Hi, 4D cubers,
Today I ticked off another item on my to-do list for Melinda's physical
2^4 puzzle: I found a 3-cycle of corners using only the "common subset"
moves** on the phys 2^4.
LF' (FO2 LO' FO2 LO FO2) LF OR2
LF' (FO2 LO' FO2 LO FO2) LF OR2
It cycles LUFI to RUFO to RDBO, and consists of 16 twists, each of which
affects half the puzzle.
Video demo on the phys 2^4 (90 secs): http://youtu.be/S8hm3CupPJA
In the video description is a link to my vid replicating the 3-cycle on
mc4d using the same colors. Or, here is the mc4d sequence in log file
format:
103,1,1 79,-1,1 79,-1,1 101,-1,1 79,-1,1 79,-1,1 101,1,1 79,-1,1 79,-1,1
103,-1,1
133,-1,1 133,-1,1 133,-1,1 133,-1,1 23,-1,2 23,-1,2 103,1,1 79,-1,1
79,-1,1 101,-1,1
79,-1,1 79,-1,1 101,1,1 79,-1,1 79,-1,1 103,-1,1 23,-1,2 23,-1,2.
** Note 1 - the meaning of "common subset": In the phys 2^4, we forbid
moves that can't be done in one click on mc4d (restacks, clamshells, and
4-piece subface twists). In mc4d, we forbid moves that can't be done
in one simple twist on the phys 2^4 (i.e. 90 degree twists or rotations
that do not hold the R-L axis fixed; all half turns are OK, even those
without an R or L as the first or second letter of their name).
** Note 2: I am 99% certain that all moves in both puzzles have either
simple moves or macros (sequences of legal moves) in the other puzzle.
The "common subset" is a proper subset of the moves available in each
puzzle, that is useful for direct translations of algorithms from one to
the other. It limits the macro/sequence length to 1 in both directions.
Algorithms not in the common subset take a greater number of moves
in one puzzle or the other.
** Note 3: The common subset cannot solve the puzzle because it cannot
fully orient each piece. It can't move stickers on or off of the
combined R+L faces. On the phys 2^4, we must add either FOro or FUro
(or something equivalent) to the common subset in order to fully solve
all 2^4 scrambles. I demoed those in my video #3 on June 10th.
Cheers
Marc
------=_NextPart_000_0028_01D2ECD2.36FF3D30
Content-Type: text/plain;
charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
Great !!
Does now existe a table which gives the "phys 2^4" moves
for each "one click on mc4d" ?
Kind regards
Ed
----- Original Message -----=20
From: Marc Ringuette ringuette@solarmirror.com [4D_Cubing]=20
To: 4D_Cubing@yahoogroups.com=20
Sent: Saturday, June 24, 2017 2:26 AM
Subject: [MC4D] 3-cycle demo on phys 2^4
=20=20=20=20
Hi, 4D cubers,
Today I ticked off another item on my to-do list for Melinda's physical=20
2^4 puzzle: I found a 3-cycle of corners using only the "common subset"=20
moves** on the phys 2^4.
LF' (FO2 LO' FO2 LO FO2) LF OR2
LF' (FO2 LO' FO2 LO FO2) LF OR2
It cycles LUFI to RUFO to RDBO, and consists of 16 twists, each of which=
=20
affects half the puzzle.
Video demo on the phys 2^4 (90 secs): http://youtu.be/S8hm3CupPJA
In the video description is a link to my vid replicating the 3-cycle on=20
mc4d using the same colors. Or, here is the mc4d sequence in log file=20
format:
103,1,1 79,-1,1 79,-1,1 101,-1,1 79,-1,1 79,-1,1 101,1,1 79,-1,1 79,-1,1=
=20
103,-1,1
133,-1,1 133,-1,1 133,-1,1 133,-1,1 23,-1,2 23,-1,2 103,1,1 79,-1,1=20
79,-1,1 101,-1,1
79,-1,1 79,-1,1 101,1,1 79,-1,1 79,-1,1 103,-1,1 23,-1,2 23,-1,2.
** Note 1 - the meaning of "common subset": In the phys 2^4, we forbid=20
moves that can't be done in one click on mc4d (restacks, clamshells, and=
=20
4-piece subface twists). In mc4d, we forbid moves that can't be done=20
in one simple twist on the phys 2^4 (i.e. 90 degree twists or rotations=20
that do not hold the R-L axis fixed; all half turns are OK, even those=20
without an R or L as the first or second letter of their name).
