on top and bottom.
(2) turn the top layer 90 degrees clockwise (a large, then a small
    piece go past the center divider).
(3) turn the bottom layer 90 degrees clockwise (viewed from the bottom
    as in Rubik's Cube!).  The kites on the top and bottom should be
    pointing in opposite directions.
(4) turn the right half 180 degrees again -- the top should show a
    "butterfly" shape (LSSLLSSL); the bottom the same shape turned 90
    degrees.
(5) turn the top layer 30 degrees clockwise, the bottom 60 degrees
    clockwise.  This should align the two butterflies vertically.
(6) turn the right half 180 degrees again.  Both top and bottom have
    the shape LLLLSSSS, aligned identically.
(7) turn the top layer 60 degrees clockwise, the bottom 120 degrees
    clockwise.
(8) turn the right layer 180 degrees clockwise once more, and the top
    layer should show the shape you want (LLSSSSSSSS), while the bottom
    layer has six Large segments.

-- Michael Keller <Wgreview@aol.com>, World Game Review

There is also a solution booklet available.  It is written and
published by Richard Snyder.  He can be contacted at:

P.O. Box 1451
Westford. MA 01886

Send him 4 dollars and he will send you a copy.  Besides the solution
the booklet contains some pretty patterns and pretty shapes and gives
details of a Square-1 newsletter !!

-- Ray Bathke <bathke@cix.compulink.co.uk>, Village Games, London,
   September 1994