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 Post subject: Rainbow Nautilus SolutionPosted: Wed Jan 09, 2013 4:26 pm

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany
Hi Puzzle Friends,
Had someone solved the Rainbow Nautilus ?

To understand this puzzle I developed a notation. Perhaps it makes no sense , to give the turns with angles. Similar to the first step of solving the cube shape on Square-1, I turned a layer until the next cutting line appears.
The angles are 20Â° 30Â° 40Â° 50Â° 60Â° 70Â° and 90Â°

I work not with angles. I work with cutting lines.
u+ means turn until the next cutting line appears.
u- means turn until the previous line appears. (u or (u+) is the inverse from u-)

u++ until the second cutting line appears.
u+++ the 3th cutting line.

u is short for u+

l= turn left half puzzle 180 degrees.
r= turn right half puzzle 180 degrees.

The pieces in the middle layer are easy to solve, similar to square-1.

I tried some sequences:
l u r u-
3 times = invert the order of top layer
7 times = invert top and bottom pieces
135 times = startposition.

sequence 2)
r u r u- r u r u r u++

x 8 = invert order of upper pieces
x 10= inverted order of up and down
x 15= startposition

r u r u r u++ r u-- r u- r u-

x 9 = startposition
The sequence are depend of the configuration of pieces.

I have no idea to solve this puzzle. Memorize many possible positions is not a good funny way , I think.

I tried this sequences with a little simple java program, sources are included, too.

Some ideas for this puzzle ?

Cheers,
Andrea

PS: It's possible to paste the sequences in the simulator. The 'return' key repeats the last input. The simulator shows no middle layer.
Perhaps this is my first unsolveable puzzle.

 Attachments: nautilus.jpg [ 26.77 KiB | Viewed 7645 times ] rnautilus.zip [5.63 KiB] Downloaded 87 times
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 Post subject: Re: Rainbow Nautilus SolutionPosted: Wed Jan 09, 2013 4:57 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Hi Andrea, I think of the Rainbow Nautilus as a heavily bandaged dihedral puzzle. Even more irregular and more bandaged than the Square-1. That said, the total number of reachable states should be very small relative to most other twisty puzzles.

Have you tried programming and iterative deepening algorithm to enumerate all states and God's number for each state?

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Wed Jan 09, 2013 5:24 pm

Joined: Fri Feb 18, 2000 8:50 am
Location: chicago, IL area U.S.A
Yeah, after playing with this just a very little bit, I realized early on that there are barely any legal possibilities compared to most puzzles. I would imagine it a program could probably come up with every possibility and solution in not much time.

I like the notation, I'm not sure if any other would make any sense. But yeah, if the Rubik's cube is to the Meffert's bandaged cube, then the Square 1 is to the Rainbow Nautilus.

-d

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Wed Jan 09, 2013 5:57 pm

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
I have not tried it yet, but...
katsmom wrote on her Blog
Roxanne Wong wrote:
It's a bandaged puzzle. The inner layer though has only one place that can turn. It's like a square one, but it doesn't change positions of the pieces as much as the square 1 does. It also reminds me of morph. It's actually a harder puzzle to mess up than to solve. When you do manage to mess it up, it doesn't take a lot to solve it. It can be done by intuition.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Wed Jan 09, 2013 6:11 pm

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
Hi Andrea,

I agree with Rox's statement that it's harder to scramble than to solve. I only found one way to separate all pieces in a full scramble. This or it's mirror:
Attachment:

Nautilus Scramble.jpg [ 537.77 KiB | Viewed 7569 times ]
To solve it, you just need to start putting pieces back next to each other and it all falls into place. I'd expect the possible permutations to be very low.

Cheers,
Burgo.

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PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Wed Jan 09, 2013 7:21 pm

Joined: Sat Sep 15, 2012 7:42 am
Thank you for making the simulator, Andrea!

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Named 3x3x3 Bandaging Patterns

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Thu Jan 10, 2013 12:29 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
EDIT: This post is incorrect. I had a bug in my program. See below.

Hi Andrea,

I've done some testing trying to enumerate all of the states of this puzzle and determine "god's algorithm" for it. It has more than 10M states so I've had to move the calculation to a machine with more memory.

From what I can tell from pictures of the puzzle, identically colored pieces are chiral. You can tell the different between the top red and bottom red (and top and bottom of all of the other colors). This has a big impact on the number of states of the puzzle. I think you should mark the top pieces differently than the bottom pieces so that distinguishable states on the physical puzzle are distinguishable on your simulator.

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Last edited by Brandon Enright on Fri Jan 11, 2013 1:53 pm, edited 1 time in total.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Thu Jan 10, 2013 12:27 pm

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany
Hi Brandon,

Quote:
I think you should mark the top pieces differently than the bottom pieces so that distinguishable states on the physical puzzle are distinguishable on your simulator.

Yes. I had done it. I added white circles to the bottom pieces of startposition.

For exploring the puzzle this is a good idea. But for calculating it multiplies the positions with the factor 256.

I was able to code the whole puzzle into a 64 bit longword, inclusive the differentation of same colored pieces.
To find a strategy with a program is ok. But it should be possible to solve it without the computer.

Memorize 3-5 positions and memorize 3-5 sequences was an optimal way.

I add the new executable jar file and source.

 Attachments: nautilus.jpg [ 35.74 KiB | Viewed 7417 times ] rnautilus1.1.zip [6.07 KiB] Downloaded 75 times
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 Post subject: Re: Rainbow Nautilus SolutionPosted: Thu Jan 10, 2013 3:53 pm

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
Andrea wrote:
...
u+ means turn until the next cutting line appears.
u- means turn until the previous line appears. (u or (u+) is the inverse from u-)

u++ until the second cutting line appears.
u+++ the 3th cutting line.

u is short for u+
....
Thank you Andrea for the simulator!
I just gave it a first try and I'm afraid I do not understand how you define the "next" or "previous" cutting line.
Do you define it independent of the direction (i. e. clockwise or counter clockwise) of the turn (i.e. "next" means the smallest angle of a u or d turn to get to a position where an l or r turn can be done.)
Or do you turn the layer always counter clockwise for u+ and clockwise for u-? (This is the case in the start position after "reset")

[German] i.e. steht fÃ¼r lateinisch "id est" = "das heiÃŸt" [/German]

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Thu Jan 10, 2013 6:19 pm

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany

Quote:
Do you define it independent of the direction (i. e. clockwise or counter clockwise) of the turn (i.e. "next" means the smallest angle of a u or d turn to get to a position where an l or r turn can be done.)

The cutting line is vertical. The direction of angles is in mathematics counterclockwise. U+ means the next possible line to turn r or l.
U- is clockwise.

