Archived Thread

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Hi,

Making (manufacturing) a physical 2^4 puzzle you have to chose the right mi=
rrored 4-color cubicle of two possible ones for each of the 16 (8 over 4) c=
olor combinations which don't contain any opponent color pair. You can also=
manufacture the mirrored physical 2^4 puzzle. Now these two physical 2^4 c=
annot be transformed on in the other with 3D operations. On the other side =
a 3D view (projection) of the virtual 2^4 can be transformed in his mirror =
! See https://www.quora.com/Is-it-possible-for-my-3-dimensional-body-to-be-=
rotated-into-its-mirror-image-via-the-4th-dimension

Is something wrong in this consideration?
If yes, what?

Please read carefully my message.

Best regards
Ed

----- Original Message -----=20
From: Marc Ringuette ringuette@solarmirror.com [4D_Cubing]=20
To: 4D_Cubing@yahoogroups.com=20
Sent: Friday, November 17, 2017 8:38 PM
Subject: Re: [MC4D] Melinda's physical 2^4 puzzle, full puzzle rotations

=20=20=20=20
On 11/16/2017 2:22 PM, 'Eduard Baumann' ed.baumann@bluewin.ch [4D_Cubing]=
wrote:

BIG question. Is it possible to change the two "inverted" colors by sim=
ple moves?

Hi, Ed,

I think you're asking about what Melinda calls a full-puzzle rotation. =
Yes, this is key to being able to solve the full 2^4 puzzle in the same way=
on both the virtual and the physical versions. You can use either of the=
se two move sequences to do it.

Melinda's new video of her classic FOro: https://www.youtube.com/watch?=
v=3Dd2Fh_1m0UVY
Marc's alternative FOro using the ROIL moveset: https://youtu.be/3gvrda5=
fMto

I call the result of these sequences FOro, because they both correspond t=
o the single click MC4D move that is a Front-Out ROtation.=20

Here's an old video of mine, cued up to 2:30 where I demonstrate Melinda'=
s classic FOro while I carefully keep the puzzle oriented on a desk. At t=
he end I show the MC4D equivalent of the rotation I just did (at about 3:25=
of the video).
https://www.youtube.com/watch?v=3DhtTTn7qY35M&t=3D150s

I wonder who is going to be the first person to solve the physical 2^4 pu=
zzle? It's not me, yet. I've been waiting for more people to have the p=
uzzle before I get back to trying to solve it. I've come up with a 3-cycle=
algorithm, though, so that's a good start. I also have a long solution to=
the "double twisted corner" problem, I think, but I really want to come to=
a more thorough understanding of the 12 orientations of the 4-color pieces=
of a 4D hypercube.=20=20

A good milestone will be when somebody generates an MC4D scramble, sets u=
p that position on their physical puzzle, and then solves it on the physica=
l puzzle while following along with MC4D, resulting in both puzzles being s=
olved.

Cheers
Marc

p.s. here's a collection of my relevant videos of the physical 2^4 puzzl=
e, so you don't have to dig for them. I've omitted the ones that use the d=
eprecated (parity violating) quarter puzzle moves. These are still marked =
"unlisted" in YouTube, at Melinda's request, until her Public Release (TBD)=
. These vids are not fantastic quality, but they're what I've got.

2 over the shoulder intro v3 https://youtu.be/_rXJr4SokC0
3 special rotations FUro and FOro https://youtu.be/htTTn7qY35M
4 overlap of natural moves https://youtu.be/Dsu-5TTDNbM
07a a 3 cycle on phys 2^4 https://youtu.be/S8hm3CupPJAd
07b same 3 cycle on mc4d https://youtu.be/XCaaXxij6bc
12 ROIL moveset intro https://youtu.be/oYeh5fK8LCI
13 ROIL version of FOro https://youtu.be/3gvrda5fMto=20
14 ROIL scrambling https://youtu.be/ymnvMzypUp8
15 ROIL strict subset vs macro equivalents https://youtu.be/Iu7MKzD4yq0
16 ROIL meaning of junctions out positions https://youtu.be/xN_vn-6RnV8

=20=20
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=EF=BB=BF

To: ps.com=20
href=3D"mailto:4D_Cubing@yahoogroups.com">4D_Cubing@yahoogroups.com <=
/DIV>

Sent: Friday, November 17, 2017 8:=
38=20
PM

Subject: Re: [MC4D] Melinda's phys=
ical=20
2^4 puzzle, full puzzle rotations

=20

On 11/16/2017 2:22 PM, 'Eduard Baumann' iated=20
href=3D"mailto:ed.baumann@bluewin.ch">ed.baumann@bluewin.ch [4D_Cubin=
g]=20
wrote:

> size=3D2 face=3DArial>BIG question. Is it possible to change the two "i=
nverted"=20
colors by simple moves?