** Note 2: I am 99% certain that all moves in both puzzles have either=20
simple moves or macros (sequences of legal moves) in the other puzzle.=20
The "common subset" is a proper subset of the moves available in each=20
puzzle, that is useful for direct translations of algorithms from one to=
=20
the other. It limits the macro/sequence length to 1 in both directions.=20
Algorithms not in the common subset take a greater number of moves=20
in one puzzle or the other.
** Note 3: The common subset cannot solve the puzzle because it cannot=20
fully orient each piece. It can't move stickers on or off of the=20
combined R+L faces. On the phys 2^4, we must add either FOro or FUro=20
(or something equivalent) to the common subset in order to fully solve=20
all 2^4 scrambles. I demoed those in my video #3 on June 10th.
Cheers
Marc
=20=20
------=_NextPart_000_0028_01D2ECD2.36FF3D30
Content-Type: text/html;
charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
=EF=BB=BF
Hi, 4D cubers,
Today I ticked off another item on my to-do list=
for=20
Melinda's physical
2^4 puzzle: I found a 3-cycle of corners using onl=
y the=20
"common subset"
moves** on the phys 2^4.
LF' (FO2 LO' FO2 LO F=
O2)=20
LF OR2
LF' (FO2 LO' FO2 LO FO2) LF OR2
It cycles LUFI to RUFO t=
o=20
RDBO, and consists of 16 twists, each of which
affects half the=20
puzzle.
Video demo on the phys 2^4 (90 secs):=20
http://youtu.be/S8hm3CupPJA
In the video description is a link to =
my=20
vid replicating the 3-cycle on
mc4d using the same colors. Or, here i=
s the=20
mc4d sequence in log file
format:
103,1,1 79,-1,1 79,-1,1 101,-1,1=
=20
79,-1,1 79,-1,1 101,1,1 79,-1,1 79,-1,1
103,-1,1
133,-1,1 133,-1,1=
=20
133,-1,1 133,-1,1 23,-1,2 23,-1,2 103,1,1 79,-1,1
79,-1,1=20
101,-1,1
79,-1,1 79,-1,1 101,1,1 79,-1,1 79,-1,1 103,-1,1 23,-1,2=20
23,-1,2.
** Note 1 - the meaning of "common subset": In the phys 2=
^4,=20
we forbid
moves that can't be done in one click on mc4d (restacks,=20
clamshells, and
4-piece subface twists). In mc4d, we forbid moves tha=
t=20
can't be done
in one simple twist on the phys 2^4 (i.e. 90 degree twi=
sts=20
or rotations
that do not hold the R-L axis fixed; all half turns are =
OK,=20
even those
without an R or L as the first or second letter of their=20
name).
** Note 2: I am 99% certain that all moves in both puzzles =
have=20
either
simple moves or macros (sequences of legal moves) in the other=
=20
puzzle.
The "common subset" is a proper subset of the moves available=
in=20
each
puzzle, that is useful for direct translations of algorithms fro=
m one=20
to
the other. It limits the macro/sequence length to 1 in both direct=
ions.=20
Algorithms not in the common subset take a greater number of moves
one puzzle or the other.
** Note 3: The common subset cannot solve=
the=20
puzzle because it cannot
fully orient each piece. It can't move stick=
ers=20
on or off of the
combined R+L faces. On the phys 2^4, we must add eit=
her=20
FOro or FUro
(or something equivalent) to the common subset in order =
to=20
fully solve
all 2^4 scrambles. I demoed those in my video #3 on June=
=20
10th.
Cheers
Marc
------=_NextPart_000_0045_01D2ECD3.B2ED5AC0
Content-Type: text/plain;
charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
For those who don't have a physical 2^4 yet:
How about a "virtual physical 2^4" ? (a nice word combination!)
Anyone has already programmed this?
