Thread: "Chronicles of a Rubik junkie's experience with the {5}x{5}"

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Hi fellow hypercubists,

I was sharing my experience on the "{5}x{5} 3" (Uniform Pentagonal Duoprism)
puzzle with Melinda, and she encouraged me to post an expanded version of it
here, and to describe how it felt while going through it :)

I hadn't thought I'd get to work on it Wednesday night since I had other
plans, but when those ended earlier than expected I gave it a start - the
impromptu beginning ended up a marathon session that still hadn't ended by
5am the next morning! I used to do all night solves like this when I first
joined the group, and it was fun to fully regress into circa 2000 obsessive
mode.

There were a number of interesting new scenarios in this puzzle. The
duoprisms are neat because they are approximately two dual interlocking tori
in 4-space (this is more obvious on the uniform duoprisms, where both "tori"
are based on the same polygon). From my start on the {5}x{4} and the
problems I had been encountering there, I knew it would be best to first
solve the rings of 2C pieces along the two main circles of revolution of the
tori. This did turn out to be the right decision because I could correct
"parity problems" (which I did encounter) earlier without having to worry
about the other 2C pieces. With the two rings done, the remaining 2C pieces
went smoothly afterward. For those, I worked my way around one torus, then
around the other.

3C pieces went smoothly as well, and this was the most enjoyable part for
me. I got in the groove (even had a beer :)), and just picked them off one
by one. I was saving my log file every 6 pieces (not sure why), which was
about 15 minutes intervals, and finished these around midnight. I ran into
another parity problem at the end of the 3C pieces, but luckily was able to
find a sequence for it pretty quickly. So I was excitedly off to the
corners feeling like I was making rapid progress...

I had trouble finding a sequence which cycled only 3 corners (and in fact
didn't do so till the next morning), but I did find a sequence that swapped
two sets of two corners. It was pretty easy to get into a rhythm with it,
and I was feeling confident to finish the whole puzzle by 2am. But the
sequence became more awkward as more pieces were solved, especially towards
the end. And with 4 pieces left, those troubles escalated. I couldn't get
these rebels to swap with correct positions or orientations. I was able to
reduce to 3 unsolved pieces, but still couldn't find a sequence for that.
Also, I could see the parity problems in these 3 (two swapped and one
disoriented, instead of a 3-cycle). The tiredness was growing, but I was on
a mission and didn't want to stop!

By 3:30am, there were just 2 pieces remaining (right positions, wrong
orientations), but by this point and during that last hour and a half,
instead of thinking about how to solve these two pieces, I was tired enough
that I was literally just guessing sequences on my practice puzzle. I got a
rush of adrenaline a couple times when I thought I was there, but each time
it didn't go quite right. Rush, disappointment, rush, despondence...

At 5am, the final pieces had given me enough strain that I knew I needed to
get some sleep first to have any hope with it, and as difficult as it was I
pulled myself away from the computer. Besides exhaustion, I think what I
was feeling at this point was a sense of urgency to have all this work and
loss of sleep count as a first! (as silly as that is) Sarah legitimately
called me a coocoo for staying up so ridiculously late, and my mind had
enough inertia going that I couldn't immediately turn it off - good thing is
while trying to fall asleep I had the idea that ended up working.

The sleep did the trick, and I was able to find the right sequence in about
15 minutes around 10am with a set of fresher eyes. Finishing felt like
solving the 4^3 for the first time! My performance at work definitely
suffered a bit on Thursday, but I think in this case it was worth it :)

As an aside, it seemed I hit every possible parity problem along the way,
and it made me wonder if the statistical chances of this were higher than on
the 4^4. I also wonder if parity problems are more prevalent on odd uniform
duoprisms. The "{4}x{4} 3", aka the 4^3, certainly doesn't do these kinds
of things. Would a {6}x{6}?

Well, I hope you enjoyed the story, and that it wasn't too long.

Take Care,
Roice

P.S. If anyone is interested in the particular sequences, just let me know.
Maybe we need a "sequence library" for each puzzle in the wiki?

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Hi fellow hypercubists,

I was sharing my experience on the "{5}=
x{5} 3" (Uniform Pentagonal Duoprism) puzzle with Melinda, and she enc=
ouraged me to post an expanded version of it here, and to describe how it f=
elt while going through it :)


I hadn't thought I'd get to work on it Wednesday night since I =
had other plans, but when those ended earlier than expected I gave it a sta=
rt - the impromptu beginning ended up a marathon session that still hadn=
9;t ended by 5am the next morning! =A0I used to do all night solves like th=
is when I first joined the group, and it was fun to fully regress into circ=
a 2000 obsessive mode.


