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Hey guys,
I'm avoiding doing my taxes and so I've had some interesting investigations
into MC4D checkerboard patterns I thought I'd share :) I was curious about
how the uncommon checkerboards on the 3D cube extended to four dimensions.
If you aren't familiar with those, it is possible to make a checkerboard
that has two 3-cycles of colors instead of three 2-cycles (the latter being
the most familiar and most easily produced pattern that exchanges opposite
colors). As far as I know, there is no easy sequence to make the 3-cycle
checkerboard, so you have to manually place all the pieces. There is also a
checkerboard pattern on the 3D cube with one 6-cycle! You can quickly make
it if you already have the 3-cycle checkerboard by then applying the
sequence of moves that normally makes the 2-cycle checkerboard from the
pristine state. While the 6-cycle is a "superposition" of a 2-cycle and a
3-cycle, these two seem to be sort of basic checkerboard patterns for the 3D
puzzle in that neither can be created by superpositions of the other. Also
of note in 3D is that you can do a single 2-cycle that only checkerboards 2
opposite faces and leaves the remaining 4 faces solid. Puzzle states that
are superpositions of this valid state are also valid (in fact, maybe it is
better to consider this a basic unit with which to develop more complicated
checkerboards over the 'three 2-cycle pattern' since the latter is just 3 of
these single guys).
With that background I wondered what types of checkerboard cycles can be
produced on MC4D and what the lowest level patterns are that can be
superimposed to make the more complicated ones. I figured some of these
might only be be creatable by manually placing pieces like in the 3D case,
which would be a lot of work, so I decided to edit log files by hand and use
Don's cool solve feature in MC4D to check whether certain puzzle states were
valid. Much of what I found was surprising and against what I might have
guessed. Here is a rundown...
*Full Checkerboards:*
puzzle states that were possible:
four 2-cycles of opposite colors (standard checkerboard like the ones in the
hall of fame)
four 2-cycles of adjacent colors
one 6-cycle and one 2-cycle
two orthogonal 4-cycles (the cycle "direction" of the first did not force a
direction of the second cycle. I can describe more about this interesting
case if anyone wants me to.)
puzzle states that were not possible:
two 3-cycles and one 2-cycle (various arrangements tested)
four 2-cycles (2 opposite, 2 adjacent)
one 8-cycle
*Partial Checkerboards:*
puzzle states that were possible:
two 2-cycles of opposite colors (4 solid faces)
two 3-cycles (2 solid faces)
puzzle states that were not possible:
one 2-cycle! (contrast this with 3D case)
two 2-cycles of adjacent colors
one 4-cycle (whether through 2 sets of opposite faces or 4 adjacent faces or
a combination)
one 4-cycle and one 2-cycle
three 2-cycles (whether opposite cycles or not)
...
Without going too far into the last category, it was clear that
superpositions of the possible partial checkerboards led to the possible
full checkerboards I found. Superpositions of impossible partial states
could lead to valid states, but didn't necessarily do so. I also noticed
one basic unit could produce all the valid full checkerboards I found. This
was the partial checkerboard with two 3-cycles (the valid 2-cycle partial
checkerboard can be made by superimposing that one, yet another contrast to
the 3D case). Since this was a trial and error approach vs. an
enumeration proof, I certainly may have missed some valid checkerboards, but
this seems like it might be it. I wondered (but doubted) if any of these
less common checkerboards could have shorter solutions than the current
record.
I found it interesting that it was not possible to do a full checkerboard
pattern having 3-cycles. I tried this in various ways (having the 2-cycle
that would go along with it be of opposite faces or adjacent faces and
playing with the twirl direction of the two 3-cycles).
I put files of the possible full checkerboards at
www.gravitation3d.com/mc4d/checkerboards if anyone wants to see what they
look like.
Take care all,
Roice
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puzzle states that were not possible:
puzzle states that were not possible:
heh, I should have known to do that. I promise I am trainable.&nbs= Anyway, here ya go! www.gravita=
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heh, I should have known to do that. I promise I am trainable. It just
takes a long time. (Sarah can vouch that both of those statements are true
:))
Anyway, here ya go!
www.gravitation3d.com/mc4d/checkerboards/
On Fri, Apr 11, 2008 at 1:21 AM, Melinda Green
wrote:
> Screenshots!!
