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 Rubik Can you master this amazing cube? Magna Cube An interesting variation on the SOMA CUBE... Rubik Twist A twisting puzzle challenge that takes the form of thousands of shapes! # Flip Flop Puzzle

## A flat version of Rubik's Cube

 Ryohei Miyadera Munetoshi Sakaguchi Daisuke Minematsu Ryota Kawazoe Toshiro Miura - Kwansei Gakuin High School I. Introduction  The Back and Front Puzzle is a flat version of Rubik's Cube. Although this is a very simple game compared to Rubik's Cube, this game is very interesting in its own way. This game was first introduced by Shigeo Takagi in Takagi . In this game the front is colored with red and the back, with yellow. Shigeo Takagi treated the b/f puzzle as if it is a game you can pick up and rotate like Rubik's Cube. Therefore two arrangements  in graph 1 opposite can be seen identical, because you can get one from another just by picking up and rotate horizontally.
We use a different approach. We fix our puzzle on a paper or screen. Therefore we treat two arrangements shown in the drawing opposite as two different ones, but our approach can lead to beautiful results that you will see later. We studied the b/f puzzle as a problem of graph theory, and found a very beautiful patterns in the puzzle. Takagi studied the game with 3 rows and 3 columns, but first we are going to study the case of 3 rows and 2 columns. Later we are going to study the case of 4 rows. The difference of the number of rows and columns turned out to be very important factor in the puzzle. II. The case of 2x3 cells  We only use five rotations for these arrangements. We name these rotations R1, R2, R3, R4 and R5.

Example 1: The following pictures show you how these rotation occur.  Example 2: (a) If you start with and use R1, then you get . As shown below. (b) After that if you use R4, then you get . See picture below. Example 3: If you start with , how many arrangements are there that you can get by using these five rotations? Here you can use rotations as many times as you want.

Answer: You can get the arrangements in diagram (2) below. diagram (2)

Example 4: It is a good way to use the theory of graphs to study the b/f puzzle. We denote each arrangement by a red vertex. If you can get an arrangement from another arrangement using only one rotation, then we connect the two vertexes corresponding to these two arrangements with a blue line.
For example, look at the vertex 5 and 9. It is easy to see that by rotation R3, we get arrangement 9 from arrangement 5. Therefore we connect them with a blue line. In the similar way we can connect other vertexes, and get graph (3).
When we made graph (3), we chose vertexes with fewer lines and located them on the first column and the last column.  We located vertexes with more lines in the middle of graph (3) below. graph (3)

A Hamiltonian path is a path between two vertexes of a graph that visits each vertex exactly  once.  Can you find a Hamiltonian path of the above graph?

Answer: A Hamiltonina path is {1, 2, 7, 4, 3, 5, 9, 13, 14, 11, 16, 15, 10, 8, 12, 6}. It is not difficult to check this using the above graph.
Perhaps it is easier to see the Hamiltonian path in the list of pictures.  Please see diagram (4) below.  The rotations you are going to use are R1, R5, R4, R3, R5, R3, R5, R1, R4, R5, R3, R5, R2, R1 and R2. Problem 1: If you start with , how many arrangements are there that you can get by using the five rotations. You can use rotations as many times as you want.

Answer to problem 1: If you start with , then there are 24 arrangement that you can get by using the five rotations. It is not difficult to check the answer once you get one. The order of arrangement in the following table looks a little bit strange, but this order is best fit to the structure of the b/f puzzle. You will see this fact later. diagram (5)

Problem 2: Can you make a graph using diagram (5) above? Can you find a Hamiltonian path?

Answer to problem 2: See graph (6) below. graph (6)

The following sequence is a Hamiltonian path. It is not difficult to check this solution {2, 6, 3, 7, 9, 10, 13, 17, 21, 22, 18, 14, 20, 24, 23, 19, 15, 11, 16, 12, 8, 4, 5, 1}. III. The case of 2x4 cells In the previous section we studied the puzzle with 2 columns and 3 rows. In this section we are going to study the puzzle with 2 columns and 4 rows. This time we have 6 rotations.

Example 5: (a) If you start with and use R1, then you get .
(b) After that if you use R5, then you get .

Problem 3: If you start with , how many arrangements are there that you can get by using the five rotations. You can use rotations as many times as you want.

Answer to problem 3:  It is not difficult to get all the 36 arrangements, but it needs some time to make a good table with these arrangements. If you get a good table, then it will make it easier to make a beautiful graph with it. The strategy is the same. It is better to locate vertexes with fewer lines in the first and the last columns. diagram (7)

Problem 4: (a) Can you make a beautiful graph using diagram (7)?
(b) Can you find a Hamiltonian path?

Answer to problem 4 (a): It is not difficult to make a beautiful graph (8) using diagram (7). It is better to locate vertexes with a lot of lines in the middle part of graph, and vertexes with fewer lines in the first column and the last columns. graph (8)

Answer to problem 4 (b): It is not easy to find a Hamiltonian path. To tell you the truth it took Ryohei Miyadera who is one of the author of this article 6 hours to find a Hamiltonian path, but it was quite an experience and gave him a lot of joy when he found one. Perhaps there are many people who are a lot better than he is in finding things. If you can use a good software like Mathematica, it will take only a few minutes to find one.
{2, 6, 3, 7, 9, 17, 10, 18, 30, 20, 32, 33, 25, 13, 21, 15, 27, 35, 28, 36, 29, 19, 31, 34, 26, 14, 22, 16, 24, 11, 23, 12, 8, 4, 5, 1} is a Hamiltonian path. Once you have an answer, it won't be difficult to check it.

Remark: We used Mathematica and the programs in S. Skiena and S. Pemmaraju  to make graphs and diagrams of this article.

References by Larry Bickford
The Flip-Flop Puzzle page could use some grammar corrections, but I surmise the authors' native language is not English. Remembering my skills in a foreign language at their age, I appreciate the excellence of their article as it is.
But two non-grammar corrections are needed. Halfway through, starting at "Problem 1", the diagram is clearly that of 2x4, not 2x3. Likewise the diagram at "Answer to Problem 1".
Also, at "Problem 3", there are six rotations, not "five". A couple of other interesting insights into the 2x3 puzzle:
From diagram 2, if you swap red and yellow, you get yet another set of 16 arrangements not in either diagram.
And there are two sets of four arrangements distinct to themselves. Within each set, R1 and R3 are always identity operators, and the graph is a square with one diagonal.
R Y R Y R Y R Y Y R Y R Y R Y R
Y Y R R R Y Y R Y Y R R R Y Y R
Y R Y R Y R Y R R Y R Y R Y R Y
There is are similarly small sets in the 2x4 case when the initial position has only opposite corners reversed, e.g.,
R Y
Y Y
Y Y
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