Message #1520

From: Andrey <andreyastrelin@yahoo.com>
Subject: Re: new hemi-cube and hemi-dodecahedron puzzles
Date: Mon, 07 Mar 2011 21:09:48 -0000

Nan,
I haven’t meet single 120-deg twisted corner in my two twists, but now, when you mentioned it, I agree - it’s really nice (but simple).
I think that Klein bottle-formed puzzles of type {6,3} are possible. It’s a matter of coloring, and if you take something like

.a.b.c.a.b.c
d.e.f.d.e.f.
.g.h.i.g.h.i
c.b.a.c.b.a.
.f.e.d.f.e.d
i.h.g.i.h.g.
.a.b.c.a.b.c

you will get what you want (but colors are not equivalent here).

I don’t see problems in 4-length - but it looks like 5-length puzzle with some missing pieces (like cell centers). And some couples of stickers are always glued together. But I never tried "kilominx", so I don’t know what I should expect.

If you twist opposite sides in same direction (relative to sphee surface), you’ll get orientable puzzle. I don’t know, if there are sphere paintings that are invariant to this transformation (i.e. order of adjacent colors of all intances of one face is the same). There is 5-color painting if icosahedron (all faces of outscribed tetrahedra have the same color), but I’m not sure that it will work.

Andrey