In this definition it's always possible to turn each layer.
If you se a u+ you turn the u layer counterclockwise until a vertical line appears.

Cheers,
Andrea

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 1:20 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
This post was wrong. I have a bug in my program. I'll edit it with less-wrong (possibly even correct!) results later.

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 4:25 am

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
Thanks Andrea!
Have you ever tried to solve it just by intuition? (Rox and Burgo suggest that it is not so hard this way.)
I guess that a "method" is harder to describe in the Nautilus case than just following your intuition. (And I suppose you are very good at this! )

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 4:35 am

Joined: Mon Feb 28, 2011 4:54 am
Thanks Andrea!
Have you ever tried to solve it just by intuition? (Rox and Burgo suggest that it is not so hard this way.)
I guess that a "method" is harder to describe in the Nautilus case than just following your intuition. (And I suppose you are very good at this! )

I have tried by intuition. I'd like to comment.

Burgo's "not so hard" doesn't count, because nothing's hard for him Rox? I'm not sure whether she's scrambled it properly. She may well have but as she hasn't posted here yet, I don't know.

By "scrambled properly" I'd mean as in Burgo's picture above. I had it "mostly" scrambled a few times and solved it without too much trouble (no real method, just fiddling), but as soon as I've had it scrambled like Burgo's picture, it has morphed into a hair-pulling impossible puzzle. I love Burgo's comment as to how "you just put the pieces back next to each other". Ha!

This is my method so far.

1. Make a triplet of the orange-yellow-green. once done, everything else is simple. (I can't do this for the fully scrambled puzzle).

2. Make sets of 90* groups (eg. orange-violet). Once done I figure I should be able to move each group around by 90* rather than only by 180* turns (or "lucky" other cuts). I haven't been able to make more than 6 such groups in my fully scrambled solve.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 6:32 am

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany
Hi Konrad, Burgo, rline, Brandon and all other,

Have you ever tried to solve it just by intuition?

I tried it with the simulator, without success. But I scrambled it very good.
I want not turn my Rainbow nautilus. Its very loose, and I want not disassemble it , to explore sequences.

The scrambling is like a tree. In some depth is very less scrambling, if only one cutting line exist. It's only a 180Â° turn possible. But sometimes there are 2 or more cutting lines. In this cases the tree expands.

I get the same position with different sequences.
This:

(r u r u- ) x 4
Attachment:

symmetrie.png [ 24.38 KiB | Viewed 7218 times ]

My (actual idea) not solve the startposition ! Solve one of this key positions.

This position has symmetry, so it appears more often.
This is the same strategy on square-1, make the cube shape.

Solve this :
Attachment:

square-1.png [ 57.56 KiB | Viewed 7218 times ]

Perhaps this idea is a startpoint for a good solution method.

Cheers,
Andrea

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 6:57 am

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
I'm pretty sure there's only one way to completely scramble it and separate all the pieces (with mirrors etc).. and there is a path `in and out`.

I believe there are a few loops within the path `in and out` (the path from solved to fully scrambled and back).. but not many.. so rline may be caught in a loop, that you have to go around and take another path out (or the same way back and a new way foreward), but the only strategy I had was to put pieces that go together `together` (or separate them). Of course there was: (R2 U2)x3 to flip an equatorial layer piece, but that's trivial for us.

I made some pictures of a scramble:
Attachment:

Nautilus scramble.jpg [ 2.15 MiB | Viewed 7202 times ]
The critical part from what I can see is making a 90* cut available so you can split the 2 large red pieces (and join them on the way back). Everything else just relies on using a second available cut.

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 7:09 am

Joined: Mon Feb 28, 2011 4:54 am
Well after my earlier rambling, I just solved it. Came on here and saw Burgo's post.

My new strategy:

1. Get the two red pieces together and stored out of the way.
2. Join an orange-yellow-green triplet.
3. Everything else just "falls into place".

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Tutorials: Pitcher Octo-Star Cube | GERANIUM | Wheel of Wisdom

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 8:01 am

Joined: Sat Jun 09, 2012 10:56 am
rline wrote:
My new strategy:

1. Get the two red pieces together and stored out of the way.
2. Join an orange-yellow-green triplet.
3. Everything else just "falls into place".

Fantastic, using your tips I just solved it really quickly, I had been going round in circles before now.
Thankyou.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 11:53 am

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
Andrea wrote:
...I want not turn my Rainbow nautilus. Its very loose, and I want not disassemble it , to explore sequences....
Hi Andrea and all other Captain Nemos ,
it is the same for me, my time schedule doesn't allow that I'm getting lost with an unsolvable puzzle, right now. I know myself too well, if I scramble it, I'll be a very unsociable person until I have solved it.

So, I do not risk a scramble currently. Maybe, your simulator helps when I have finished some other projects.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 1:52 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
I wrote a program to enumerate all states and find "god's algorithm" for the Rainbow Nautilus. For a while I had a subtle bug that made me think the puzzle had a lot more states than I initially anticipated.

First, if you treat rotations of the top and bottom layers as free and all the same state then this puzzle has 978 states. The middle layer has 3 states. If you ignore the middle layer then the puzzle has 326 states. Each of these 326 states appears with each of the 3 middle layer states so 326 * 3 = 978.

There are actually only 326 possible configurations of the top and bottom layers and they are not independent. For each configuration of the top layer, the bottom layer is fully defined and forced into a single state.

Counting only "flips" where you swap pieces between the top and bottom layers as a move, then there are 32 states that require 13 flips. The table is:

Code:
Depth     # of States
0         1
1         32
2         66
3         232
4         106
5         152
6         78
7         124
8         28
9         32
10        16
11        48
12        32
13        32

You can get the full table of states at http://www.brandonenright.net/~bmenrigh/rbow_flip_states.txt.

Since there are so few states I'm going to look into other ways of viewing the data.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 3:56 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
I made a graph representing all of the states of the Rainbow Nautilus where the edges represent the minimal path length from state to state to reach any particular position.

The node numbers are in "top - middle - bottom" format where the top and bottom number represent which of the 326 possible states it is in. Since each top and bottom state always appear together the numbers are linked together, making the bottom number redundant.
Attachment:

rbow_state_graph.png [ 1.71 MiB | Viewed 7054 times ]

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 6:47 pm

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
Hi Brandon,

Is it possible for you to highlight the path to the fully scrambled state in my picture? And I'm curious if you found another `fully scrambled state`?

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 8:04 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Burgo wrote:
Hi Brandon,

Is it possible for you to highlight the path to the fully scrambled state in my picture? And I'm curious if you found another `fully scrambled state`?

There are a handful of them. 8/32 are full scrambles at depth 12 and fully 16/32 at depth 13.