Kind regards
Ed
----- Original Message -----=20
From: Marc Ringuette ringuette@solarmirror.com [4D_Cubing]=20
To: 4D_Cubing@yahoogroups.com=20
Sent: Saturday, June 24, 2017 2:26 AM
Subject: [MC4D] 3-cycle demo on phys 2^4
=20=20=20=20
Hi, 4D cubers,
Today I ticked off another item on my to-do list for Melinda's physical=20
2^4 puzzle: I found a 3-cycle of corners using only the "common subset"=20
moves** on the phys 2^4.
LF' (FO2 LO' FO2 LO FO2) LF OR2
LF' (FO2 LO' FO2 LO FO2) LF OR2
It cycles LUFI to RUFO to RDBO, and consists of 16 twists, each of which=
=20
affects half the puzzle.
Video demo on the phys 2^4 (90 secs): http://youtu.be/S8hm3CupPJA
In the video description is a link to my vid replicating the 3-cycle on=20
mc4d using the same colors. Or, here is the mc4d sequence in log file=20
format:
103,1,1 79,-1,1 79,-1,1 101,-1,1 79,-1,1 79,-1,1 101,1,1 79,-1,1 79,-1,1=
=20
103,-1,1
133,-1,1 133,-1,1 133,-1,1 133,-1,1 23,-1,2 23,-1,2 103,1,1 79,-1,1=20
79,-1,1 101,-1,1
79,-1,1 79,-1,1 101,1,1 79,-1,1 79,-1,1 103,-1,1 23,-1,2 23,-1,2.
** Note 1 - the meaning of "common subset": In the phys 2^4, we forbid=20
moves that can't be done in one click on mc4d (restacks, clamshells, and=
=20
4-piece subface twists). In mc4d, we forbid moves that can't be done=20
in one simple twist on the phys 2^4 (i.e. 90 degree twists or rotations=20
that do not hold the R-L axis fixed; all half turns are OK, even those=20
without an R or L as the first or second letter of their name).
** Note 2: I am 99% certain that all moves in both puzzles have either=20
simple moves or macros (sequences of legal moves) in the other puzzle.=20
The "common subset" is a proper subset of the moves available in each=20
puzzle, that is useful for direct translations of algorithms from one to=
=20
the other. It limits the macro/sequence length to 1 in both directions.=20
Algorithms not in the common subset take a greater number of moves=20
in one puzzle or the other.
** Note 3: The common subset cannot solve the puzzle because it cannot=20
fully orient each piece. It can't move stickers on or off of the=20
combined R+L faces. On the phys 2^4, we must add either FOro or FUro=20
(or something equivalent) to the common subset in order to fully solve=20
all 2^4 scrambles. I demoed those in my video #3 on June 10th.
Cheers
Marc
=20=20
------=_NextPart_000_0045_01D2ECD3.B2ED5AC0
Content-Type: text/html;
charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
=EF=BB=BF
Hi, 4D cubers,
Today I ticked off another item on my to-do list=
for=20
Melinda's physical
2^4 puzzle: I found a 3-cycle of corners using onl=
y the=20
"common subset"
moves** on the phys 2^4.
LF' (FO2 LO' FO2 LO F=
O2)=20
LF OR2
LF' (FO2 LO' FO2 LO FO2) LF OR2
It cycles LUFI to RUFO t=
o=20
RDBO, and consists of 16 twists, each of which
affects half the=20
puzzle.
Video demo on the phys 2^4 (90 secs):=20
http://youtu.be/S8hm3CupPJA
In the video description is a link to =
my=20
vid replicating the 3-cycle on
mc4d using the same colors. Or, here i=
s the=20
mc4d sequence in log file
format:
103,1,1 79,-1,1 79,-1,1 101,-1,1=
=20
79,-1,1 79,-1,1 101,1,1 79,-1,1 79,-1,1
103,-1,1
133,-1,1 133,-1,1=
=20
133,-1,1 133,-1,1 23,-1,2 23,-1,2 103,1,1 79,-1,1
79,-1,1=20
101,-1,1
79,-1,1 79,-1,1 101,1,1 79,-1,1 79,-1,1 103,-1,1 23,-1,2=20
23,-1,2.
** Note 1 - the meaning of "common subset": In the phys 2=
^4,=20
we forbid
moves that can't be done in one click on mc4d (restacks,=20
clamshells, and
4-piece subface twists). In mc4d, we forbid moves tha=
t=20
can't be done
in one simple twist on the phys 2^4 (i.e. 90 degree twi=
sts=20
or rotations
that do not hold the R-L axis fixed; all half turns are =
OK,=20
even those
without an R or L as the first or second letter of their=20
name).