There were a number of interesting new scenarios in this puzzle.=A0 The=
duoprisms are neat because they are approximately two dual interlocking to=
ri in 4-space (this is more obvious on the uniform duoprisms,=A0where both =
"tori" are based on the same polygon).=A0 From my start on the {5=
}x{4} and the problems I had been encountering there, I knew it would be be=
st to first solve the rings of 2C pieces along the two main circles of revo=
lution of the tori.=A0 This did turn out to be the right decision because I=
could correct "parity problems" (which I did encounter) earlier =
without having to worry about the other 2C pieces.=A0 With the two rings do=
ne, the remaining 2C pieces went smoothly afterward. =A0For those, I worked=
my way around one torus, then around the other.


3C pieces went smoothly as well, and this was the most enjoyable part f=
or me.=A0 I got in the groove (even had a beer :)), and just picked them of=
f one by one.=A0 I was saving my log file every 6 pieces (not sure why), wh=
ich was about 15 minutes intervals, and finished these around midnight.=A0 =
I ran into another parity problem at the end of the 3C pieces, but luckily =
was able to find a sequence for it pretty quickly. =A0So I was excitedly of=
f to the corners feeling like I was making rapid progress...

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Because I like responding to myself...

As an aside, it seemed I hit every possible parity problem along the way,
> and it made me wonder if the statistical chances of this were higher than on
> the 4^4. I also wonder if parity problems are more prevalent on odd uniform
> duoprisms. The "{4}x{4} 3", aka the 4^3, certainly doesn't do these kinds
> of things. Would a {6}x{6}?
>

I wanted to be a little more clear on this. The issues I ran into with the
3C and 4C pieces did not require undoing previous work (I could find
sequences to solve the pieces). So I am perhaps using the term "parity
problem" too loosely, but what I meant is that when 2 or 3 pieces were left,
their positions/orientations were in states that were very strange looking
compared to the 4^3.

On the 2C pieces, I think you can get yourself into more a genuine "parity
problem" if you don't solve the 2C pieces along the rings first. And by
this I mean, you'd have to undo previous work to fix the issue...

All the best,
Roice

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Because I like responding to myself...

And yes, I loved your description of your experience. Thanks so much for
writing it up. I'm sure that everyone else appreciates it as much as I
do. :D

-Melinda

Roice Nelson wrote:
> Hi fellow hypercubists,
>
> I was sharing my experience on the "{5}x{5} 3" (Uniform Pentagonal Duoprism)
> puzzle with Melinda, and she encouraged me to post an expanded version of it
> here, and to describe how it felt while going through it :)
>
> I hadn't thought I'd get to work on it Wednesday night since I had other
> plans, but when those ended earlier than expected I gave it a start - the
> impromptu beginning ended up a marathon session that still hadn't ended by
> 5am the next morning! I used to do all night solves like this when I first
> joined the group, and it was fun to fully regress into circa 2000 obsessive
> mode.
>
> There were a number of interesting new scenarios in this puzzle. The
> duoprisms are neat because they are approximately two dual interlocking tori
> in 4-space (this is more obvious on the uniform duoprisms, where both "tori"
> are based on the same polygon). From my start on the {5}x{4} and the
> problems I had been encountering there, I knew it would be best to first
> solve the rings of 2C pieces along the two main circles of revolution of the
> tori. This did turn out to be the right decision because I could correct
> "parity problems" (which I did encounter) earlier without having to worry
> about the other 2C pieces. With the two rings done, the remaining 2C pieces
> went smoothly afterward. For those, I worked my way around one torus, then
> around the other.
>
> 3C pieces went smoothly as well, and this was the most enjoyable part for
> me. I got in the groove (even had a beer :)), and just picked them off one
> by one. I was saving my log file every 6 pieces (not sure why), which was
> about 15 minutes intervals, and finished these around midnight. I ran into
> another parity problem at the end of the 3C pieces, but luckily was able to
> find a sequence for it pretty quickly. So I was excitedly off to the
> corners feeling like I was making rapid progress...
>
> I had trouble finding a sequence which cycled only 3 corners (and in fact
> didn't do so till the next morning), but I did find a sequence that swapped
> two sets of two corners. It was pretty easy to get into a rhythm with it,
> and I was feeling confident to finish the whole puzzle by 2am. But the
> sequence became more awkward as more pieces were solved, especially towards
> the end. And with 4 pieces left, those troubles escalated. I couldn't get
> these rebels to swap with correct positions or orientations. I was able to
> reduce to 3 unsolved pieces, but still couldn't find a sequence for that.
> Also, I could see the parity problems in these 3 (two swapped and one
> disoriented, instead of a 3-cycle). The tiredness was growing, but I was on
> a mission and didn't want to stop!
>
> By 3:30am, there were just 2 pieces remaining (right positions, wrong
> orientations), but by this point and during that last hour and a half,
> instead of thinking about how to solve these two pieces, I was tired enough
> that I was literally just guessing sequences on my practice puzzle. I got a
> rush of adrenaline a couple times when I thought I was there, but each time
> it didn't go quite right. Rush, disappointment, rush, despondence...
>
> At 5am, the final pieces had given me enough strain that I knew I needed to
> get some sleep first to have any hope with it, and as difficult as it was I
> pulled myself away from the computer. Besides exhaustion, I think what I
> was feeling at this point was a sense of urgency to have all this work and
> loss of sleep count as a first! (as silly as that is) Sarah legitimately
> called me a coocoo for staying up so ridiculously late, and my mind had
> enough inertia going that I couldn't immediately turn it off - good thing is
> while trying to fall asleep I had the idea that ended up working.
>
> The sleep did the trick, and I was able to find the right sequence in about
> 15 minutes around 10am with a set of fresher eyes. Finishing felt like
> solving the 4^3 for the first time! My performance at work definitely
> suffered a bit on Thursday, but I think in this case it was worth it :)
>
> As an aside, it seemed I hit every possible parity problem along the way,
> and it made me wonder if the statistical chances of this were higher than on
> the 4^4. I also wonder if parity problems are more prevalent on odd uniform
> duoprisms. The "{4}x{4} 3", aka the 4^3, certainly doesn't do these kinds
> of things. Would a {6}x{6}?
>
> Well, I hope you enjoyed the story, and that it wasn't too long.
>
> Take Care,
> Roice
>
> P.S. If anyone is interested in the particular sequences, just let me know.
> Maybe we need a "sequence library" for each puzzle in the wiki?
>
>