Code:
Full scrambles at depth 12:
[B]{ooooo}{ll}[GGG]{}[YYYY]{b}{yyyy}[RRRRRRR][OOOOO][YYYY][GGG][LL][B][][LL][RRRRRRR]{ggg}{rrrrrrr}[OOOOO][]
[B]{ggg}[RRRRRRR][LL][YYYY]{b}{yyyy}[RRRRRRR][OOOOO][YYYY][GGG][LL][B][][GGG]{ll}{ooooo}{rrrrrrr}[OOOOO][]{}
[GGG]{}[][OOOOO]{rrrrrrr}{ooooo}{ll}[RRRRRRR][OOOOO][YYYY][GGG][LL][B][][B]{yyyy}{b}[YYYY][LL][RRRRRRR]{ggg}
[LL][][OOOOO]{rrrrrrr}{ggg}[RRRRRRR][RRRRRRR][OOOOO][YYYY][GGG][LL][B][][B]{yyyy}{b}[YYYY]{}[GGG]{ll}{ooooo}
[B]{yyyy}{ll}{rrrrrrr}[GGG]{b}[YYYY][RRRRRRR][OOOOO][YYYY][GGG][LL][B][][LL]{ggg}[]{}{ooooo}[RRRRRRR][OOOOO]
[B]{yyyy}[]{ggg}[LL][OOOOO]{b}[YYYY][RRRRRRR][OOOOO][YYYY][GGG][LL][B][][GGG]{rrrrrrr}{ll}{}{ooooo}[RRRRRRR]
[GGG][RRRRRRR]{ooooo}{}{ll}{rrrrrrr}[RRRRRRR][OOOOO][YYYY][GGG][LL][B][][B][YYYY]{b}[OOOOO][LL]{ggg}[]{yyyy}
[LL][OOOOO][RRRRRRR]{ooooo}{}[]{ggg}[RRRRRRR][OOOOO][YYYY][GGG][LL][B][][B][YYYY]{b}[GGG]{rrrrrrr}{ll}{yyyy}

Full scrambles at depth 13:
[B]{ooooo}{ll}[GGG]{}[YYYY]{b}{yyyy}[RRRRRRR][GGG][YYYY][OOOOO][LL][B][][LL][RRRRRRR]{ggg}{rrrrrrr}[OOOOO][]
[B]{ggg}[RRRRRRR][LL][YYYY]{b}{yyyy}[RRRRRRR][GGG][YYYY][OOOOO][LL][B][][GGG]{ll}{ooooo}{rrrrrrr}[OOOOO][]{}
[B]{ggg}[RRRRRRR][LL][YYYY]{b}{yyyy}[RRRRRRR][OOOOO][B][LL][GGG][YYYY][][GGG]{ll}{ooooo}{rrrrrrr}[OOOOO][]{}
[B]{ooooo}{ll}[GGG]{}[YYYY]{b}{yyyy}[RRRRRRR][OOOOO][B][LL][GGG][YYYY][][LL][RRRRRRR]{ggg}{rrrrrrr}[OOOOO][]
[GGG]{}[][OOOOO]{rrrrrrr}{ooooo}{ll}[RRRRRRR][GGG][YYYY][OOOOO][LL][B][][B]{yyyy}{b}[YYYY][LL][RRRRRRR]{ggg}
[LL][][OOOOO]{rrrrrrr}{ggg}[RRRRRRR][RRRRRRR][GGG][YYYY][OOOOO][LL][B][][B]{yyyy}{b}[YYYY]{}[GGG]{ll}{ooooo}
[LL][][OOOOO]{rrrrrrr}{ggg}[RRRRRRR][RRRRRRR][OOOOO][B][LL][GGG][YYYY][][B]{yyyy}{b}[YYYY]{}[GGG]{ll}{ooooo}
[GGG]{}[][OOOOO]{rrrrrrr}{ooooo}{ll}[RRRRRRR][OOOOO][B][LL][GGG][YYYY][][B]{yyyy}{b}[YYYY][LL][RRRRRRR]{ggg}
[B]{yyyy}{ll}{rrrrrrr}[GGG]{b}[YYYY][RRRRRRR][GGG][YYYY][OOOOO][LL][B][][LL]{ggg}[]{}{ooooo}[RRRRRRR][OOOOO]
[B]{yyyy}[]{ggg}[LL][OOOOO]{b}[YYYY][RRRRRRR][GGG][YYYY][OOOOO][LL][B][][GGG]{rrrrrrr}{ll}{}{ooooo}[RRRRRRR]
[B]{yyyy}{ll}{rrrrrrr}[GGG]{b}[YYYY][RRRRRRR][OOOOO][B][LL][GGG][YYYY][][LL]{ggg}[]{}{ooooo}[RRRRRRR][OOOOO]
[B]{yyyy}[]{ggg}[LL][OOOOO]{b}[YYYY][RRRRRRR][OOOOO][B][LL][GGG][YYYY][][GGG]{rrrrrrr}{ll}{}{ooooo}[RRRRRRR]
[GGG][RRRRRRR]{ooooo}{}{ll}{rrrrrrr}[RRRRRRR][GGG][YYYY][OOOOO][LL][B][][B][YYYY]{b}[OOOOO][LL]{ggg}[]{yyyy}
[LL][OOOOO][RRRRRRR]{ooooo}{}[]{ggg}[RRRRRRR][GGG][YYYY][OOOOO][LL][B][][B][YYYY]{b}[GGG]{rrrrrrr}{ll}{yyyy}
[GGG][RRRRRRR]{ooooo}{}{ll}{rrrrrrr}[RRRRRRR][OOOOO][B][LL][GGG][YYYY][][B][YYYY]{b}[OOOOO][LL]{ggg}[]{yyyy}
[LL][OOOOO][RRRRRRR]{ooooo}{}[]{ggg}[RRRRRRR][OOOOO][B][LL][GGG][YYYY][][B][YYYY]{b}[GGG]{rrrrrrr}{ll}{yyyy}

The path highlighting is a bit hard because I generated the graph "forward" from solved to scrambled. Instead I'd have to build the whole graph in memory so that I have the reverse paths to track back to the solved state.

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 8:27 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Burgo wrote:
Is it possible for you to highlight the path to the fully scrambled state in my picture?

I decided to generate another graph that shows the depth better.

