From:

Subject:

MONET01@mizzou1.missouri.edu writes of a cube that

``looks like someone took a knife to a normal solved cube and cut

a diagonal 'x' through each face and folded the flaps back down

the sides. This leads to a cube where opposing centers have an

'x' that has four colors in a mirror image. (It is hard to

describe, sorry.)''

I would appreciate a few more details. I think the color scheme of

each face you describe is something like

+-----+-----+-----+ |.1111|11111|1111.| |44.11|11111|11.22| |4444.|11111|.2222| +-----+-----+-----+ |44444|.111.|22222| |44444|44.22|22222| |44444|.333.|22222| +-----+-----+-----+ |4444.|33333|.2222| |44.33|33333|33.22| |.3333|33333|3333.| +-----+-----+-----+

where 1,2,3, and 4 are distinct colors, but there are still several

ways to make the colors on different faces match up. Look at a

corner, where the colors are

+-------+ /a.bbbbb/c\ /aaa.bbb/ccc\ /aaaaa.b/ccccc\ +-------+.......+ \fffff.e\ddddd/ \fff.eee\ddd/ \f.eeeee\d/ +-------+

That is, one corner is colored a/b, another c/d, and the third e/f,

where I expect some of a,b,c,d,e,f will be the same color.

One possibility was pictured in Hofstatder's Scientific American

article of February, 1981. It had b=c,d=e,f=a and used twelve colors.

Jim Saxe and I were impressed by its wasteful use of color and its

failure to exhibit edge orientation. From your remarks about turning

it over, I suspect this isn't what you mean.

You may be talking about the cube in which a=d,b=e,c=f which uses six

colors. I would say it is as if you cut an 'x' on a cube and

exchanged each triangle with the other triangle on the same edge of

the cube. That is a reasonably good coloring. It isn't really

necessary to solve it twice, though. To find out whether a given

corner goes on the top or bottom, look at the two colors that the

corner shares with the top face center. Either the corner will have

the two colors in the same order as the top, or they will be reversed,

and that determines whether that corner goes on the top or bottom.

That tells you where the third color on that corner goes, and the

last color is determined by elimination.

There is an even more interesting coloring that uses only four colors.

In this coloring a=c=e and the other three colors are distinct. Jim

Saxe and I came up with this coloring in our discussions of

Hofstatder's article. It isn't quite symmetric enough, since its

reflection is a coloring in which b=d=f, a slightly different pattern.

Our discussions then led to the Tartan coloring we talked about in our

article of 16 February 1981.

The only cube in the archives called the Ultimate Cube is the one that

has ``over 43 quintillion solutions.'' It has all six sides colored

the same.

Dan Hoey

Hoey@AIC.NRL.Navy.Mil