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Hello Roice, and congrats on solving the {5}x{5} 3, and for sharing your
story with us!

I took inspiration from the fact that last night, when you uploaded your
solution to {5}x{5} 3, that there was still puzzles other than the {5}x{4}
which I failed to finish first, and as such, was able to start a second
puzzle. I was quite fascinated by how the uniform duoprisms worked,
especially how whereas the {5}x{4} has multiple kinds of pieces based on
which faces they are touching (like there are 3c pieces that touch two 5 and
one 4 block, and 3c pieces that touch one 5 and two 4 blocks, each requiring
different sequences), the uniform duoprisms only have one kind of piece
since the two torii that make up the duoprism are formed by the same kinds
of blocks (fun trying to 'visualize' it as two interlocking torii :D). This
meant that I would only need algorithms for 2c(within torus), 2c(between
torus), 3c, 4c.

So I decided to try to solve {6}x{6}, it being the next largest uniform
duoprism. From my experience with the {5}x{4}, I was able to rather quickly
solve all 2c pieces without macros, and for 3c and 4c, was able to rather
quickly find new algorithms. Basically, all I ended up needing was a
3-cycle for 3c pieces, and a 3-cycle for 4c pieces. Also, along the way you
notice that the colors of each torus, stay on it's respective torus. So if
you have a white face on one torus, you won't find a white sticker on the
other torus. Helps to keep this in mind when you are working away. You are
able to solve all kinds of parity situations with just these moves by
careful usage of conjugation. For instance, if on one face you have all 3c
in place except you need to swap two, you can bring down one from an
unsolved layer, 3 cycle it into the face you're working on (my 3-cycle macro
only does swaps in a plane, but conjugation can make it do almost anything),
then put that piece back up into the face it came from but in an adjacent
position, then take the piece you want to bring back down, and pull it
down. The result being you do two pair swaps, one in the face you're
working on, one in the other face you don't care about. For orientation,
you can do a similar thing, but by commuting with a twist of one of the
surrounding torii's faces (it temporarily messes up a couple 2c(within
torus) pieces, but the commutation fixes that right up). It really helps to
also have scrap paper to use and carefully keep track of where you move
pieces and whatnot when you are trying to find the proper conjugations
needed, but after a while you can start to see the bigger picture and do
these fixes on the go.

Interestingly, the exact same methodology applied for the {5}x{4} puzzle
too, only I needed separate algorithms for the two kinds of 3c pieces each.
And upon solving the {5}x{4} 3 and {6}x{6} 3, I have the feeling that the
same algorithms can be adapted to solve {n}x{n} 3 for any size duoprism of
length 3, it would just obviously take more time. So to answer your
question Roice, I think that while these are definately parity cases we run
into, since they can be solved by 3-cycles and conjugation for both the
uniform and non-uniform duoprisms I solved, they are probably a prevalent
feature of all duoprisms. But unlike {4}x{4} 3, I never ran into the case
where a single 4c piece needs to be flipped. It might be that I was lucky,
but the whole puzzle just felt like complicated parity cases like that are
non-existent. Again, I could be wrong, but that's just what I felt (hard to
draw accurate results from a single trial though :D). As for 4 length
puzzles... the parity possibilities with that scare me!

Another note, Roice, I two typically find my double swap 4c macro first
also, but it's fairly simple to take that and make it into a straight-up
3-cycle by putting it into a commutator I discovered.

In conclusion, most of these length 3 puzzles I feel can be solved fully
once you isolate a 3-cycle for each of the corresponding pieces. I had more
algorithms for when I did {5}x{4} 3, but if I did it again I'd probably do
it similar to how I solved the {6}x{6} 3 and cut down on my algorithms
greatly (I had one realllly crazy 174 move algorithm just to do a 3-cycle of
one kind of 3c pieces, but it can probably be greatly shortened if I start
from scratch :D).

