# Message #2454

From: Eduard Baumann <baumann@mcnet.ch>

Subject: Re: [MC4D] MagicTile Solving

Date: Sat, 03 Nov 2012 12:00:24 +0100

Color graphs of MagicTile puzzles.

Actually I study the graphs of the MT irp {4,5} a30 and b30.

It’s a good graph study exercice.

These are regular graphs (same degre for all vertices).

I started to establish the adjacencylist (for each vertex the list of vertices which are joined) of the graph for a30.

This is not easy and you can make a lot of errors.

Here is the adjacencylist of a30:

1 2 3 4 5

2 1 6 7 8

3 1 9 10 11

4 1 12 14 13

5 1 15 16 17

6 2 9 18 19

7 2 20 21 22

8 2 15 23 24

9 25 6 3 26

10 3 13 27 24

11 3 29 20 28

12 4 23 20 19

13 4 30 21 10

14 4 26 22 17

15 8 28 30 5

16 25 20 28 5

17 5 29 18 14

18 17 6 21 27

19 29 6 30 12

20 7 11 12 16

21 18 13 7 27

22 25 7 24 14

23 8 27 12 26

24 10 8 22 29

25 9 22 16 30

26 14 23 28 9

27 10 16 23 18

28 21 15 26 11

29 24 17 19 11

30 25 19 13 15

Next I tried to embed this graph manually. It is very hard to find a nice embedding. I didn’t succeed so far.

Now I want try the intelligent procedure "spring embedding" in Mathematica presented in "Computational Discret Mathematics" from Pemmaraju et al, Cambridge.

I’m wondering what will be the result and how compare the two graphs a30 and b30.

Questions

Are all 4-30-regular graphs playable in MT irp {4,5} x30 ?

How many exist ?

Springembedding often let emerge the picture of a polyhedron. Will it here let emgerge an IRP ?

Regards

Ed

—– Original Message —–

From: Melinda Green

To: 4D_Cubing@yahoogroups.com

Sent: Friday, November 02, 2012 11:52 PM

Subject: Re: [MC4D] MagicTile Solving

Ah, I missed the ‘6’, thank you for the correction. This is one of the 3 IRPs that are as perfectly symmetric as the Platonic solids in every way. It is also the IRP twin of the original Rubik’s cube. I would still like to know why Nan’s solution is so much shorter.

I also do not understand why you see the IRP 4-5 b30 f001 as a warm-up exercise to the IRP {4,5} a30 F 0:0:1. True they both have 30 colors and genus 4, but they have different symmetries which I would guess would make the ‘a’ puzzle the simpler of the two.

-Melinda

On 11/2/2012 2:05 PM, Eduard Baumann wrote:

```
Wait.
The similar puzzle I mentioned is <br>
NOT<br>
MT irp {4,5} a30 F 0:0:1<br>
BUT<br>
MT irp {4,6} 12 F 0:0:1
I will attack <br>
MT irp {4,5} a30 F 0:0:1<br>
next time but I wanted study before he color topology of a30 and b30.
Ed
----- Original Message ----- <br>
From: Melinda Green <br>
To: 4D_Cubing@yahoogroups.com <br>
Sent: Friday, November 02, 2012 9:53 PM<br>
Subject: Re: [MC4D] MagicTile Solving
```

{4,5} a30 is one of my favorite IRPs. I find it to be quite beautiful and symmetric. It is the one that I showcase on the main geometry page to introduce the subject. (Third image down.) The ‘b’ puzzle that surprised you is less symmetric but is still a fascinating structure. It looks very much like an apartment complex. I would like to know why Nan was able to solve it with such a smaller number of twists. Unless your macros are extremely long, it doesn’t seem like that can be the only difference. What do you think, Nan?

-Melinda

On 11/2/2012 11:17 AM, Eduard wrote:

Solving of MT irp {4,5} b30 F 0:0:1 —– || 11/02/2012 || 2393

Remark:

Over 2000 twists. I worked without macros this time. Not low hanging fruit. Here 30 colors. In the similar puzzle "irp 4-6 12 f001" with 12 colors I worked with macros and needed 21’000 twists (Nan only 400 !!).