The solved state is red. There is a bluish progression from depth 1 to 9.
Depth 10 is yellow, 11 is green, 12 is blue, and 13 is purple. The 24 "fully scrambled" states are orange.
Attachment:

rbow_state_depth_graph.png [ 2.11 MiB | Viewed 7002 times ]

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 11, 2013 10:44 pm

Joined: Tue Feb 08, 2011 3:17 am
Location: Australia
Hi Brandon,

These 2 groups must be mirrored groups. The same type of mirroring must be relevant to the rest of the states. Taking mirrors into account (like excluding equatorial layers): Does this reduce the reachable states to only 163?
Attachment:

rbow_state_depth_graph.png [ 2.12 MiB | Viewed 6974 times ]

Do you have the physical puzzle? Because when I get to that `fully scrambled state`, one twist can make another `fully scrambled state (here):
Attachment:

Untitled-1.jpg [ 228.84 KiB | Viewed 6974 times ]
.. does this condition fit the graph? (your graph seems to me to suggest that all `fully scrambled states` are `at the end of the road`).

I'm interested to see the other `area` of fully scrambled states (I wonder if they look identical or mirrored).. there must be a divergent path about 3 twists away? Anyone got any photos?

It's clear that there is only one path `in and out` of these states.. nice job on making the chart visually understandable. Interesting that greater depth is not necessarily `more scrambled` in the traditional sense.

Cheers,
Burgo.

_________________
1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sat Jan 12, 2013 12:20 am

Joined: Mon Feb 28, 2011 4:54 am
SunnWinder wrote:
rline wrote:
My new strategy:

1. Get the two red pieces together and stored out of the way.
2. Join an orange-yellow-green triplet.
3. Everything else just "falls into place".

Fantastic, using your tips I just solved it really quickly, I had been going round in circles before now.
Thankyou.

Hey, thanks Sunnwinder!

For anyone who wants to see this "method" I made a video. It contains 3 solves from different scrambles. The last solve is of the full scramble in Burgo's first picture above.

I titled it "How to Solve the Rainbow Nautilus Puzzle" but it really should be "How I solve it...". Maybe from the video you'll see my general technique, although it's about as far from a rigorous method as any video I've made. it's not a rigorous how to take it from any state to any other, but rather a view to how I go about solving it.

Also, there's at least one time where I picked up later that I'd made something I was trying to make but didn't spot it during the recording (if that makes sense). I'd ask your forbearance with this. For anyone who has made videos for public viewing without multiple cuts and takes, you'll know that sometimes with the concentration involved on one particular thing, you just don't see other things that seem quite obvious later on.

Anyway, hope it's useful.

How to Solve the Rainbow Nautilus Puzzle

or

http://twistypuzzling.blogspot.com/2013 ... uzzle.html (Blog)

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Last edited by rline on Sat Jan 12, 2013 1:59 am, edited 2 times in total.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sat Jan 12, 2013 12:27 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Well there you had to go and make a bunch more work for me

So you made a very good point about mirroring so I decided to eliminate states in steps. To eliminate states, as I'm doing iterative deepening, the first "essentially unique" state I hit becomes the only allowed copy of that state. I will add more and more "essentially unique" checks to eliminate more and more states.

First, as I said before, each state comes in three copies, one for each middle state. Here is the graph where I've only included one copy of each state when ignoring the middle. There are 326 states left:
Attachment:

rbow_state_sym_no_middle.png [ 591.94 KiB | Viewed 6933 times ]

Now as you pointed out, there can be "mirroring". One such mirroring is reversing the order of every piece on a side. Instead of red left of orange you'd have orange left of red. After adding this symmetry there are 163 states left:
Attachment:

rbow_state_sym_no_middle_no_mirror.png [ 247.2 KiB | Viewed 6933 times ]

But the graph still looks like it has a mirror symmetry. In fact, it does. If you swap all piece colors (right chiral red becomes left chiral red, etc.) then the states are further reduced to 93:
Attachment:

rbow_state_sym_no_middle_no_mirror_transpose.png [ 127.17 KiB | Viewed 6933 times ]

This means there are 23 "pieces not interchangable" states and then 70 pieces interchangeable. By only retaining one copy of the interchanged piece states there are only 93 essentially different states in the puzzle.

The reason why you don't see an arrow between the two "fully scrambled" states is that I'm only drawing arrows for the shortest path from solved to each state. It may be that many of these states are connected to each other in different ways but if those ways aren't the shortest path then they aren't shown.

The two unique "full scramble" states are:
[B]{ggg}[RRRRRRR][LL][YYYY]{b}{yyyy}[RRRRRRR][OOOOO][YYYY][GGG][LL][B][][GGG]{ll}{ooooo}{rrrrrrr}[OOOOO][]{}
and
[B]{ooooo}{ll}[GGG]{}[YYYY]{b}{yyyy}[RRRRRRR][OOOOO][YYYY][GGG][LL][B][][LL][RRRRRRR]{ggg}{rrrrrrr}[OOOOO][]

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sat Jan 12, 2013 6:51 am

Joined: Sun Nov 23, 2008 2:18 am
Given the small number of states, am I correct to assume that calculating this puzzle's Devil's Algorythm is possible in a reasonable amount of time. Though, with the extreme bandaging of this puzzle, is it possible to find a sequence that visits every state once and only once or would visiting every state require visiting some states multiple times?

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sat Jan 12, 2013 3:11 pm

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
bmenrigh wrote:
...The two unique "full scramble" states are:
[B]{ggg}[RRRRRRR][LL][YYYY]{b}{yyyy}[RRRRRRR][OOOOO][YYYY][GGG][LL][B][][GGG]{ll}{ooooo}{rrrrrrr}[OOOOO][]{}
and
[B]{ooooo}{ll}[GGG]{}[YYYY]{b}{yyyy}[RRRRRRR][OOOOO][YYYY][GGG][LL][B][][LL][RRRRRRR]{ggg}{rrrrrrr}[OOOOO][]
Am I supposed to understand this code?
I did not see an explanation above. Seems that colours could be encoded (R = Red; l = ? lilac?) The different bracket types could express chirality? It is a real puzzle to me

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sat Jan 12, 2013 4:47 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Am I supposed to understand this code?
I did not see an explanation above. Seems that colours could be encoded (R = Red; l = ? lilac?) The different bracket types could express chirality? It is a real puzzle to me

Indeed you have it. I divided each layer of the puzzle into 36 slices, each 10 degrees wide.

The starting state is:

R is the red piece and it is 90 degrees wide so there is a [ then 7 Rs and then a ] where [] and {} indicated the edges of a piece. The starting state is three layers encoded together:

[RRRRRRR][OOOOO][YYYY][GGG][LL][B][] <- TOP LAYER
[RRRRRRR][OOOOO][YYYY][GGG][LL][B][] <- MIDDLE LAYER
{rrrrrrr}{ooooo}{yyyy}{ggg}{ll}{b}{} <- BOTTOM LAYER

I used [] for the top layer pieces and {} for the bottom layer to distinguish their chirality.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sat Jan 12, 2013 4:55 pm

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
Thank you!
Aah, so L stands for "light blue". I was confused by the seven Rs

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sat Jan 12, 2013 10:06 pm

Joined: Wed Jan 14, 2009 6:37 pm
Location: The Great White North
Wow. You guys are way ahead of me. I just got my rainbow nautilus this week, and today I of course could not resist scrambling it just to see if I can solve it.