** Note 2: I am 99% certain that all moves in both puzzles =
have=20
either
simple moves or macros (sequences of legal moves) in the other=
=20
puzzle.
The "common subset" is a proper subset of the moves available=
in=20
each
puzzle, that is useful for direct translations of algorithms fro=
m one=20
to
the other. It limits the macro/sequence length to 1 in both direct=
ions.=20
Algorithms not in the common subset take a greater number of moves
one puzzle or the other.
** Note 3: The common subset cannot solve=
the=20
puzzle because it cannot
fully orient each piece. It can't move stick=
ers=20
on or off of the
combined R+L faces. On the phys 2^4, we must add eit=
her=20
FOro or FUro
(or something equivalent) to the common subset in order =
to=20
fully solve
all 2^4 scrambles. I demoed those in my video #3 on June=
=20
10th.
Cheers
Marc
--------------E980DC34B8CC9C767D3A7CB2
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: quoted-printable
Hello Ed,
Burkard has bought two puzzles and has also made a video on the Buddhabrot =
ebted to him. He has intended to do another on the 2x2x2x2 but hasn't gotte=
n around to it. He mainly wants a good simulator to generate graphics. Feel=
free to contact him yourself or simply leave suggestions in the comments o=
f his new videos to nudge him. At the moment Shapeways raised their 3D prin=
ting costs by 20% which pretty much killed puzzle sales, so I won't be read=
y to sell many until I get it mass produced anyway. He has solved the puzzl=
e but hasn't made a solution video. Of course a mathologer video on the top=
ic (or follow-up like he did for MC4D solution) could be an ideal solution,=
and I expect that is his intention. In short, I think this is likely to ha=
ppen eventually.
Thanks for all your support,
-Melinda
On 5/3/2019 7:53 AM, 'Eduard Baumann' ed.baumann@bluewin.ch [4D_Cubing] wro=
te:
> =EF=BB=BF
>
> Hi Marc,
> _Mathologer_ hast made beautifull presentations of
>
> * the 4D Rubic and
> * MagicTile
>
> Marc, you have collectet a considable amount of stuff about Melinda's phy=
sical 2x2x2x2.
> I think it would be eminently worthfull to have an animated view of an in=
terested non-insider.
> Who has the e-mail address of the Mathologer?
> Kind regards
> Ed
> https://www.youtube.com/watch?v=3DyhPH1369OWc&t=3D602s
> https://www.youtube.com/watch?v=3DDvZnh7-nslo&feature=3Dyoutu.be
> https://www.youtube.com/watch?v=3DiOla7WPfCvA&feature=3Dyoutu.be
>
> ----- Original Message -----
> *From:* Marc Ringuette ringuette@solarmirror.com [4D_Cubing]
> *To:* 4D_Cubing@yahoogroups.com
> *Sent:* Sunday, June 25, 2017 7:24 PM
> *Subject:* Re: [MC4D] 3-cycle demo on phys 2^4
>
> (welcome, Okko!)
>
> Hey, Ed, regarding Melinda's physical 2^4 puzzle,
>
> =C2=A0 (1) Nope, I haven't written down a big table of move equivalen=
ts, although my videos will give you enough to figure it out.=C2=A0=C2=A0 T=
he one-sentence version is:=C2=A0 any mc4d move that leaves the stickers on=
the R+L faces still on the R+L faces, has a simple twist in the phys 2^4 p=
uzzle.=C2=A0=C2=A0 With the addition of a single rotation to this common su=
bset -- any rotation that moves the R+L faces onto a different pair of face=
s, FOro and FUro being the two phys 2^4 sequences discovered so far to do t=
his -- the entire mc4d state space can be reached on the phys 2^4 and vice =
versa.
>
> =C2=A0 (2) Nope, nobody else has written down the move table either.=
=C2=A0 That's not too surprising, since nobody else but Melinda and me have=
made physical versions of the puzzle yet.=C2=A0=C2=A0 C'mon, gang, get bui=
lding!=C2=A0 It's fun, and it'll only take you a couple of days.=C2=A0 :)=
=C2=A0=C2=A0=C2=A0=C2=A0 I suspect people are waiting=C2=A0 for Melinda or =
me to provide some nice step-by-step instructions that can be completed in =
3-12 hours, or even better, just wheedle one of us into building them one.
