Message #2650

From: schuma <mananself@gmail.com>
Subject: Re: Parity aspects in skew MagicTile
Date: Mon, 04 Feb 2013 23:49:52 -0000

Congrats for finding another parity!

OK my name becomes Schumacher in this table. But I’m happy with it. The reason I use schuma for online forums etc is because I’m a big fan of Michael Schumacher.

Nan

— In 4D_Cubing@yahoogroups.com, "Eduard" wrote:
>
>
> 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
> twists
> * The 4 edge orbits in PitDeeDom have checkerboard pattern
> * In the smaller PitDeeDom (16) I was lucky enough to not encounter
> the parity problem
>