To my surprise, I found that it's not hard to solve at all. It's true what somebody has said in this thread: it's harder to scramble than to solve! Often, just fiddling around with a seemingly-scrambled state ends up in an almost-solved state. There are some states where it's easy to get stuck in a loop, but once you figure out how to exit to loop, it's actually not hard to solve.

I haven't solved it that many times yet, but I've just started playing with it today and I've already solved it at least 3-4 times from various random states. So far, the only way I've figured out to separate yellow from green leads to the same group of states, which dead-ends with one pair of blue/cyan still together. I didn't know if it was possible to separate all pieces, but the bandaging sure makes it hard to scramble!

Anyway, I didn't know the exact relative proportions of the pieces before I read this thread, so I did a little math exercise in which I made a matrix of 7 unknowns and filled it in based on which combinations of pieces add up to 180Â° (which is easy to check, because that's when a twist is permissible). Turns out I can only get 5 equations out of the 7 needed for a full solution from first principles; based on visual observation, I made another guess that blue = 2*purple, which added a 6th equation. However, I still could not get the 7th independent equation to solve the entire matrix.

Why is this interesting? It's not just for the sake of finding the angles -- I could've just looked it up, after all. But it's interesting because it shows how severely bandaged the puzzle is -- there aren't enough reachable permutations of pieces to provide enough independent equations that would allow one to solve for all 7 angles. Based on what I did manage to solve, it seems that you could easily vary two of the angles arbitrarily and still have a puzzle with exactly the same states as the Rainbow Nautilus -- there are 2 degrees of freedom due to too many unreachable piece combinations. Now, granted, I may have missed some combinations due to not having explored all possible states -- but I suspect that 5 equations may be the max because of the severe bandaging.

Anyway, just throwing this out there, in case somebody finds this interesting.

Last of all, I have to thank all the people responsible for this puzzle -- the inventor, and everyone who made it possible to produce this puzzle -- it's a really beautiful puzzle. It's definitely a keeper for my collection!

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sat Jan 12, 2013 10:27 pm

Joined: Fri Feb 18, 2000 8:50 am
Location: chicago, IL area U.S.A
Jeffery Mewtamer wrote:
Given the small number of states, am I correct to assume that calculating this puzzle's Devil's Algorythm is possible in a reasonable amount of time. Though, with the extreme bandaging of this puzzle, is it possible to find a sequence that visits every state once and only once or would visiting every state require visiting some states multiple times?

I thought about this and my gut reaction is that it's not possible due to all of the blocked sequences unlike the 1x3x3. That said, I have no proof.

-d

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sat Jan 12, 2013 10:40 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
darryl wrote:
Jeffery Mewtamer wrote:
Given the small number of states, am I correct to assume that calculating this puzzle's Devil's Algorythm is possible in a reasonable amount of time. Though, with the extreme bandaging of this puzzle, is it possible to find a sequence that visits every state once and only once or would visiting every state require visiting some states multiple times?

I thought about this and my gut reaction is that it's not possible due to all of the blocked sequences unlike the 1x3x3. That said, I have no proof.

978 states (and way more edges) is a really big graph for an NP-complete problem. Even if one existed I doubt it could be found. I really doubt a solution exists.

I could probably produce a vertex and edge list for every state but I haven't found any software for Hamiltonian cycle finding and the fastest algorithms are all more complicated than something I feel like programming.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sat Jan 12, 2013 11:00 pm

Joined: Wed Jan 14, 2009 6:37 pm
Location: The Great White North
quickfur wrote:
[...]
Anyway, I didn't know the exact relative proportions of the pieces before I read this thread, so I did a little math exercise in which I made a matrix of 7 unknowns and filled it in based on which combinations of pieces add up to 180Â° (which is easy to check, because that's when a twist is permissible). Turns out I can only get 5 equations out of the 7 needed for a full solution from first principles; based on visual observation, I made another guess that blue = 2*purple, which added a 6th equation. However, I still could not get the 7th independent equation to solve the entire matrix.

Why is this interesting? It's not just for the sake of finding the angles -- I could've just looked it up, after all. But it's interesting because it shows how severely bandaged the puzzle is -- there aren't enough reachable permutations of pieces to provide enough independent equations that would allow one to solve for all 7 angles. Based on what I did manage to solve, it seems that you could easily vary two of the angles arbitrarily and still have a puzzle with exactly the same states as the Rainbow Nautilus -- there are 2 degrees of freedom due to too many unreachable piece combinations. Now, granted, I may have missed some combinations due to not having explored all possible states -- but I suspect that 5 equations may be the max because of the severe bandaging.[...]

Actually, nevermind what I wrote, apparently I missed some combinations because the bandaging segregated some of the states so that I have to almost-solve the puzzle before I can get to more unique piece combinations. It is possible to derive the angles from first principles after all. To prevent careless mistakes, I plugged the equations into a linear solver and got 90Â°, 70Â°, 60Â°, 50Â°, 40Â°, 30Â°, and 20Â°.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sun Jan 13, 2013 12:13 pm

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany
Hi Brandon,
Brandon wrote:
Have you tried programming and iterative deepening algorithm to enumerate all states and God's number for each state?

Yes, now I had finished that program in Standard C.
(edit)

Perhaps the program has errors. I was not able to try some positions with the simulator.

Last edited by Andrea on Mon Jan 14, 2013 4:57 pm, edited 1 time in total.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sun Jan 13, 2013 1:47 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Andrea wrote:
My results are similar to yours.
I programed a "breadth first generating" tree.
It has 325 nodes / states.

Hi Andrea. Our results are similar but slightly different.

I count 326 states when you ignore the middle layer. I assume the off-by-one is the starting solved state which I'm counting and perhaps you aren't?

Also, you say each state requires at most 8 moves but I count it as 12 when you ignore the middle layer.

Here are my numbers at each depth:
Code:
Starting with 1 state at depth 0
Added 16 states at depth 1
Added 49 states at depth 2
Added 66 states at depth 3
Added 40 states at depth 4
Added 36 states at depth 5
Added 42 states at depth 6
Added 20 states at depth 7
Added 8 states at depth 8
Added 8 states at depth 9
Added 8 states at depth 10
Added 16 states at depth 11
Added 16 states at depth 12

I consider a move any turn of the top and bottom followed by a flip. That is, I'm really only counting flips.