Hi, Ed,

I think y=
ou're=20
asking about what Melinda calls a full-puzzle rotation.   Yes, =
this=20
is key to being able to solve the full 2^4 puzzle in the same way on both=
the=20
virtual and the physical versions.   You can use either of thes=
e two=20
move sequences to do it.

Melinda's new video of her classic=20
FOro:   href=3D"https://www.youtube.com/watch?v=3Dd2Fh_1m0UVY">https://www.youtub=
e.com/watch?v=3Dd2Fh_1m0UVY
Marc's=20
alternative FOro using the ROIL moveset: eetext=20
href=3D"https://youtu.be/3gvrda5fMto">https://youtu.be/3gvrda5fMto>
I=20
call the result of these sequences FOro, because they both correspond to =
the=20
single click MC4D move that is a Front-Out ROtation.

Here's an ol=
d=20
video of mine, cued up to 2:30 where I demonstrate Melinda's classic FOro=
=20
while I carefully keep the puzzle oriented on a desk. At the end I=
show=20
the MC4D equivalent of the rotation I just did (at about 3:25 of the=20
video).
href=3D"https://www.youtube.com/watch?v=3DhtTTn7qY35M&t=3D150s">https=
://www.youtube.com/watch?v=3DhtTTn7qY35M&t=3D150s

I=20
wonder who is going to be the first person to solve the physical 2^4=20
puzzle?   It's not me, yet.   I've been waiting for m=
ore=20
people to have the puzzle before I get back to trying to solve it. =
I've=20
come up with a 3-cycle algorithm, though, so that's a good start. I=
also=20
have a long solution to the "double twisted corner" problem, I think, but=
I=20
really want to come to a more thorough understanding of the 12 orientatio=
ns of=20
the 4-color pieces of a 4D hypercube.

A good milestone will=
be=20
when somebody generates an MC4D scramble, sets up that position on their=
=20
physical puzzle, and then solves it on the physical puzzle while followin=
g=20
along with MC4D, resulting in both puzzles being=20
solved.

Cheers
Marc

p.s. here's a collecti=
on of=20
my relevant videos of the physical 2^4 puzzle, so you don't have to dig f=
or=20
them. I've omitted the ones that use the deprecated (parity violati=
ng)=20
quarter puzzle moves. These are still marked "unlisted" in YouTube,=
at=20
Melinda's request, until her Public Release (TBD).   These vids=
are=20
not fantastic quality, but they're what I've got.

2 over the shoul=
der=20
intro v3 href=3D"https://youtu.be/_rXJr4SokC0">https://youtu.be/_rXJr4SokC0>3=20
special rotations FUro and FOro href=3D"https://youtu.be/htTTn7qY35M">https://youtu.be/htTTn7qY35M>4=20
overlap of natural moves href=3D"https://youtu.be/Dsu-5TTDNbM">https://youtu.be/Dsu-5TTDNbM>07a a=20
3 cycle on phys 2^4 href=3D"https://youtu.be/S8hm3CupPJAd">https://youtu.be/S8hm3CupPJAd<=
BR>07b=20
same 3 cycle on mc4d href=3D"https://youtu.be/XCaaXxij6bc">https://youtu.be/XCaaXxij6bc>12=20
ROIL moveset intro href=3D"https://youtu.be/oYeh5fK8LCI">https://youtu.be/oYeh5fK8LCI>13=20
ROIL version of FOro href=3D"https://youtu.be/3gvrda5fMto">https://youtu.be/3gvrda5fMto R>14=20
ROIL scrambling   href=3D"https://youtu.be/ymnvMzypUp8">https://youtu.be/ymnvMzypUp8>15=20
ROIL strict subset vs macro equivalents etext=20
href=3D"https://youtu.be/Iu7MKzD4yq0">https://youtu.be/Iu7MKzD4yq0>16=20
ROIL meaning of junctions out positions etext=20
href=3D"https://youtu.be/xN_vn-6RnV8">https://youtu.be/xN_vn-6RnV8>