>
> Roice Nelson wrote:
>
> > Hey guys,
> > I'm avoiding doing my taxes and so I've had some interesting
> > investigations into MC4D checkerboard patterns I thought I'd share :) I was
> > curious about how the uncommon checkerboards on the 3D cube extended to four
> > dimensions. If you aren't familiar with those, it is possible to make a
> > checkerboard that has two 3-cycles of colors instead of three 2-cycles (the
> > latter being the most familiar and most easily produced pattern that
> > exchanges opposite colors). As far as I know, there is no easy sequence to
> > make the 3-cycle checkerboard, so you have to manually place all the pieces.
> > There is also a checkerboard pattern on the 3D cube with one 6-cycle! You
> > can quickly make it if you already have the 3-cycle checkerboard by then
> > applying the sequence of moves that normally makes the 2-cycle checkerboard
> > from the pristine state. While the 6-cycle is a "superposition" of a
> > 2-cycle and a 3-cycle, these two seem to be sort of basic checkerboard
> > patterns for the 3D puzzle in that neither can be created by superpositions
> > of the other. Also of note in 3D is that you can do a single 2-cycle that
> > only checkerboards 2 opposite faces and leaves the remaining 4 faces solid.
> > Puzzle states that are superpositions of this valid state are also valid
> > (in fact, maybe it is better to consider this a basic unit with which to
> > develop more complicated checkerboards over the 'three 2-cycle pattern'
> > since the latter is just 3 of these single guys).
> > With that background I wondered what types of checkerboard cycles can
> > be produced on MC4D and what the lowest level patterns are that can be
> > superimposed to make the more complicated ones. I figured some of these
> > might only be be creatable by manually placing pieces like in the 3D case,
> > which would be a lot of work, so I decided to edit log files by hand and use
> > Don's cool solve feature in MC4D to check whether certain puzzle states were
> > valid. Much of what I found was surprising and against what I might have
> > guessed. Here is a rundown...
> > *Full Checkerboards:*
> > puzzle states that were possible:
> > four 2-cycles of opposite colors (standard checkerboard like the ones
> > in the hall of fame)
> > four 2-cycles of adjacent colors
> > one 6-cycle and one 2-cycle
> > two orthogonal 4-cycles (the cycle "direction" of the first did not
> > force a direction of the second cycle. I can describe more about this
> > interesting case if anyone wants me to.)
> >
> > puzzle states that were not possible:
> >
> > two 3-cycles and one 2-cycle (various arrangements tested)
> > four 2-cycles (2 opposite, 2 adjacent)
> > one 8-cycle
> > *Partial Checkerboards:*
> > puzzle states that were possible:
> > two 2-cycles of opposite colors (4 solid faces)
> > two 3-cycles (2 solid faces)
> >
> > puzzle states that were not possible:
> >
> > one 2-cycle! (contrast this with 3D case)
> > two 2-cycles of adjacent colors
> > one 4-cycle (whether through 2 sets of opposite faces or 4 adjacent
> > faces or a combination)
> > one 4-cycle and one 2-cycle
> > three 2-cycles (whether opposite cycles or not)
> > ...
> > Without going too far into the last category, it was clear that
> > superpositions of the possible partial checkerboards led to the possible
> > full checkerboards I found. Superpositions of impossible partial states
> > could lead to valid states, but didn't necessarily do so. I also noticed
> > one basic unit could produce all the valid full checkerboards I found. This
> > was the partial checkerboard with two 3-cycles (the valid 2-cycle partial
> > checkerboard can be made by superimposing that one, yet another contrast to
> > the 3D case). Since this was a trial and error approach vs. an enumeration
> > proof, I certainly may have missed some valid checkerboards, but this seems
> > like it might be it. I wondered (but doubted) if any of these less common
> > checkerboards could have shorter solutions than the current record.
> > I found it interesting that it was not possible to do a full
> > checkerboard pattern having 3-cycles. I tried this in various ways (having
> > the 2-cycle that would go along with it be of opposite faces or adjacent
> > faces and playing with the twirl direction of the two 3-cycles).
> > I put files of the possible full checkerboards at
> > www.gravitation3d.com/mc4d/checkerboards <
> > http://www.gravitation3d.com/mc4d/checkerboards> if anyone wants to see
> > what they look like.
> > Take care all,
> > Roice
>
>
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p; It just takes a long time. (Sarah can vouch that both of those sta=
tements are true :))
tion3d.com/mc4d/checkerboards/
lt;melinda@superliminal.com=
> wrote:px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid">Screenshots!!
