# Message #2405

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] MT {8,3} 10 colors

Date: Thu, 04 Oct 2012 22:15:52 -0500

Cool stuff! Taking a look, the underlying abstract shape has 10 faces, 24

edges, and 16 vertices. So its Euler Characteristic is 2, and it has the

topology of the sphere. This means the graph of it can be drawn on the

plane:

```
http://www.gravitation3d.com/magictile/pics/83/83-10_graph.png
```

Here is the unrolled version for reference:

```
http://www.gravitation3d.com/magictile/pics/83/83-10_unrolled.png
```

The first pic nicely shows how by starting your solution with the digons

(the "order 2" faces), it will be similar to solving a 3^3 starting with

the middle layer.

The "irregular" octagonal faces are interesting. I initially thought these

faces were hexagons in the abstract, until I realized they shared multiple

disjoint edges with the same neighbor. I hadn’t seen anything like that

before.

Cheers,

Roice

On Fri, Sep 28, 2012 at 4:02 AM, Andrey <andreyastrelin@yahoo.com> wrote:

> {8,3} 10 colors puzzle is an another strange beast. It has four faces of

> order 2 (i.e. each of them has only 2 neighbors), two faces of order 4 and

> 4 "irregular" faces. And if you start to solve it from order 2 faces (that

> is good idea because puzzle is the most dense there), you find yourself in

> situation where you have two disjoint unsolved "layers" - around order 4

> faces - and have to sort pieces and exchange parity/orientation between

> them (like when you solve 3^3 starting with the middle layer).

> And there is a chance to meet parity problem: some 2C pieces are

> identical and if odd number of pairs are swapped, you’ll need to solve it

> (by swapping some pair once more). And repeat sorting of order 4 layers

> again :)

> Nice thing :)

>

> Andrey