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From: Marc Ringuette <ringuette@solarmirror.com>
Date: Sat, 18 Nov 2017 08:03:44 -0800
Subject: Re: [MC4D] Melinda's physical 2^4 puzzle, mirrors

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Hi, Ed,

(1) There are 16 = 2*2*2*2 color combinations.   Each piece has either
W/Y, R/O, B/G, P/V.   There is one piece of each combination.
(2) The pieces each have one of the two possible handednesses. So,
people assembling their own puzzle must take care to get the handedness
of each piece correct.
(3) This is also true for the 2^4 hypercube: there are 16 four-color
pieces, in 2*2*2*2 color combinations, and there is only one of the two
handednesses of each. The MC4D program arbitrarily chose a
handedness.   Even if the 4th dimension allows any 3D face to be turned
"inside out" to reach the other handedness, the twists and rotations
allowed for the n^4 hypercube puzzle do not allow access to that other
handedness.
(4) Your version of mc4d is fine. When I built my first physical
puzzle, I created a text file, facecolors.txt, with some custom colors
to match the virtual with the physical version.   If you save the
attached file in the same folder with mc4d.jar, then restart MC4D and
open the 2^4 hypercube, the colors will hopefully match. This is not
well documented, but I saw it mentioned somewhere deep down in some old
release notes for MC4D, and messed with it until I made it look OK for me.

Cheers
Marc

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From: Marc Ringuette <ringuette@solarmirror.com>
Date: Sat, 18 Nov 2017 17:24:18 +0100
Subject: Re: [MC4D] Melinda's physical 2^4 puzzle, mirrors

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Thank for the answer and the hint for customized colors.
Best regards
Ed

----- Original Message -----=20
From: Marc Ringuette ringuette@solarmirror.com [4D_Cubing]=20
To: 4D_Cubing@yahoogroups.com=20
Sent: Saturday, November 18, 2017 5:03 PM
Subject: Re: [MC4D] Melinda's physical 2^4 puzzle, mirrors [1 Attachment]

=20=20=20=20
[Attachment(s) from Marc Ringuette included below]
=20=20=20
Hi, Ed,

(1) There are 16 =3D 2*2*2*2 color combinations. Each piece has either=
=20
W/Y, R/O, B/G, P/V. There is one piece of each combination.
(2) The pieces each have one of the two possible handednesses. So,=20
people assembling their own puzzle must take care to get the handedness=20
of each piece correct.
(3) This is also true for the 2^4 hypercube: there are 16 four-color=20
pieces, in 2*2*2*2 color combinations, and there is only one of the two=20
handednesses of each. The MC4D program arbitrarily chose a=20
handedness. Even if the 4th dimension allows any 3D face to be turned=20
"inside out" to reach the other handedness, the twists and rotations=20
allowed for the n^4 hypercube puzzle do not allow access to that other=20
handedness.
(4) Your version of mc4d is fine. When I built my first physical=20
puzzle, I created a text file, facecolors.txt, with some custom colors=20
to match the virtual with the physical version. If you save the=20
attached file in the same folder with mc4d.jar, then restart MC4D and=20
open the 2^4 hypercube, the colors will hopefully match. This is not=20
well documented, but I saw it mentioned somewhere deep down in some old=20
release notes for MC4D, and messed with it until I made it look OK for me=
.

Cheers
Marc

=20=20
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=EF=BB=BF

Thank for the answer and the hint for cust=
omized=20
colors.

Best regards

style=3D"BORDER-LEFT: #000000 2px solid; PADDING-LEFT: 5px; PADDING-RIGHT: =
0px; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px">

----- Original Message -----

style=3D"FONT: 10pt arial; BACKGROUND: #e4e4e4; font-color: black">Fro=
m:=20
href=3D"mailto:ringuette@solarmirror.com [4D_Cubing]">Marc Ringuette=20
ringuette@solarmirror.com [4D_Cubing]

To: ps.com=20
href=3D"mailto:4D_Cubing@yahoogroups.com">4D_Cubing@yahoogroups.com <=
/DIV>