Roice Nelso=
n wrote:BORDER-LEFT: #ccc 1px solid">
I'm avoiding doing my taxes an=
d so I've had some interesting investigations into MC4D checkerboard pa=
tterns I thought I'd share :) I was curious about how the uncommo=
n checkerboards on the 3D cube extended to four dimensions. If you ar=
en't familiar with those, it is possible to make a checkerboard that ha=
s two 3-cycles of colors instead of three 2-cycles (the latter being the mo=
st familiar and most easily produced pattern that exchanges opposite colors=
). As far as I know, there is no easy sequence to make the 3-cycle ch=
eckerboard, so you have to manually place all the pieces. There is al=
so a checkerboard pattern on the 3D cube with one 6-cycle! You can qu=
ickly make it if you already have the 3-cycle checkerboard by then applying=
the sequence of moves that normally makes the 2-cycle checkerboard from th=
e pristine state. While the 6-cycle is a "superposition" of=
a 2-cycle and a 3-cycle, these two seem to be sort of basic checkerboard p=
atterns for the 3D puzzle in that neither can be created by superpositions =
of the other. Also of note in 3D is that you can do a single 2-cycle =
that only checkerboards 2 opposite faces and leaves the remaining 4 faces s=
olid. Puzzle states that are superpositions of this valid state are a=
lso valid (in fact, maybe it is better to consider this a basic unit with w=
hich to develop more complicated checkerboards over the 'three 2-cycle =
pattern' since the latter is just 3 of these single guys).
With that background I wondered what types of checkerboard cycles can=
be produced on MC4D and what the lowest level patterns are that can be sup=
erimposed to make the more complicated ones. I figured some of these =
might only be be creatable by manually placing pieces like in the 3D case, =
which would be a lot of work, so I decided to edit log files by hand and us=
e Don's cool solve feature in MC4D to check whether certain puzzle stat=
es were valid. Much of what I found was surprising and against what I=
might have guessed. Here is a rundown...
*Full Checkerboards:*
puzzle states that were possible:
&=
nbsp;four 2-cycles of opposite colors (standard checkerboard like the ones =
in the hall of fame)
four 2-cycles of adjacent colors
one 6-cycle and=
one 2-cycle
two orthogonal 4-cycles (the cycle "direction" of the first did n=
ot force a direction of the second cycle. I can describe more about this in=
teresting case if anyone wants me to.)
puzzle states that were not p=
ossible:
two 3-cycles and one 2-cycle (various arrangements tested)
four 2-cy=
cles (2 opposite, 2 adjacent)
one 8-cycle
*Partial Checkerboard=
s:*
puzzle states that were possible:
two 2-cycles of opp=
osite colors (4 solid faces)
two 3-cycles (2 solid faces)
puzzle states that were not possible:
one 2-cycle! (contrast this with 3D case)
two 2-cycles of adjacent=
colors
one 4-cycle (whether through 2 sets of opposite faces or 4 adjac=
ent faces or a combination)
one 4-cycle and one 2-cycle
three 2-cycles (whether opposite cycles or n=
ot)
...
Without going too far into the last category, it was cl=
ear that superpositions of the possible partial checkerboards led to the po=
ssible full checkerboards I found. Superpositions of impossible parti=
al states could lead to valid states, but didn't necessarily do so. &nb=
sp;I also noticed one basic unit could produce all the valid full checkerbo=
ards I found. This was the partial checkerboard with two 3-cycles (th=
e valid 2-cycle partial checkerboard can be made by superimposing that one,=
yet another contrast to the 3D case). Since this was a trial and err=
or approach vs. an enumeration proof, I certainly may have missed some vali=
d checkerboards, but this seems like it might be it. I wondered (but =
doubted) if any of these less common checkerboards could have shorter solut=
ions than the current record.
I found it interesting that it was not possible to do a full checkerb=
oard pattern having 3-cycles. I tried this in various ways (having th=
e 2-cycle that would go along with it be of opposite faces or adjacent face=
s and playing with the twirl direction of the two 3-cycles).
gravitation3d.com/mc4d/checkerboards <ion3d.com/mc4d/checkerboards" target=3D"_blank">http://www.gravitation3d.co=
m/mc4d/checkerboards> if anyone wants to see what they look like.
Take care all,
Roice
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