Anyway, hopefully my train of thought put into text makes sense ^^

Chris

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Hello Roice, and congrats on solving the {5}x{5} 3, and for sharing your st=
ory with us!

I took inspiration from the fact that last night, when =
you uploaded your solution to {5}x{5} 3, that there was still puzzles other=
than the {5}x{4} which I failed to finish first, and as such, was able to =
start a second puzzle.=C2=A0 I was quite fascinated by how the uniform duop=
risms worked, especially how whereas the {5}x{4} has multiple kinds of piec=
es based on which faces they are touching (like there are 3c pieces that to=
uch two 5 and one 4 block, and 3c pieces that touch one 5 and two 4 blocks,=
each requiring different sequences), the uniform duoprisms only have one k=
ind of piece since the two torii that make up the duoprism are formed by th=
e same kinds of blocks (fun trying to 'visualize' it as two interlo=
cking torii :D).=C2=A0 This meant that I would only need algorithms for 2c(=
within torus), 2c(between torus), 3c, 4c.


So I decided to try to solve {6}x{6}, it being the next largest uniform=
duoprism.=C2=A0 From my experience with the {5}x{4}, I was able to rather =
quickly solve all 2c pieces without macros, and for 3c and 4c, was able to =
rather quickly find new algorithms.=C2=A0 Basically, all I ended up needing=
was a 3-cycle for 3c pieces, and a 3-cycle for 4c pieces.=C2=A0 Also, alon=
g the way you notice that the colors of each torus, stay on it's respec=
tive torus.=C2=A0 So if you have a white face on one torus, you won't f=
ind a white sticker on the other torus.=C2=A0 Helps to keep this in mind wh=
en you are working away.=C2=A0 You are able to solve all kinds of parity si=
tuations with just these moves by careful usage of conjugation.=C2=A0 For i=
nstance, if on one face you have all 3c in place except you need to swap tw=
o, you can bring down one from an unsolved layer, 3 cycle it into the face =
you're working on (my 3-cycle macro only does swaps in a plane, but con=
jugation can make it do almost anything), then put that piece back up into =
the face it came from but in an adjacent position, then take the piece you =
want to bring back down, and pull it down.=C2=A0 The result being you do tw=
o pair swaps, one in the face you're working on, one in the other face =
you don't care about.=C2=A0 For orientation, you can do a similar thing=
, but by commuting with a twist of one of the surrounding torii's faces=
(it temporarily messes up a couple 2c(within torus) pieces, but the commut=
ation fixes that right up).=C2=A0 It really helps to also have scrap paper =
to use and carefully keep track of where you move pieces and whatnot when y=
ou are trying to find the proper conjugations needed, but after a while you=
can start to see the bigger picture and do these fixes on the go.


Interestingly, the exact same methodology applied for the {5}x{4} puzzl=
e too, only I needed separate algorithms for the two kinds of 3c pieces eac=
h.=C2=A0 And upon solving the {5}x{4} 3 and {6}x{6} 3, I have the feeling t=
hat the same algorithms can be adapted to solve {n}x{n} 3 for any size duop=
rism of length 3, it would just obviously take more time.=C2=A0 So to answe=
r your question Roice, I think that while these are definately parity cases=
we run into, since they can be solved by 3-cycles and conjugation for both=
the uniform and non-uniform duoprisms I solved, they are probably a preval=
ent feature of all duoprisms.=C2=A0 But unlike {4}x{4} 3, I never ran into =
the case where a single 4c piece needs to be flipped.=C2=A0 It might be tha=
t I was lucky, but the whole puzzle just felt like complicated parity cases=
like that are non-existent.=C2=A0 Again, I could be wrong, but that's =
just what I felt (hard to draw accurate results from a single trial though =
:D).=C2=A0 As for 4 length puzzles... the parity possibilities with that sc=
are me!


Another note, Roice, I two typically find my double swap 4c macro first=
also, but it's fairly simple to take that and make it into a straight-=
up 3-cycle by putting it into a commutator I discovered.

In conclusi=
on, most of these length 3 puzzles I feel can be solved fully once you isol=
ate a 3-cycle for each of the corresponding pieces.=C2=A0 I had more algori=
thms for when I did {5}x{4} 3, but if I did it again I'd probably do it=
similar to how I solved the {6}x{6} 3 and cut down on my algorithms greatl=
y (I had one realllly crazy 174 move algorithm just to do a 3-cycle of one =
kind of 3c pieces, but it can probably be greatly shortened if I start from=
scratch :D).