Sent: Saturday, November 18, 2017 =
5:03=20
PM

Subject: Re: [MC4D] Melinda's phys=
ical=20
2^4 puzzle, mirrors [1 Attachment]

=20

Hi, Ed,

(1) There are 16 =3D 2*2*2*2 color combinations. &=
nbsp;=20
Each piece has either
W/Y, R/O, B/G, P/V.   There is one pi=
ece=20
of each combination.
(2) The pieces each have one of the two possible=
=20
handednesses. So,
people assembling their own puzzle must take =
care=20
to get the handedness
of each piece correct.
(3) This is also true=
for=20
the 2^4 hypercube: there are 16 four-color
pieces, in 2*2*2*2 color=20
combinations, and there is only one of the two
handednesses of each.&=
nbsp;=20
The MC4D program arbitrarily chose a
handedness.   Even if =
the=20
4th dimension allows any 3D face to be turned
"inside out" to reach t=
he=20
other handedness, the twists and rotations
allowed for the n^4 hyperc=
ube=20
puzzle do not allow access to that other
handedness.
(4) Your vers=
ion=20
of mc4d is fine. When I built my first physical
puzzle, I creat=
ed a=20
text file, facecolors.txt, with some custom colors
to match the virtu=
al=20
with the physical version.   If you save the
attached file =
in=20
the same folder with mc4d.jar, then restart MC4D and
open the 2^4=20
hypercube, the colors will hopefully match. This is not
well document=
ed,=20
but I saw it mentioned somewhere deep down in some old
release notes =
for=20
MC4D, and messed with it until I made it look OK for=20
me.

Cheers
Marc

>

------=_NextPart_000_0006_01D36092.121C69A0--

From: Melinda Green <melinda@superliminal.com>
Date: Sun, 19 Nov 2017 17:00:53 -0800
Subject: Re: [MC4D] Melinda's physical 2^4 puzzle, mirrors

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It's not so much an issue as a useful quirk of 3D space into which MC4D has=
been projected because it lets us specify twists according to clockwise an=
d counterclockwise rotations about an axis. That only works in 3D. In gener=
al you need to specify a 2D plane about which to rotate such that no point =
ends up at a different distance from that plane than it started. I don't kn=
ow if the concept of handedness can be meaningfully extended to higher dime=
nsions, but I barely feel like I have a grip on the concept in 3D. I'll als=
o note that the original Rubik's cube has the same issues. If you require 6=
particular colors in 3 particular pairs, then after you've stickered 2 pai=
rs of faces, you'll be faced with a choice of 2 ways to add the final pair.=
So for example, if you look for the corner with red, white, and blue stick=
ers, they will be arranged clockwise in one arrangement and counterclockwis=
e in it's mirrored twin.

