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138 competition/games/square-1.p


This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

138 competition/games/square-1.p

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



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.


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

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


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