# Message #1397

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] Interesting object

Date: Wed, 09 Feb 2011 19:40:01 -0600

On Tue, Feb 8, 2011 at 11:05 AM, Roice Nelson <roice3@gmail.com> wrote:

> A coincident aside related to these hyperbolic translations is that I found

> out yesterday I missed a class of checkerboards on KQ which are due to

> symmetries of this kind. You can have six translational 4-cycles of KQ,

> much like what we are discussing here

>

Bummer, it turns out the class of hyperbolic translation KQ symmetries don’t

lead to valid checkerboards. The symmetries are "fixed point free", and all

84 edges move in 4-cycles. That’s 21 4-cycles, an odd number of odd

permutations, so no go (unless we cheat and pop the puzzle apart :D).

On the plus side, the translation checkerboards should work on the {8,3}

puzzles, both the existing 12-colored one and the 24-colored one which needs

to be implemented.

24-colored

There are 8*24/2 = 96 edges, so when they are all 4-cycled, we’ll have 24

sets.

12-colored

There are 8*12/2 = 48 edges, and the translations are 3-cycles in this case,

16 sets of them.

The latter should be a fun exercise, and if anyone would like to try, here

is a sequence to 3-cycle edge pieces without affecting anything else, using

clicks on only two faces.

```
( R2 L' R' L R' L R ) ( L2' R L R' L R' L' )
```

Swapping the two sub-sequences in parenthesis is another useful variant, and

these are what I used to make the KQ checkerboards.

Best,

Roice