-Melinda

On 11/19/2017 2:05 AM, 'Eduard Baumann' ed.baumann@bluewin.ch [4D_Cubing] w=
rote:
> =EF=BB=BF
>
> 2^4 MCD4 is handed=C2=A0 (is that a programming issue?)
> *_/Not an easy/_* thema.
>
>
> Foundation Mathematics for Computer Science: A Visual Approach ps://books.google.ch/books?id=3DWNlECgAAQBAJ&pg=3DPA100&lpg=3DPA100&dq=3Dha=
ndedness+in+4-dimensions&source=3Dbl&ots=3DIDHCOUZsk6&sig=3DGt11AUaUQXX9avv=
l12U1CyZ1_fs&hl=3Dde&sa=3DX&ved=3D0ahUKEwiwtMqpsMrXAhWLPxQKHV8PBrE4ChDoAQg4=
MAM>
>
> https://books.google.ch/books?isbn=3D3319214373-Diese Seite =C3=BCbersetz=
en .google.ch/books%3Fid%3DWNlECgAAQBAJ%26pg%3DPA100%26lpg%3DPA100%26dq%3Dhand=
edness%2Bin%2B4-dimensions%26source%3Dbl%26ots%3DIDHCOUZsk6%26sig%3DGt11AUa=
UQXX9avvl12U1CyZ1_fs&prev=3Dsearch>
> John Vince 4&biw=3D1024&bih=3D649&tbm=3Dbks&q=3Dinauthor:%22John+Vince%22&ved=3D0ahUKE=
wiwtMqpsMrXAhWLPxQKHV8PBrE4ChD0CAg7MAM>- 2015 - =E2=80=8EComputers
> 6.5 a A/left-handed/system. b A right-handed system Fig. ..._It also wort=
h noting that/handedness /has no meaning in spaces with/4 dimensions/or mor=
e_.
> But see also:
> https://math.stackexchange.com/questions/1884953/what-are-the-higher-dime=
nsional-analogues-of-left-and-right-handedness
> Best regards
> Ed
>
> ----- Original Message -----
> *From:* Marc Ringuette ringuette@solarmirror.com [4D_Cubing] ringuette@solarmirror.com%20[4D_Cubing]>
> *To:* 4D_Cubing@yahoogroups.com
> *Sent:* Saturday, November 18, 2017 5:03 PM
> *Subject:* Re: [MC4D] Melinda's physical 2^4 puzzle, mirrors [1 Attac=
hment]
>
> Hi, Ed,
>
> (1) There are 16 =3D 2*2*2*2 color combinations.=C2=A0=C2=A0 Each pie=
ce has either
> W/Y, R/O, B/G, P/V.=C2=A0=C2=A0 There is one piece of each combinatio=
n.
> (2) The pieces each have one of the two possible handednesses.=C2=A0 =
So,
> people assembling their own puzzle must take care to get the handedne=
ss
> of each piece correct.
> (3) This is also true for the 2^4 hypercube: there are 16 four-color
> pieces, in 2*2*2*2 color combinations, and there is only one of the t=
wo
> handednesses of each.=C2=A0 The MC4D program arbitrarily chose a
> handedness.=C2=A0=C2=A0 Even if the 4th dimension allows any 3D face =
to be turned
> "inside out" to reach the other handedness, the twists and rotations
> allowed for the n^4 hypercube puzzle do not allow access to that othe=
r
> handedness.
> (4) Your version of mc4d is fine.=C2=A0 When I built my first physica=
l
> puzzle, I created a text file, facecolors.txt, with some custom color=
s
> to match the virtual with the physical version.=C2=A0=C2=A0 If you sa=
ve the
> attached file in the same folder with mc4d.jar, then restart MC4D and
> open the 2^4 hypercube, the colors will hopefully match. This is not
> well documented, but I saw it mentioned somewhere deep down in some o=
ld
> release notes for MC4D, and messed with it until I made it look OK fo=
r me.
>
> Cheers
> Marc
>
>
>
>=20=20=20=20=20
>

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Content-Transfer-Encoding: quoted-printable

">

It's not so much an issue as a useful quirk of 3D space into which
MC4D has been projected because it lets us specify twists according
to clockwise and counterclockwise rotations about an axis. That only
works in 3D. In general you need to specify a 2D plane about which
to rotate such that no point ends up at a different distance from
that plane than it started. I don't know if the concept of
handedness can be meaningfully extended to higher dimensions, but I
barely feel like I have a grip on the concept in 3D. I'll also note
that the original Rubik's cube has the same issues. If you require 6
particular colors in 3 particular pairs, then after you've stickered
2 pairs of faces, you'll be faced with a choice of 2 ways to add the
final pair. So for example, if you look for the corner with red,
white, and blue stickers, they will be arranged clockwise in one
arrangement and counterclockwise in it's mirrored twin.

-Melinda

On 11/19/2017 2:05 AM, 'Eduard Baumann'
in.ch">ed.baumann@bluewin.ch [4D_Cubing] wrote:

cite=3D"mid:A84B866124044318A2AC58D6DE8D833C@LAB">
=EF=BB=BF
pe">

=20=20=20=20=20=20

2^4 MCD4 is handed=C2=A0 (is tha=
t a
programming issue?)

=C2=A0

Not an easy<=
/u>
thema.

