"Hi to everyone from the... 4th dimension. (this is where I have been lost the past months,
and I am not sure how to get out).

So... I present you the... Matrix! - using what I call as the "matrix-mech". As usual, I prefer
new ideas which are as simple and different as possible. And by using matrix-related permutations,
this one hits Group Theory on its very elemental base.

Puzzle Goal: To use two permutation-matrices and their inverses, to arrange the four marbles/beads
(each of different color) to their correct order. By definition, each permutation can be translated into
a matrix, which will give more than enough clues regarding how to send each marble to a specific capsule.
There are two solutions, and it is easy to go from one to another as long as you know how the full
symmetric algebraic group S4 acts on the colored marbles.



I will be a bit more technical now:

The "Matrix" puzzle is made of two cylinders, one external and one internal.
The external cylinder has two sets of four capsules C1 and C2. Each capsule-set
may accommodate four marbles (one blue, one green, one red, and one yellow).
Below each capsule set, there is a set of solution-stickers with a specific color order.
(S1 for C1 which is blue-green-red-yellow, and S2 for C2 which is blue-red-green-yellow).



The internal cylinder has two sets of four pipes P1 and P2.
By rotating the internal cylinder inside the external cylinder, each pipe-set takes
the four marbles from one capsule set, rearranges their order, and sends them
(with gravity) to the other capsule set. This corresponds to using a permutation matrix.
Pipe-set P1 implements the matrix permutation M1=(1234) (also translated as
the 4x4 matrix [0 0 0 1; 1 0 0 0; 0 1 0 0; 0 0 1 0] ) and its inverse (1432).
Pipe-set P2 implements the matrix permutation M1=(1324) (also translated as
the 4x4 matrix [0 0 1 0; 0 0 0 1; 0 1 0 0; 1 0 0 0] ) and its inverse (1423).
For example, M1 rearranges the ordered colors [Blue-Green-Red-Yellow] to
[Green-Red-Yellow-Blue], and M2 rearranges the ordered colors
[Blue-Green-Red-Yellow] to [Yellow-Red-Blue-Green],



The goal is to place the marbles inside the capsules in such a way, that their colors
match the solution-stickers S1 or S2. Another challenge is going from S1 to S2 and vice versa.

SOLUTION: Identify M1 and M2 to find either S1 or S2. Since the generators are based
on the full symmetric group on four elements Symm(4), there are only 4! = 24 different
arrangements (12 for each capsule), so it shouldn't be hard to reach one of the two solutions.
As an example of using the pipe-sets P1 and P2 to go from S1 to S2, we use the following moves:
(M1) - (M2) - (M1). To go back, (inverse M1) - (inverse M2) -(inverse M1).



In the video, I am using a map which connects each one of the four elements in four ways.
And to go from any one element to any other, only three or less moves are allowed, that is,
if you know how to get there!


Video here




Hypothetically, by using 48 marbles (instead of 4), and 6 sets of pipes (instead of two), then we could
construct a Matrix which can emulate (in a cylindrical way) the Rubik's Cube, and for which I call... Rubik's Tube!
(an image is shown in the video).

In fact, since the matrix concept is based on permutations which are the fundamental elements of groups,
this cylindrical "matrix" structure can emulate ANY algebraic puzzle in existence. The internal pipes may
be made in such a way where they are only one direction, different size of marbles could create blocked
moves etc etc etc. The possibilities are virtually endless, but here I am focusing on a cute group (S4)
which only has 24 different states (not compared to Rubik cube's 4.3*10^19 different states).

This concept was an idea I had in the early 90s when I was at the University of Crete in Greece (Maths Department).
The accurate 3d models (there are two, one big and one small) were magnificently made by the magnificent,
extraordinary, unbelievable, amazing... Gregoire Pfennig!!! Many thanks!.


Pantazis"

Very Interesting Puzzle! I had no idea it was going to be that big Are you going to make any more complex versions of this?

I can't check out the video until I get home tonight but I really like this idea. To make sure I understand the Rubik's Tube idea...