Anyway, hopefully my train of thought put into text makes sense ^^
<=
br>Chris


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Chris Locke wrote:
> Hello Roice, and congrats on solving the {5}x{5} 3, and for sharing your
> story with us!
>
> I took inspiration from the fact that last night, when you uploaded your
> solution to {5}x{5} 3, that there was still puzzles other than the {5}x{4}
> which I failed to finish first, and as such, was able to start a second
> puzzle. I was quite fascinated by how the uniform duoprisms worked,
> especially how whereas the {5}x{4} has multiple kinds of pieces based on
> which faces they are touching (like there are 3c pieces that touch two 5 and
> one 4 block, and 3c pieces that touch one 5 and two 4 blocks, each requiring
> different sequences), the uniform duoprisms only have one kind of piece
> since the two torii that make up the duoprism are formed by the same kinds
> of blocks (fun trying to 'visualize' it as two interlocking torii :D). This
> meant that I would only need algorithms for 2c(within torus), 2c(between
> torus), 3c, 4c.
>

I hadn't noticed it before but you're totally right that there are two
different kinds of 3c pieces! That means that sometimes we may need to
be clear on which type of 3c piece we're talking about. That's very cool
and I bet you're right that this will be why the non-uniform duoprisms
will be harder to solve than the uniform ones.

> So I decided to try to solve {6}x{6}, it being the next largest uniform
> duoprism. From my experience with the {5}x{4}, I was able to rather quickly
> solve all 2c pieces without macros, and for 3c and 4c, was able to rather
> quickly find new algorithms. Basically, all I ended up needing was a
> 3-cycle for 3c pieces, and a 3-cycle for 4c pieces. Also, along the way you
> notice that the colors of each torus, stay on it's respective torus. So if
> you have a white face on one torus, you won't find a white sticker on the
> other torus.
>

Yes, I'd noticed that when I was reimplementing scrambling. I just
happened to end up with mostly yellow/red colors on one torus and
blue/green colors on the other. Then when I performed a full scramble, I
ended up with two beautifully speckled toruses, one in each color
scheme. [See attached screen shot.] At first I was sure that I simply
had a bug in my scrambling algorithm. It took me a while before I
understood what you just pointed out.

Here's a 12-color specification in case anybody wants to reproduce this.
Just be sure to unwrap the formatting and put this all on one line in
your facecolors.txt file:
255,0,204 153,0,153 255,51,0 255,102,102 255,153,0 255,255,0 102,153,0
0,255,0 0,204,153 0,255,255 0,0,255
102,102,255

> Helps to keep this in mind when you are working away. You are
> able to solve all kinds of parity situations with just these moves by
> careful usage of conjugation. For instance, if on one face you have all 3c
> in place except you need to swap two, you can bring down one from an
> unsolved layer, 3 cycle it into the face you're working on (my 3-cycle macro
> only does swaps in a plane, but conjugation can make it do almost anything),
> then put that piece back up into the face it came from but in an adjacent
> position, then take the piece you want to bring back down, and pull it
> down. The result being you do two pair swaps, one in the face you're
> working on, one in the other face you don't care about. For orientation,
> you can do a similar thing, but by commuting with a twist of one of the
> surrounding torii's faces (it temporarily messes up a couple 2c(within
> torus) pieces, but the commutation fixes that right up). It really helps to
> also have scrap paper to use and carefully keep track of where you move
> pieces and whatnot when you are trying to find the proper conjugations
> needed, but after a while you can start to see the bigger picture and do
> these fixes on the go.
>
> Interestingly, the exact same methodology applied for the {5}x{4} puzzle
> too, only I needed separate algorithms for the two kinds of 3c pieces each.
> And upon solving the {5}x{4} 3 and {6}x{6} 3, I have the feeling that the
> same algorithms can be adapted to solve {n}x{n} 3 for any size duoprism of
> length 3, it would just obviously take more time. So to answer your
> question Roice, I think that while these are definately parity cases we run
> into, since they can be solved by 3-cycles and conjugation for both the
> uniform and non-uniform duoprisms I solved, they are probably a prevalent
> feature of all duoprisms. But unlike {4}x{4} 3, I never ran into the case
> where a single 4c piece needs to be flipped. It might be that I was lucky,
> but the whole puzzle just felt like complicated parity cases like that are
> non-existent. Again, I could be wrong, but that's just what I felt (hard to
> draw accurate results from a single trial though :D). As for 4 length
> puzzles... the parity possibilities with that scare me!
>

This suggests to me that one of the next big prizes up for grabs by
those patient cubists among you who like big puzzles will be the {5}x{5}
5. It has the same sort of notational appeal as the 5^5, and the parity
issues it brings should make it suitably frightening in that dimension
as well. To the couple of masochist among you who have been asking us to
implement 6^4 and larger: even though you now finally have those cubes
available, I suggest that you focus on this plumb new prize first! ;-)

> Another note, Roice, I two typically find my double swap 4c macro first
> also, but it's fairly simple to take that and make it into a straight-up
> 3-cycle by putting it into a commutator I discovered.
>
> In conclusion, most of these length 3 puzzles I feel can be solved fully
> once you isolate a 3-cycle for each of the corresponding pieces. I had more
> algorithms for when I did {5}x{4} 3, but if I did it again I'd probably do
> it similar to how I solved the {6}x{6} 3 and cut down on my algorithms
> greatly (I had one realllly crazy 174 move algorithm just to do a 3-cycle of
> one kind of 3c pieces, but it can probably be greatly shortened if I start
> from scratch :D).
>
> Anyway, hopefully my train of thought put into text makes sense ^^

The thing that I really like is that even though I'm a better coder than
a cubist, I found that that actually did make sense to me! Either I'm
learning, or you're a good writer. Probably both!