It's hard to double-check one's own code due to biases and assumptions. You encode pieces in a different order than I do but with some work I probably could translate each of your states into one of mine to figure out where we diverge.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sun Jan 13, 2013 7:32 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Brandon, if I could trouble you, I'm wondering if you could maybe make another similar analysis for the original concept version of the puzzle, where the 90 was split into 10 and 80. I'd guess it doesn't actually change much, but I'd really like to know.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sun Jan 13, 2013 8:14 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Jared wrote:
Brandon, if I could trouble you, I'm wondering if you could maybe make another similar analysis for the original concept version of the puzzle, where the 90 was split into 10 and 80. I'd guess it doesn't actually change much, but I'd really like to know.

I thought this was a good idea and I was about to do it until I realized by code requires pieces to have the [] or {} end caps on them. I can't handle a 10 degree piece. I should have spaced out my pieces like Andrea did but since I didn't, I can't help you without re-working a lot of code.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Mon Jan 14, 2013 6:32 am

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany
Sorry, the text file has impossible cases. I deleted It.

Last edited by Andrea on Mon Jan 14, 2013 4:54 pm, edited 1 time in total.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Mon Jan 14, 2013 6:47 am

Joined: Mon Feb 28, 2011 4:54 am
If I could ask what might seem like a stupid question: I don't really understand any of the stuff in the last few posts, but can someone explain how to interpret it to get from, for example, the solved state to the stage in Burgo's first picture above, or vice versa? And am I right in thinking you're saying there's a maximum of only 8 turns to get from any state to solved?

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Mon Jan 14, 2013 7:22 am

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany
My posted nautilus textfile has errors, I think.

It is not possible to make this turns. Perhaps the program has a bug.

Sorry.

Andrea

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Mon Jan 14, 2013 10:29 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
bmenrigh wrote:
I thought this was a good idea and I was about to do it until I realized by code requires pieces to have the [] or {} end caps on them. I can't handle a 10 degree piece. I should have spaced out my pieces like Andrea did but since I didn't, I can't help you without re-working a lot of code.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Tue Jan 15, 2013 10:23 am

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany
The solution of rainbow Nautilus !
Second try. I give not up

The updated gods number is the lucky 13 !

The intention of the program was:
1) find the number of states.
2) find the gods number.
3) find the sequence of most scrambled state.

The first program was without errors. The definition of move/turn was wrong.
New definition:
Turn up face , turn left halve 180 degrees, turn down face.
The turn of zero degrees is included.
The godsnumber goes up, the state number goes down!
Number of possible position is 310. Godsnumber is 13.

The most scrambled position is the last in this table.
Attachment:
nautilus1.txt.zip [5.15 KiB]

It's possible to find it and create sequences for real puzzle or simulator.
Entry NO 309 is the most scrampled state.
Use the "find" function in the text editor to find the first entry (connector) with the 309.
This is the line with the entry no 298.
It's possible to continue until you arrive enry 0 = startposition.
For every entry you can find the turns with simulator.

Code:
0 GGGG YYYYY OOOOOO RRRRRRRR P II BBB gggg yyyyy oooooo rrrrrrrr p ii bbb   1   2   3   4
2 oooooo yyyyy gggg GGGG YYYYY OOOOOO p ii bbb BBB II P RRRRRRRR rrrrrrrr   9  10
10 P II BBB bbb ii p GGGG YYYYY OOOOOO RRRRRRRR rrrrrrrr gggg yyyyy oooooo  31  16  32  33
31 rrrrrrrr RRRRRRRR BBB bbb ii p GGGG yyyyy oooooo II P OOOOOO YYYYY gggg  71  72
71 P II oooooo yyyyy rrrrrrrr RRRRRRRR YYYYY gggg GGGG p ii bbb BBB OOOOOO 126 127
126 p GGGG gggg YYYYY P II oooooo yyyyy ii bbb BBB OOOOOO RRRRRRRR rrrrrrrr 178 179 128 180 181 182
178 OOOOOO BBB bbb ii YYYYY P II oooooo RRRRRRRR rrrrrrrr gggg GGGG p yyyyy 214 183 215 216
215 rrrrrrrr RRRRRRRR oooooo OOOOOO BBB GGGG p yyyyy II P YYYYY ii bbb gggg 245 246
246 P II yyyyy p GGGG oooooo OOOOOO BBB bbb gggg RRRRRRRR rrrrrrrr YYYYY ii 260 261
261 RRRRRRRR gggg bbb oooooo OOOOOO BBB rrrrrrrr YYYYY ii GGGG p yyyyy II P 272 259 273 270
273 ii YYYYY rrrrrrrr BBB RRRRRRRR gggg yyyyy II P OOOOOO oooooo bbb GGGG p 282 283
282 OOOOOO P II yyyyy ii YYYYY rrrrrrrr oooooo bbb GGGG p gggg RRRRRRRR BBB 297 279 298 299
298 p GGGG bbb oooooo rrrrrrrr OOOOOO P gggg RRRRRRRR BBB YYYYY ii yyyyy II 292 309
309 BBB RRRRRRRR gggg rrrrrrrr OOOOOO P YYYYY ii yyyyy II oooooo bbb GGGG p 305 282

u++ l d
l d
u l d
u l d
u l d
u l d
u- l d
l d
l d
u- l d
u l d
u- l d
l d

13 moves = 35 turns
( 310 states)

Sequence for the simulator:
u++ l d l d u l d u l d u l d u l d u- l d l d l d u- l d u l d u- l d l d

This is an animation of the sequence to get the most scrambled position:
9 seconds per frame:
Attachment:

snautilus.gif [ 77.51 KiB | Viewed 6385 times ]

Here the standard C code of the generating program:

Code:
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <ctype.h>
#ifdef WINDOWS
#define _CLR_ "cls"
#else
#define _CLR_ "clear"
#endif
#define MAXNAUT 2000
#define NAUTSIZE 14

int th[] = {5,6,7,9,2,3,4,5,6,7,9,2,3,4};
char *color = "pibgyorr";
int count=0, last=0;
char **nauts;

int turnableu(char *naut) {
int i, cnt=0;
for (i=0; i<14; i++) {
if(naut[i]==0 || naut[i]==18) cnt++;
}
return((cnt==2));
}

int turnabled(char *naut) {
int i, cnt=0;
for (i=0; i<14; i++) {
if(naut[i]==36 || naut[i]==54) cnt++;
}
return((cnt==2));
}

void turnu(char *naut) {
int x,i;
for(i=0; i<14; i++) {
x = naut[i];
if(x<36)
naut[i]=(x+1)%36;
}
}

void turnd(char *naut) {
int x,i;
for(i=0; i<14; i++) {
x = naut[i];
if(x>=36)
naut[i] = (x+1)%36+36;
}
}