=C2=A0

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class=3D"r"> pointer; TEXT-DECORATION: none"
href=3D"https://books.google.ch/books?id=3DWNlECgAAQBAJ&pg=3DPA100&=
lpg=3DPA100&dq=3Dhandedness+in+4-dimensions&source=3Dbl&ots=3DI=
DHCOUZsk6&sig=3DGt11AUaUQXX9avvl12U1CyZ1_fs&hl=3Dde&sa=3DX&=
ved=3D0ahUKEwiwtMqpsMrXAhWLPxQKHV8PBrE4ChDoAQg4MAM"
moz-do-not-send=3D"true">Foundation Mathematics for
Computer Science: A Visual Approach

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ttps://books.google.ch/books%3Fid%3DWNlECgAAQBAJ%26pg%3DPA100%26lpg%3DPA100=
%26dq%3Dhandedness%2Bin%2B4-dimensions%26source%3Dbl%26ots%3DIDHCOUZsk6%26s=
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class=3D"slp f">:
pointer; TEXT-DECORATION: none" class=3D"fl"
href=3D"https://www.google.ch/search?sa=3DN&rlz=3D1C1RNRC_enCH504CH504&=
amp;biw=3D1024&bih=3D649&tbm=3Dbks&q=3Dinauthor:%22John+Vince%2=
2&ved=3D0ahUKEwiwtMqpsMrXAhWLPxQKHV8PBrE4ChD0CAg7MAM"
moz-do-not-send=3D"true">John Vince class=3D"Apple-converted-space">=C2=A0- 2015 -
=E2=80=8EComputers

class=3D"st">6.5 a A>=C2=A0 style=3D"FONT-STYLE: normal; COLOR: rgb(106,106,106);
FONT-WEIGHT: bold">left-handed class=3D"Apple-converted-space">=C2=A0system. b =
A
right-handed system Fig. ... It also worth noting
that=C2=A0=
style=3D"FONT-STYLE: normal; COLOR:
rgb(106,106,106); FONT-WEIGHT: bold">handedness =
has
no meaning in spaces with class=3D"Apple-converted-space">=C2=A0 style=3D"FONT-STYLE: normal; COLOR:
rgb(106,106,106); FONT-WEIGHT: bold">4 dimensions> class=3D"Apple-converted-space">=C2=A0or more<=
/u>.

=C2=A0

But see also:

href=3D"https://math.stackexchange.com/questions/1884953/what-are-the-highe=
r-dimensional-analogues-of-left-and-right-handedness"
moz-do-not-send=3D"true">https://math.stackexchange.com/quest=
ions/1884953/what-are-the-higher-dimensional-analogues-of-left-and-right-ha=
ndedness

=C2=A0

Best regards

Ed

=C2=A0

5px; PADDING-RIGHT: 0px; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px">

----- Original Message -----

black">From: href=3D"mailto:ringuette@solarmirror.com%20[4D_Cubing]"
moz-do-not-send=3D"true">Marc Ringuette
ringuette@solarmirror.com [4D_Cubing]

To: title=3D"4D_Cubing@yahoogroups.com"
href=3D"mailto:4D_Cubing@yahoogroups.com"
moz-do-not-send=3D"true">4D_Cubing@yahoogroups.com

Sent: Saturday, November
18, 2017 5:03 PM

Subject: Re: [MC4D]
Melinda's physical 2^4 puzzle, mirrors [1 Attachment]

=C2=A0

Hi, Ed,

(1) There are 16 =3D 2*2*2*2 color combinations.=C2=A0=C2=A0 Ea=
ch piece
has either

W/Y, R/O, B/G, P/V.=C2=A0=C2=A0 There is one piece of each
combination.

(2) The pieces each have one of the two possible
handednesses.=C2=A0 So,

people assembling their own puzzle must take care to get the
handedness

of each piece correct.

(3) This is also true for the 2^4 hypercube: there are 16
four-color

pieces, in 2*2*2*2 color combinations, and there is only one
of the two

handednesses of each.=C2=A0 The MC4D program arbitrarily chose =
a

handedness.=C2=A0=C2=A0 Even if the 4th dimension allows any 3D=
face
to be turned

"inside out" to reach the other handedness, the twists and
rotations

allowed for the n^4 hypercube puzzle do not allow access to
that other

handedness.

(4) Your version of mc4d is fine.=C2=A0 When I built my first
physical

puzzle, I created a text file, facecolors.txt, with some
custom colors

to match the virtual with the physical version.=C2=A0=C2=A0 If =
you
save the

attached file in the same folder with mc4d.jar, then restart
MC4D and

open the 2^4 hypercube, the colors will hopefully match.
This is not

well documented, but I saw it mentioned somewhere deep down
in some old

release notes for MC4D, and messed with it until I made it
look OK for me.

Cheers

Marc

=20=20=20=20=20=20=20=20

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