A Rubik’s Snake is a toy that was invented over 40 years ago together with the more famous Rubik’s Cube. It can be twisted to many interesting shapes including knots. Four blocks can form a trivial knot. Previously we have studied the shortest paths for Rubik’s Snake prime knots with up to 5 crossings. In this paper we study how many blocks are needed to form prime knots with 6 crossings. There are three different types of such knots. The results are classified using the DT (Dowker-Thistlethwaite) code. We also apply our findings to roller coaster design by using the tube version of the Rubik’s snake.

Keywords: Rubik’s Snake, prime knot, shortest path

The Rubik’s Snake is a toy invented by Prof. Rubik over 40 years ago.¹ It consists of right isosceles triangular prisms (called blocks) that, except for the first and last block, are connected to two other blocks at the centers of the square faces. The first design of Rubik’s snake came with 24 blocks, which were twisted to form many solid bodies. (Bird, cat, caterpillar, frog, flower, spiral etc.) However, more complicated design needs more blocks.

The Rubik’s Snake has been used for the study of protein  folding² and also for the construction of reconfigurable modular robots.^3–5 There are more applications of robots  presented in.^{6, 7} Some ideas in the study of Rubik’s Snake such as the use of rotation matrix is also shared in the study of rigid Origami folding.^8,9 In previous papers that the first author published with others, some strategies have been given for the design of a Rubik’s Snake,¹⁰ and some mathematical problems concerning a Rubik’s Snake have been studied.¹¹ Rotations that are not 0^◦ (or 360^◦), 90^◦, 180^◦, or 270^◦ are mentioned in¹⁰ but not much theoretical work is presented. On the other hand, there are quite some theoretical work but only concerned with integer multiple of 90 degree rotations in.¹¹ In,¹² Rubik’s snakes with general rotation angles were studied with theoretical work presented. In,¹³ theorems about Mo¨bius, palindromic and periodic Rubik’s snakes were proved.

Knot theory has been extensively studied.¹⁴ However, how could a Rubik’s Snake form a knot is not reported in the literature until our recent publication in which possible shortest Rubik’s snake trefoil knot paths with 34 blocks and possible shortest tube version (with general rotation angles not limited to integer multiple of 90 degrees) of Rubik’s snake trefoil knot path were presented.¹⁵

For a trivial knot, 4 blocks of a Rubik’s snake can form a closed loop, which has the shortest path. In this paper, we study the shortest path for a Rubik’s snake non-trivial prime knot with 6 crossings. Knots are classified according to their isotopy classes. For non-trivial prime knots up to 6 crossings, we have 3₁ (trefoil), 4₁, 5₁, 5₂, 6₁, 6₂ and 6₃. The 3₁ (trefoil) knot was studied in our previous paper¹⁵ and 4₁, 5₁ and 5₂  studied in our previous work.¹⁶ We now focus on 6₁, 6₂ and 6₃. The idea is to use certain key patterns: 6₁ comes from certain 4₁ path with an extra hole as key pattern, 6₂ comes from a different 4₁ path with an extra hole as key pattern, and 6₃ based on certain 3₁ path with an extra hole as key pattern. These are combined with local searches.

The rest of the paper is organized as follows: in Section 2, we summarize our previous work on non-trivial prime knots up to 5 crossings. In Section 3, we consider prime knots with 6 crossings. All the results are verified by the DT code to make sure they are indeed the knot we claimed they are. In Section 4, we apply our findings to roller coaster design by using the tube version of the Rubik’s Snake. We conclude in Section 5.

The Rubik’s Snake prime knots with up to 5 crossings with integer multiple of 90 degree rotations has been studied by us in the past. We just briefly summarize the results. In,¹⁵ 36 solutions of possible shortest Rubik’s Snake trefoil knots were given. Among those solutions, 6 of them are palindromic, the 30 non-palindromic solutions come in pairs and if we do not count the reverse order as a new solution, then effectively those are 15 non-palindromic solutions, making it 21 solutions overall in this sense. What have been verified are the following: there are no Rubik’s Snake trefoil knots with less than 34 blocks that has period more than 1 (period 2 and 3 are verified¹⁵ and trefoil knot cannot have period more than 3). There are no Rubik’s Snake trefoil knots with less than 34 blocks that are palindromic. In the construction of the solutions, the key pattern appears to be optimal. Based on that, we believe there is no solution with less than 34 blocks. In,¹⁶ we found possible shortest Rubik’s Snake 4₁ knot with 46 blocks, 5₁ with 52 blocks and 5₂ with 56 blocks. For nontrivial knots other than 3₁, even exhausting period 2 or palindromic Rubik’s snakes would be too expensive. The idea was to use certain key patterns combined with local searches.