void flip(char *naut) { // turn left angles from 0 to 180
int i, x, y, d;
for(i=0; i<14; i++) {
x = naut[i]%36;
y = naut[i]-x;
d = th[i];
if(x<18) {
y = 36 - y;
x = 18 - d - x;
naut[i] = x + y;
}
}
}

void printnaut(char *naut) {
int i, j, x, d;
char a[73];
a[72] = '\0';
for(j=0; j<14; j++) {
x = naut[j];
d = th[j];
a[x] = ' ';
for(i=1; i<d; i++) {
if(j < 7)
a[x+i] = toupper(color[d-2]);
else
a[x+i] = color[d-2];
}
}
printf("%s", a);
}

int isequal(char* a, char *b) {
int i;
for(i=0; i<14; i++)
if(a[i]!= b[i]) return 0;
return 1;
}

int cmpnaut(char *nauta, char * nautb) {
int i, j;
char *nautc = (char*)malloc(NAUTSIZE);
memcpy(nautc, nautb, NAUTSIZE);
for(i=0; i<36; i++) {
turnu(nautc);
if(!turnableu(nautc)) continue;
for(j=0; j<36; j++) {
turnd(nautc);
if(!turnabled(nautc)) continue;
if(isequal(nautc,nauta))
{
free(nautc);
return 1;
}
}
}
free(nautc);
return 0;
}

int findnaut(char *naut,int len) {
int i;
for(i=0; i<len; i++) {
if(cmpnaut(naut,nauts[i]))
/*if(isequal(naut,nauts[i]))*/ {
return (i);
}
}
return -1;
}

int addnodes() { /* returna 0 if more nodes exist */
/* -1 full memory */
/* 1 tree is ready */
int i, j, len, f,lastnr;
char *naut; /* temporary naut */
if(count >= MAXNAUT) return -1; /*  full memory */
naut = (char*) malloc(NAUTSIZE); /* work node */
len = count; /* "count" changes dynamically */
while(last < len) { /* last added nodes */
memcpy(naut, nauts[last++], NAUTSIZE); /* walk throug last nodes */
printf("\n%3d", last-1);
printnaut(naut);
lastnr=-1;
for (i=0; i<36; i++) { /* walk throug all possible turns */
turnu(naut);
if( !turnableu(naut) ) continue;
flip(naut);
for(j=0; j<36; j++) {
turnd(naut);
if( !turnabled(naut) ) continue;
//flip( naut);
if((f = findnaut(naut, count))>=0) {
if(f!=lastnr) {
printf(" %3d", f);
lastnr=f;
}
continue; /* node exist */
}
nauts[count] = (char*)malloc(NAUTSIZE);
if(count!=lastnr) {
printf(" %3d", count);
lastnr=count;
}
memcpy( nauts[count++], naut, NAUTSIZE);
if( count >= MAXNAUT ) break;
}
flip(naut);
if(count >= MAXNAUT) break;
}
if(count >= MAXNAUT) break;
}
free( naut );
if( count >= MAXNAUT) return(-1); /* full memory */
if( len==count ) return (0); /* no node added , ready */
return (1);
}

int generate() {
int i, k=0;
return k;
}

int main( int argc, char *argv[]) {
char naut[] = {0,5,11,18,27,29,32,36,41,47,54,63,65,68};
nauts  = (char**) malloc( MAXNAUT * sizeof(char*) );
int g;
/*****************************
* root = startposition
* build start node
*****************************/
nauts[count] = (char *) malloc(NAUTSIZE);
memcpy( nauts[count++], naut, NAUTSIZE);
g=generate();
printf("\n%d\n",g);

while(count) free( nauts[--count] ); // clear table
free( nauts );
return 0;
}

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Tue Jan 15, 2013 11:44 am

Joined: Wed Apr 13, 2011 8:37 am
Location: Germany
The result of this work is the most scrambled puzzle.
Perhaps other people had found similar results.
Attachment:

nautilusu.png [ 708.56 KiB | Viewed 6363 times ]

Attachment:

nautilusd.png [ 714.89 KiB | Viewed 6363 times ]

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sun Jan 20, 2013 1:16 pm

Joined: Thu Aug 30, 2007 7:54 am
Location: Yorkshire, UK
Hi folks, I am really amazed to see such in depth and detailed analysis of the solution(s) to this puzzle. As has been mentioned, it was originally designed to have a 10 degree piece to make a nice arithmetic progression of angles from 10 to 80 degrees. It was almost too much of a cooincidence that the triangle number for 8 was 36! However the fact that the 10 degree piece was just too fragile (I tried 3D printing it, and casting the result in eurethane), meant that there were seven rather than eight pieces, which gave the idea of the rainbow colours. So... you gain some and lose some with each success and failure.

The reason I posted was that Andrea's original post showed circle segments exactly the same as the original cardboards piece I cut out to see if the puzzle was sufficiently scarmbleable. I quickly got in such a muddle using the simulation that I concluded it was!

It would be interesting to see further analysis using the 10 degree piece, and to see if such a piece would be physically workeable on a larger puzzle.

I still only have the one copy of the Nautilus that I ordered as soon as it came out. I ame waiting for my complimentary copies to arrive (they are on their way), as I have not yet solved the puzzle myself. I will follow this thread with interest.

Cheers!

Tim

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Sun Jan 20, 2013 8:07 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
timselkirk wrote:
It would be interesting to see further analysis using the 10 degree piece, and to see ifsuch a piece would be physically workeable on a larger puzzle.

See? Don't do it for me, do it for the inventor!

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Mon Jan 21, 2013 11:26 am

Joined: Wed Jan 14, 2009 6:37 pm
Location: The Great White North
timselkirk wrote:
[...]
It would be interesting to see further analysis using the 10 degree piece, and to see if such a piece would be physically workeable on a larger puzzle.
[...]

One way to make a 10Â° piece workable might be to reverse the pairing of radius to size. That is, instead of the 80Â° piece being the one with the largest radius, make it the smallest radius, and make the 10Â° piece the one with the largest radius (so it will be like large disc). That will hopefully give it enough substance to be less fragile, perhaps? (Or will it make the problem worse because now it sticks out too much if it's put next to two small pieces?)

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Thu Jan 24, 2013 12:24 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Andrea wrote:
The solution of rainbow Nautilus !
Second try. I give not up

The updated gods number is the lucky 13 !

The intention of the program was:
1) find the number of states.
2) find the gods number.
3) find the sequence of most scrambled state.

The first program was without errors. The definition of move/turn was wrong.
New definition:
Turn up face , turn left halve 180 degrees, turn down face.
The turn of zero degrees is included.
The godsnumber goes up, the state number goes down!
Number of possible position is 310. Godsnumber is 13.

Hi Andrea, sorry for the delayed response but only now was I able to find the time to figure out the source of the discrepancy between our programs. Parallel clean-room implementations really are a great thing to have!