>
> =C2=A0 (3)=C2=A0 The "virtual physical" 2^4.=C2=A0=C2=A0 It feels sli=
ghtly goofy to take Melinda's puzzle, whose key feature is the ability for =
a physical realization, and then make a simulation of it. =C2=A0 Heh.=C2=A0=
=C2=A0 However, it would actually be really useful, given that a lot of peo=
ple won't manage to surmount the energy barrier associated with building th=
e physical puzzles.
>
> =C2=A0(4)=C2=A0 Thinking about it, it would be really useful to creat=
e a virtual physical 2^4 puzzle, in Javascript, shown side-by-side with the=
MC4D style simulation, with each move being executed simultaneously on bot=
h.=C2=A0=C2=A0 And as a side-effect of creating the MC4d style rendering in=
Javascript, we would easily be able to also produce a web-browser-compatib=
le version of all of MC4D's n^4 hypercube puzzles that would make them that=
much more accessible to people who reluctant to use the Java app. Such a v=
ersion could be far less general and tricky than MC4D itself, because it wo=
uld be so much more limited (just 4d, in a particular simplified/hacked pro=
jection).=C2=A0=C2=A0 Sort of a "gateway drug" to full MC4D.=C2=A0=C2=A0 It=
would be FANTASTIC if somebody programmed this.
>
> =C2=A0 (5)=C2=A0 A teaser:=C2=A0 I've been trying to work out an exte=
nsion of Melinda's physical 2^4 puzzle into a physical 3^4, and making some=
good progress.=C2=A0=C2=A0 However, with 81 pieces and 972 magnets, it's f=
airly impractical to create physically in a way that can actually be operat=
ed by hand.=C2=A0=C2=A0 Pretty much the only way it is likely to come to ex=
ist is in a "virtual physical" version in Javascript.
>
>
> Cheers
> Marc
>
>
> On 6/24/2017 1:21 AM, 'Eduard Baumann' ed.baumann@bluewin.ch [4D_Cubi=
ng] wrote:
>>
>> =EF=BB=BF[Is there a] table which gives the "phys 2^4" moves for eac=
h "one click on mc4d" ?
>>
>
>> =EF=BB=BFHow about a "virtual physical 2^4" ?
>>
>
>
>
>=20=20=20=20=20
>
--------------E980DC34B8CC9C767D3A7CB2
Content-Type: text/html; charset=utf-8
Content-Transfer-Encoding: quoted-printable
">
Hello Ed,
Burkard has bought two puzzles and has also made a video on the
href=3D"https://www.youtube.com/watch?v=3D9gk_8mQuerg">Buddhabrot=
,
so I am already deeply indebted to him. He has intended to do
another on the 2x2x2x2 but hasn't gotten around to it. He mainly
wants a good simulator to generate graphics. Feel free to contact
him yourself or simply leave suggestions in the comments of his new
videos to nudge him. At the moment Shapeways raised their 3D
printing costs by 20% which pretty much killed puzzle sales, so I
won't be ready to sell many until I get it mass produced anyway. He
has solved the puzzle but hasn't made a solution video. Of course a
mathologer video on the topic (or follow-up like he did for MC4D
solution) could be an ideal solution, and I expect that is his
intention. In short, I think this is likely to happen eventually.
Thanks for all your support,
-Melinda
cite=3D"mid:2E8A125A2DB240CD8B791727B28F8952@LAB">
-8">
=EF=BB=BF
pe">
=20=20=20=20=20=20
beautifull presentations of
considable amount of stuff about Melinda's physical 2x2x2x2.t>
worthfull to have an animated view of an interested
non-insider.
e
Mathologer?