It is 48 marble as you want to use 1 for each non-face center sticker... correct?

If so that makes me think of this thread and I suspect you could create the same problem with fewer marbles.

Based on this post by Taus I suspect you could make the 2x2x2 version with just 8 marbles and 3 sets of pipes. If so, I don't think that would be too big of a jump up from where you are now.

Carl

More like 24 marbles.

Did you read the post of Taus that I linked to above? In it the claim in made that you only need 8 stickers on a 2x2x2 to guarantee a unique solution when the actual color scheme is known in advance. Since Pantazis's Matrix S4 has colored stckers next to the pockets for the marbles this should tell us the puzzles color scheme. So I think tracking any more then 8 marbles would be redundant... at least assuming I understand everything correctly so far.

Carl

You can't emulate a 2x2x2 with just the permutations of 8 marbles. The fact that all states of a 2x2x2 can be distinguished by only 8 specific stickers only works because you not only get information from the permutation of the 8 stickers (which is not even something you can properly define) you also get information from where the stickers are! Each sticker can go in 24 different places so you actually have 24!/16! different positions (ignoring some stickers are on the same cubie/have the same color so you actually get slightly less information).
8! is only 40.320 but the 2x2x2 has far more than 3 million positions, so clearly it cannot be done with just 8 marbles. How many you do need is interesting.

Why not 8 marbles and 24 places? (might as well have all the 24 marbles in though)
I don't see a way to get away with less than 24 places if each marble represents a sticker.

I am sure you can get away with less than 24 places. For once, you can just get rid of 4 because once you know where all other 5 colors are, you also know where the 6th color stickers go. I am sure you can remove many more stickers before this fails.
But perhaps we can do it by replacing marbles with triangular prisms? That way each prism represents a piece (so we need only 8) and we can keep track of the orientation.

But anyway: this is an awesome concept and great execution.

I think we can get rid of 3 places by considering that piece fixed. But removing 4 places representing a face won't work, it's as limiting saying the white face can't have other colored stickers on it. The marbles can go though, no marbles is a type of marble.

Having things with orientation in the pipes helps though. So when do we see a matrix Big Chop?

Ok... just realized the error in my logic. I'll still contend that it can be done with just 8 marbles, but yes you need 24 places those marbles can end up. So this doesn't help all that much.

Tom's idea of finding the minimum number of places needed is also very interesting. Oh I just love puzzles which make one think about stuff like this.Yes, I too think the max may be 3. Yes, if you have 24 places you can remove 4 marbles of the same color and deduce where they are but in this context we are talking marbles again and not places.

Carl

Good to hear Pantazis is still making puzzles!

I was expecting a much smaller puzzle, glad I clicked on the video.

-d

I love this idea and the execution is great.

As has been pointed out, with more pipes and more spots, you could simulate a 2x2x2.

But, you could actually simulate any twisty puzzle, even ones that aren't physically constructible. The complex 3x3x3 or even then 3x3x3x3 for example. Yes the number of tubes would get really complicated for these but it could be done.

Perhaps someone should start with simulating a 2x2x2 and then 3x3x3 first

It looks like some kind of crazy water heating device.

Interesting! Seems kind of like a Twisty Puzzle Turing Machine.

Super idea!
And using a ball in stead of a cylinder could give an extra dimension.

**************************************************
"The directed Cayley Graph of the puzzle's group S4 can be used to guide us to go from
any one position to any other position. This graph has vertices the elements of the group S4
(disguised into all the possible positions that the marbles can take on the puzzle),
while the generating set (for the directed graph) is made of the permutations matrices
(1234) and (1324). The first one is presented as the internal cylinder's blue labelled sticker,
and the second one as the red. The actual graph is not directed as by turning the cylinder
by 180 degrees we automatically obtain the inverses for both (1234) and (1324), but I am
using the directions to identify the orientations when applying each generator.

And now, consider how the corresponding Cayley graph for the Rubik's Cube would look like!!! :-S "
**************************************************

Above text is from Pantazis

Updated Version Video

Pictures on Pantazis's facebook