My program finds 326 states when you ignore the middle layer. Your current incarnation finds 310. My program finds all 326 moves within 12 flips of solved, yours takes 13. I wrote a script to convert your output into the format my program uses and determined that your program finds a subset of states of mine. This is a good sign since if we diverged it would indicate a more drastic issue.

Here are the 16 states my program finds that yours does not (middle layer ignored):
Code:
[B][LL][OOOOO][GGG][YYYY]{}{rrrrrrr}[RRRRRRR]{ggg}{yyyy}{b}{ll}{ooooo}[]
[B][LL][RRRRRRR][][GGG][YYYY]{ooooo}[OOOOO]{rrrrrrr}{}{ggg}{yyyy}{b}{ll}
[B][LL][RRRRRRR][]{yyyy}{ggg}{ooooo}[GGG][OOOOO]{b}{ll}{rrrrrrr}{}[YYYY]
[B][LL][RRRRRRR][]{}{rrrrrrr}[OOOOO][GGG]{ggg}{yyyy}{b}{ll}{ooooo}[YYYY]
[B][LL][RRRRRRR][]{}{rrrrrrr}{ll}{b}[GGG][OOOOO]{ooooo}{ggg}{yyyy}[YYYY]
[B][LL]{ll}{b}{yyyy}{ggg}{}{rrrrrrr}[GGG][OOOOO]{ooooo}[][RRRRRRR][YYYY]
[B][LL]{yyyy}{ggg}[RRRRRRR][][OOOOO][GGG]{ooooo}{b}{ll}{rrrrrrr}{}[YYYY]
[B][LL]{yyyy}{ggg}{}{rrrrrrr}{ll}{b}[GGG]{ooooo}[OOOOO][][RRRRRRR][YYYY]
[GGG][YYYY][RRRRRRR][][OOOOO]{ooooo}[B]{b}{ll}{rrrrrrr}{}{ggg}{yyyy}[LL]
[GGG][YYYY][RRRRRRR][]{ooooo}[OOOOO][B]{rrrrrrr}{}{ggg}{yyyy}{b}{ll}[LL]
[GGG][YYYY]{ooooo}{ll}{b}{yyyy}{ggg}[B][OOOOO]{rrrrrrr}{}[][RRRRRRR][LL]
[GGG][YYYY]{yyyy}{ggg}{ooooo}[OOOOO][B]{b}{ll}{rrrrrrr}{}[][RRRRRRR][LL]
[GGG][YYYY]{}{rrrrrrr}{ll}{b}[OOOOO][B]{ooooo}{ggg}{yyyy}[][RRRRRRR][LL]
[GGG][YYYY]{}{rrrrrrr}{ll}{b}{ooooo}[B][OOOOO][][RRRRRRR]{ggg}{yyyy}[LL]
[OOOOO]{ll}{b}{yyyy}{ggg}{}{rrrrrrr}[B]{ooooo}[YYYY][GGG][][RRRRRRR][LL]
[RRRRRRR][]{ooooo}{ll}{b}{yyyy}{ggg}[B]{rrrrrrr}{}[YYYY][GGG][OOOOO][LL]

You wrote "The first program was without errors. The definition of move/turn was wrong." however I think this statement contradicts the fact that the number of states changed. If the only issue was the definition of what constitutes a turn then the new program should find all of the states the old one found, just in a different number of moves.

I took a look at the code you provided and spotted the difference between our programs. If you apply the patch below the output and number of state and "god's number" all match between my program and yours.

Code:
brenrigh@lambda ~ \$ diff -u naut.c new_naut.c
--- naut.c   2013-01-24 05:07:40.389229560 +0000
+++ new_naut.c   2013-01-24 05:08:21.562562648 +0000
@@ -137,16 +137,16 @@
for (i=0; i<36; i++) { /* walk throug all possible turns */
turnu(naut);
if( !turnableu(naut) ) continue;
-            flip(naut);
for(j=0; j<36; j++) {
turnd(naut);
if( !turnabled(naut) ) continue;
-                //flip( naut);
+                flip( naut);
if((f = findnaut(naut, count))>=0) {
if(f!=lastnr) {
printf(" %3d", f);
lastnr=f;
}
+          flip(naut);
continue; /* node exist */
}
nauts[count] = (char*)malloc(NAUTSIZE);
@@ -155,9 +155,9 @@
lastnr=count;
}
memcpy( nauts[count++], naut, NAUTSIZE);
+      flip(naut);
if( count >= MAXNAUT ) break;
}
-            flip(naut);
if(count >= MAXNAUT) break;
}
if(count >= MAXNAUT) break;

This goes back to using the definition of a move as any combination of turns of the top and bottom followed by a flip.

While thinking about this issue and trying to figure out why our results differed I came up with some interesting ideas. If they pan out I'm going to have some more interesting stuff to say about this puzzle soon

_________________
Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 25, 2013 12:23 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Here is the adjacency matrix for the 978 states of the Rainbow Nautilus. Because 978 x 978 is a big matrix in ASCII here it is as an image:
Attachment:

rbow_naut_adj_m.png [ 2.86 KiB | Viewed 5771 times ]

The solved state is [0, 0]. Understand that you can arbitrarily permute columns + rows in an adjacency matrix so the particular view shown in this image is mostly showing the order my program finds the states rather than the definitive view of the data. Also, adjacency matrices for undirected graphs are always mirror-symmetrical about the diagonal.

EDIT: updated image to enforce a more strict ordering of the rows + columns based on the order they are found.

_________________
Prior to using my real name I posted under the account named bmenrigh.

Last edited by Brandon Enright on Fri Jan 25, 2013 2:09 am, edited 2 times in total.

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 Post subject: Re: Rainbow Nautilus SolutionPosted: Fri Jan 25, 2013 12:55 am

Joined: Wed Jan 14, 2009 6:37 pm
Location: The Great White North
bmenrigh wrote:
Here is the adjacency matrix for the 978 states of the Rainbow Nautilus. Because 978*978 is a big matrix in ASCII here it is as an image:
Attachment:

The solved state is [0, 0]. Understand that you can arbitrarily permute columns + rows in an adjacency matrix so the particular view shown in this image is mostly showing the order my program finds the states rather than the definitive view of the data. Also, adjacency matrices for undirected graphs are always mirror-symmetrical about the diagonal.

This is really cool! Also quite telling, in that the matrix is so sparse. Jives with the fact that the rainbow nautilus is essentially a very highly bandaged 18-prism. Also kinda sad that it's so hard to scramble, though; it's such a beautiful puzzle I was hoping it could have some impressive scrambled states, but as it is, I'm having trouble figuring out how to get to the 13-twist state.

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