Thanks for the report and the instruction. Oh, and a huge
congratulations on snagging the first {6}x{6} 3 solution, Chris!!
-Melinda

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My thanks to Chris for the cool insights about the duoprisms! And thanks t=
o
Melinda for the picture, dramatically showing the way stickers are slaved t=
o
their respective tori. I hadn't mentally put some of this together as
cleanly as in these emails, even after solving the {5}x{5}-3. Great stuff
:D

One thing that was neat for me was to think about while reading was how the
{4}x{4} fit into it all. Since it has only one type of 2C piece, there
seems to be a succession of symmetry breaking that happens with the
progression of puzzles - {4}x{4} is more symmetrical than general {n}x{n},
which in turn is more symmetrical than general {n}x{m}.

In some ways, it's hard to say which puzzles are more difficult. The
{4}x{4} might take less sequences, but it's stickers can move amongst both
tori, so the scrambling result is more complicated (it wouldn't produce a
neat picture of dual tori like the {6}x{6} did for Melinda). But even if
there is somewhat of a conservation-of-difficulty effect with the tradeoff
between number of piece types and scrambling capacity, I guess it still
feels like difficulty rises more quickly as the duoprisms become more
general. Chris could say better of course.

As an aside: the {3}x{3} (which we aren't officially supporting yet because
we're still considering how the twisting works on it) also only has 1 type
of 2C piece, but it is degenerate when it comes to 2C pieces around the
"rings". And so unlike the flexible {4}x{4}, it's pieces are again slaved
to their respective tori. I always knew the 4^3 was special :)

Chris, congrats to you as well on all the firsts your snagging :) And much
thanks for all the issue updates and feedback along the way!

All the best,
Roice

P.S. I agree with Melinda about the {5}x{5}-5 being a big prize. Not sure
I'll ever tackle it, but that had been my favorite from very early on.


On Sat, Oct 31, 2009 at 12:42 AM, Melinda Green w=
rote:

