# Message #2649

From: Eduard <ed.baumann@bluewin.ch>

Subject: Parity aspects in skew MagicTile

Date: Mon, 04 Feb 2013 23:33:03 -0000

NEW parity aspect in skew MagicTile!

The even Duoprismes are also interesting and different.

2nd theorem of Baumann, "PitDeeDom"

Unlike in odd cases here in the even case the / edges are separated in

two orbits. Dito the \ edges!

I encountered a bad parity situation where I had exactly one edge swap

left in each of these 4 orbits.

If I do 4 twists in most compact constellation (corners of a small

square with horizontal and vertical sides), I hit 4 orbits with 12

diamond face elements exactly twice. This can be undone by a 3-cycle.

And I get one edge swap in each of the 4 edge orbits (plus edge swap

pairs).

This repairs the parity.

Recapitulation of parity aspects in skew MagicTile

theorem

name

@

restore parity with

twist

number

puzzle

Astrelin

PitDvoRom

odd

turn whole by 60

0

{4,6|3} 30 v020 runcinated

1st Baumann

PitDeoBom

odd

turn whole by 90

0

{6,4|3} 20 e010 bitruncated

Schumacher

PitDeoDom

odd

big X

14

{4,4|7} 49 e 1.41 duoprisme

2nd Baumann

PitDeeDom

even

small square

4

{4,4|6} 36 e 1.41 duoprisme

Remarks

```
* In the smaller PitDeoDoms (9 and 25) the big X needs only 6 and 10<br> twists<br>
* The 4 edge orbits in PitDeeDom have checkerboard pattern<br>
* In the smaller PitDeeDom (16) I was lucky enough to not encounter<br> the parity problem
```