02s"
moz-do-not-send=3D"true">https://www.youtube.com/watch?v=3DyhPH=
1369OWc&t=3D602s
re=3Dyoutu.be"
moz-do-not-send=3D"true">https://www.youtube.com/watch?v=3DDvZn=
h7-nslo&feature=3Dyoutu.be
re=3Dyoutu.be"
moz-do-not-send=3D"true">https://www.youtube.com/watch?v=3DiOla=
7WPfCvA&feature=3Dyoutu.be
BACKGROUND-COLOR: rgb(244,239,233); TEXT-INDENT: 0px; MARGIN:
0px; PADDING-LEFT: 0px; PADDING-RIGHT: 0px; FONT: 12px/18px
Arial, Helvetica, sans-serif; WHITE-SPACE: normal;
LETTER-SPACING: normal; COLOR: rgb(102,102,102); WORD-SPACING:
0px; PADDING-TOP: 0px; font-stretch: normal;
-webkit-text-stroke-width: 0px" class=3D"MsoNormal">
5px; PADDING-RIGHT: 0px; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px">
black">From: href=3D"mailto:ringuette@solarmirror.com [4D_Cubing]"
moz-do-not-send=3D"true">Marc Ringuette
ringuette@solarmirror.com [4D_Cubing]
(welcome, Okko!)
Hey, Ed, regarding Melinda's physical 2^4 puzzle,
=C2=A0 (1) Nope, I haven't written down a big table of move
equivalents, although my videos will give you enough to
figure it out.=C2=A0=C2=A0 The one-sentence version is:=C2=A0 a=
ny mc4d move
that leaves the stickers on the R+L faces still on the R+L
faces, has a simple twist in the phys 2^4 puzzle.=C2=A0=C2=A0 W=
ith the
addition of a single rotation to this common subset -- any
rotation that moves the R+L faces onto a different pair of
faces, FOro and FUro being the two phys 2^4 sequences
discovered so far to do this -- the entire mc4d state space
can be reached on the phys 2^4 and vice versa.
=C2=A0 (2) Nope, nobody else has written down the move table
either.=C2=A0 That's not too surprising, since nobody else but
Melinda and me have made physical versions of the puzzle
yet.=C2=A0=C2=A0 C'mon, gang, get building!=C2=A0 It's fun, and=
it'll only
take you a couple of days.=C2=A0 :)=C2=A0=C2=A0=C2=A0=C2=A0 I s=
uspect people are
waiting=C2=A0 for Melinda or me to provide some nice step-by-st=
ep
instructions that can be completed in 3-12 hours, or even
better, just wheedle one of us into building them one.=C2=A0=C2=
=A0
=C2=A0 (3)=C2=A0 The "virtual physical" 2^4.=C2=A0=C2=A0 It fee=
ls slightly goofy
to take Melinda's puzzle, whose key feature is the ability
for a physical realization, and then make a simulation of
it. =C2=A0 Heh.=C2=A0=C2=A0 However, it would actually be reall=
y useful,
given that a lot of people won't manage to surmount the
energy barrier associated with building the physical
puzzles.
=C2=A0(4)=C2=A0 Thinking about it, it would be really useful to=
create
a virtual physical 2^4 puzzle, in Javascript, shown
side-by-side with the MC4D style simulation, with each move
being executed simultaneously on both.=C2=A0=C2=A0 And as a
side-effect of creating the MC4d style rendering in
Javascript, we would easily be able to also produce a
web-browser-compatible version of all of MC4D's n^4
hypercube puzzles that would make them that much more
accessible to people who reluctant to use the Java app.=C2=A0=
=C2=A0
Such a version could be far less general and tricky than
MC4D itself, because it would be so much more limited (just
4d, in a particular simplified/hacked projection).=C2=A0=C2=A0 =
Sort of
a "gateway drug" to full MC4D.=C2=A0=C2=A0 It would be FANTASTI=
C if
somebody programmed this.
=C2=A0 (5)=C2=A0 A teaser:=C2=A0 I've been trying to work out a=
n extension
of Melinda's physical 2^4 puzzle into a physical 3^4, and
making some good progress.=C2=A0=C2=A0 However, with 81 pieces =
and 972
magnets, it's fairly impractical to create physically in a
way that can actually be operated by hand.=C2=A0=C2=A0 Pretty m=
uch the
only way it is likely to come to exist is in a "virtual
physical" version in Javascript.
Cheers
Marc
type=3D"cite">
=EF=BB=BF[Is there a] =
table
which gives the "phys 2^4" moves for each "one
click on mc4d" ?
type=3D"cite">
=EF=BB=BFHow about a "=
virtual
physical 2^4" ?