>
> [Attachment(s) <#124a91e52cd6e093_TopText> from Melinda Green included
> below]
>
> Chris Locke wrote:
> > Hello Roice, and congrats on solving the {5}x{5} 3, and for sharing you=
r
> > story with us!
> >
> > I took inspiration from the fact that last night, when you uploaded you=
r
> > solution to {5}x{5} 3, that there was still puzzles other than the
> {5}x{4}
> > which I failed to finish first, and as such, was able to start a second
> > puzzle. I was quite fascinated by how the uniform duoprisms worked,
> > especially how whereas the {5}x{4} has multiple kinds of pieces based o=
n
> > which faces they are touching (like there are 3c pieces that touch two =
5
> and
> > one 4 block, and 3c pieces that touch one 5 and two 4 blocks, each
> requiring
> > different sequences), the uniform duoprisms only have one kind of piece
> > since the two torii that make up the duoprism are formed by the same
> kinds
> > of blocks (fun trying to 'visualize' it as two interlocking torii :D).
> This
> > meant that I would only need algorithms for 2c(within torus), 2c(betwee=
n
> > torus), 3c, 4c.
> >
>
> I hadn't noticed it before but you're totally right that there are two
> different kinds of 3c pieces! That means that sometimes we may need to
> be clear on which type of 3c piece we're talking about. That's very cool
> and I bet you're right that this will be why the non-uniform duoprisms
> will be harder to solve than the uniform ones.
>
> > So I decided to try to solve {6}x{6}, it being the next largest uniform
> > duoprism. From my experience with the {5}x{4}, I was able to rather
> quickly
> > solve all 2c pieces without macros, and for 3c and 4c, was able to rath=
er
> > quickly find new algorithms. Basically, all I ended up needing was a
> > 3-cycle for 3c pieces, and a 3-cycle for 4c pieces. Also, along the way
> you
> > notice that the colors of each torus, stay on it's respective torus. So
> if
> > you have a white face on one torus, you won't find a white sticker on t=
he
> > other torus.
> >
>
> Yes, I'd noticed that when I was reimplementing scrambling. I just
> happened to end up with mostly yellow/red colors on one torus and
> blue/green colors on the other. Then when I performed a full scramble, I
> ended up with two beautifully speckled toruses, one in each color
> scheme. [See attached screen shot.] At first I was sure that I simply
> had a bug in my scrambling algorithm. It took me a while before I
> understood what you just pointed out.
>
> Here's a 12-color specification in case anybody wants to reproduce this.
> Just be sure to unwrap the formatting and put this all on one line in
> your facecolors.txt file:
> 255,0,204 153,0,153 255,51,0 255,102,102 255,153,0 255,255,0 102,153,0
> 0,255,0 0,204,153 0,255,255 0,0,255
> 102,102,255
>
> > Helps to keep this in mind when you are working away. You are
> > able to solve all kinds of parity situations with just these moves by
> > careful usage of conjugation. For instance, if on one face you have all
> 3c
> > in place except you need to swap two, you can bring down one from an
> > unsolved layer, 3 cycle it into the face you're working on (my 3-cycle
> macro
> > only does swaps in a plane, but conjugation can make it do almost
> anything),
> > then put that piece back up into the face it came from but in an adjace=
nt
> > position, then take the piece you want to bring back down, and pull it
> > down. The result being you do two pair swaps, one in the face you're
> > working on, one in the other face you don't care about. For orientation=
,
> > you can do a similar thing, but by commuting with a twist of one of the
> > surrounding torii's faces (it temporarily messes up a couple 2c(within
> > torus) pieces, but the commutation fixes that right up). It really help=
s
> to
> > also have scrap paper to use and carefully keep track of where you move
> > pieces and whatnot when you are trying to find the proper conjugations
> > needed, but after a while you can start to see the bigger picture and d=
o
> > these fixes on the go.
> >
> > Interestingly, the exact same methodology applied for the {5}x{4} puzzl=
e
> > too, only I needed separate algorithms for the two kinds of 3c pieces
> each.
> > And upon solving the {5}x{4} 3 and {6}x{6} 3, I have the feeling that t=
he
> > same algorithms can be adapted to solve {n}x{n} 3 for any size duoprism
> of
> > length 3, it would just obviously take more time. So to answer your
> > question Roice, I think that while these are definately parity cases we
> run
> > into, since they can be solved by 3-cycles and conjugation for both the
> > uniform and non-uniform duoprisms I solved, they are probably a prevale=
nt
> > feature of all duoprisms. But unlike {4}x{4} 3, I never ran into the ca=
se
> > where a single 4c piece needs to be flipped. It might be that I was
> lucky,
> > but the whole puzzle just felt like complicated parity cases like that
> are
> > non-existent. Again, I could be wrong, but that's just what I felt (har=
d
> to
> > draw accurate results from a single trial though :D). As for 4 length
> > puzzles... the parity possibilities with that scare me!
> >
>
> This suggests to me that one of the next big prizes up for grabs by
> those patient cubists among you who like big puzzles will be the {5}x{5}
> 5. It has the same sort of notational appeal as the 5^5, and the parity
> issues it brings should make it suitably frightening in that dimension
> as well. To the couple of masochist among you who have been asking us to
> implement 6^4 and larger: even though you now finally have those cubes
> available, I suggest that you focus on this plumb new prize first! ;-)
>
> > Another note, Roice, I two typically find my double swap 4c macro first
> > also, but it's fairly simple to take that and make it into a straight-u=
p
> > 3-cycle by putting it into a commutator I discovered.
> >
> > In conclusion, most of these length 3 puzzles I feel can be solved full=
y
> > once you isolate a 3-cycle for each of the corresponding pieces. I had
> more
> > algorithms for when I did {5}x{4} 3, but if I did it again I'd probably
> do
> > it similar to how I solved the {6}x{6} 3 and cut down on my algorithms
> > greatly (I had one realllly crazy 174 move algorithm just to do a 3-cyc=
le
> of
> > one kind of 3c pieces, but it can probably be greatly shortened if I
> start
> > from scratch :D).
> >
> > Anyway, hopefully my train of thought put into text makes sense ^^
>
> The thing that I really like is that even though I'm a better coder than
> a cubist, I found that that actually did make sense to me! Either I'm
> learning, or you're a good writer. Probably both!
>
> Thanks for the report and the instruction. Oh, and a huge
> congratulations on snagging the first {6}x{6} 3 solution, Chris!!
> -Melinda
>=20=20
>

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And yeah, it seems that 4^3 is unique in this regard. What about naming
conventions though? How do you decide what to use to notate the standard
hypercube, {4}x{4} or {4,3,3}? I'm not sure if there are other degeneracies
in naming conventions, so it's possible this is the only case like this.


Yeah, the multiple names for a single object seems it may be confusing at
times. My personal thought on this was that I'd use {4}x{4} or {4,3,3} or
3^4 as needed depending on the context. I was favoring the first in my last
email since the discussion was all centered around duoprisms. I also see I
mistyped the 3^4 as 4^3 at one point. Oops, hopefully the mistake was
obvious and my intended meaning clear to those reading.

I have a question too: how do we decide which puzzles will be recorded on
the wiki record page? Since it's a wiki, are we going to allow any puzzle?
I only ask because I realized that there are an infinity of duoprisms
possible to solve by using the "invent my own" feature, limited only by the
solver's patience (and possibly computer specs :P). Couple this with the
fact that as you go higher, I don't think the puzzles get fundamentally more
difficult, just much longer. I suspect that the algorithms can be
generalized for the duoprisms of higher order, much like how if you can
solve 3^4, 4^4, and 5^4, you should be able to adapt the algorithms to solve
any n^4 hypercube with enough time. Nevertheless, it is still quite a feat
to be able to solve these higher order puzzles, and I would not be opposed
to allowing such higher order records to stand.


