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.
Matrix_pipe_set.gif [ 28.87 KiB | Viewed 2656 times ]
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).
Matrix_1.jpg [ 229.99 KiB | Viewed 2656 times ]
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],
Matrix_2.jpg [ 347.43 KiB | Viewed 2656 times ]
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).
Matrix_3.jpg [ 252.98 KiB | Viewed 2656 times ]
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!.