My vote is to go ahead and add these to the wiki, and after seeing your
{6}x{6}, I did wonder if we should perhaps try to officially support uniform
duoprism up to {10}x{10}, and by that I mean show them in the UI (entered
now as issue 80
if anyone
wants to weigh in). In any case, I think it is safe to say the
puzzle engine can safely handle any {n}x{m} with n and m between 4 and 10
right now, which is a lot of firsts for people to grab.

Also, will the next release allow all possible twists for length 2 puzzles?
What will happen to the records of these puzzles set with the more limited
twists? And yes, I realize both of these questions put 3 of my 4 firsts on
thin ice, which was actual part of my motivation for going for the
dodecahedral prism this weekend, since that should be a permanent one :D.


This is trickier, and I doubt we'll do it for 4.0. We did enter the thought
to perhaps mark the current puzzles in a special way to note their
limitation in the mean time (see issue
72).
That way, if the puzzles are ever enhanced in the future, we could at least
be clear on what solutions were done with what puzzle behaviors. If anyone
has opinions on this issue, we'd love your feedback as well. The
fundamental problem here stems from the fact that length-2 puzzles don't
have enough stickers to represent all the grips of the parent polytope. On
the 2^4, this was handled by making where you click on the sticker do
different things (each sticker maps to multiple grips instead of just one).
This has caused a number of complications in the coding though, and for
these reasons Melinda and I are not inclined to further propagate this
design to the other puzzles, at least for now (we have even been considering
removing the special 2^4 behavior, though it will likely be good to keep
around for legacy reasons).

I will say that the unsupported triangle puzzles are more dangerous to
consider stable at this point. While your {6}x{6} is fully safe as a first,
I'm not confident that there won't be meaningful changes in triangle puzzles
like the {3}x{3}, so people might take that in consideration in regards to
what you tackle with "invent my own". However, I encourage you to play with
triangle puzzles and give us feedback on the behavior! (If we are pointed
to specific problems, it will be easier to look at).

Anyway, those are some of the thoughts that have been floating around...

Take Care,
Roice

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border: none; padding: 0px;">And yeah, it seems that 4^3 is unique in this=
regard.=A0 What about naming conventions though?=A0 How do you decide what=
to use to notate the standard hypercube, {4}x{4} or {4,3,3}?=A0 I'm no=
t sure if there are other degeneracies in naming conventions, so it's p=
ossible this is the only case like this.

It was pretty easy for me to notice the torus locking stickers for me
because of my custom color scheme.  For instance, here is the longest list I
have for 14 colors:

black, white, red, green, blue, yellow, magenta, teal, grey, brown, pink,
turquoise, dark green, peach

So you can see, the first torus was basically all primary/secondary colors,
and the second torus was all the more complicated tertiary colors, so the
pattern REALLY jumped out at me.  The reason for this, is that unlike the
3^4 case, where every facet can be turned by a quarter turn in any axis, the
duoprisms are limited by geometry to only allow quarter turns within the
respective torus.  And yeah, it seems that 4^3 is unique in this regard.
What about naming conventions though?  How do you decide what to use to
notate the standard hypercube, {4}x{4} or {4,3,3}?  I'm not sure if there
are other degeneracies in naming conventions, so it's possible this is the
only case like this.
  

I have a question too: how do we decide which puzzles will be recorded on
the wiki record page?  Since it's a wiki, are we going to allow any puzzle?
I only ask because I realized that there are an infinity of duoprisms
possible to solve by using the "invent my own" feature, limited only by the
solver's patience (and possibly computer specs :P).  Couple this with the
fact that as you go higher, I don't think the puzzles get fundamentally more
difficult, just much longer.  I suspect that the algorithms can be
generalized for the duoprisms of higher order, much like how if you can
solve 3^4, 4^4, and 5^4, you should be able to adapt the algorithms to solve
any n^4 hypercube with enough time.  Nevertheless, it is still quite a feat
to be able to solve these higher order puzzles, and I would not be opposed
to allowing such higher order records to stand.  

Also, will the next release
allow all possible twists for length 2 puzzles?  What will happen to the
records of these puzzles set with the more limited twists?  

And yes, I
realize both of these questions put 3 of my 4 firsts on thin ice, which was
actual part of my motivation for going for the dodecahedral prism this
weekend, since that should be a permanent one :D.  I need a break for a bit
now from all that cubing over the weekend (^_^')
  

And yeah, the {5}x{5} 5 is one monster.  If I still had my notes from when I
solved 5^4, I might be able to adapt the algorithms to solve some of the
interior pieces, but upon inspection, there are also MANY kinds of interior
pieces, all which will probably need separate, and again probably unique,
algorithms to solve.  Beastly indeed.
  

2009/11/2 Roice Nelson <roice3@gmail.com>