Now consider the knot 6₁. The shortest we found has 64 blocks shown in Figure 2 with its line representation in Figure 1:

[1, 1, 0, 1, 1, 3, 3, 1, 1, 0, 0, 1, 0, 3, 0, 0, 1, 2, 0, 0, 3, 0, 0, 1, 0, 3, 3, 2, 3, 0, 3, 3, 0, 2, 3, 0, 0, 0, 3, 0, 0, 1, 2, 0, 1, 1, 2, 1, 1, 1, 3, 0, 3, 3, 0, 3, 1, 0, 0, 0, 1, 0, 1, 1]

Figure 1 The line representation of a Rubik’s snake 6₁ knot with 64 blocks

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Figure 2 A Rubik’s snake 6₁ knot with 64 blocks.

The path is based on the following 4₁ with an extra room   in   a   hole   to   allow   crossing   another time. [1, 0, 1, 1, 3, 3, 1, 1, 0, 0, 1, 0, 3, 0, 0, 1, 2, 0, 0, 3, 0, 0, 1, 0, 3, 3, 2, 3, 0, 3, 3, 0, 2, 3, 0, 0, 0, 3, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0]. Note that the shortest 4₁ we found has 46 blocks but does not have this extra room. This sequence with 48 blocks should be the shortest with such extra room needed for 6₁.

To better understand the construction, we observe that the 4₁ sequence has a common subsequence with the 2nd to the 46th of the 6₁. The green color in Figure 1 represents the first 16 blocks and the red color represents the 49th to the 64th blocks, and the black color is for the 33th to 47th. That means if we remove the red color and some green and black, then connect the green with black, we will get 4₁. That is exactly what happened.

The DT code of our 6₁ after the self-crossing is removed is [2, −10, −12, 14, −4, −16, 8, 6] (not the best reduction of DT but indeed corresponds to 6₁).  The DT code for the 4₁ is [4, −8, 10, −2, 6] (not the best reduction of DT code but it indeed corresponds to 4₁). The two crossings saved are those two below between red and blue lines. To summarize, the construction of 6₁ is based on 4₁, going through a hole another time.

Now consider the knot 6₂. The shortest we found has 62 blocks shown in Figure 4 with its line representation in Figure 3:

[2, 1, 0, 0, 0, 3, 0, 0, 1, 2, 0, 1, 1, 1, 1, 1, 3, 3, 0, 0, 1, 1, 3, 3, 1, 1, 0, 0, 1, 0, 3, 1, 1, 0, 1, 3, 0, 0, 1, 0, 3, 3, 2, 3, 0, 3, 0, 0, 2, 1, 0, 0, 1, 2, 0, 1, 1, 0, 1, 2, 1, 1]

Figure 3 The line representation of a Rubik’s snake 6₂ knot with 62 blocks.

Figure 4 A Rubik’s snake 6₂ knot with 62 blocks.

The path is based on the following 4₁ with an extra room   in   a   hole   to   allow   crossing   another   time: [2, 3, 0, 0, 0, 3, 0, 0, 1, 2, 0, 1, 1, 1, 1, 1, 3, 3, 0, 0, 1, 1, 3, 3, 1, 1, 0, 0, 1, 0, 3, 1, 1, 0, 1, 3, 0, 0, 1, 0, 3, 3, 2, 3, 0, 3, 3, 0]. Note that the shortest 4₁ we found has 46 blocks but does not have this extra room. This sequence with 48 blocks should be the shortest with such extra room needed for 6₂.

To better understand the construction, we observe that the 4₁ sequence has a common subsequence with the 3rd to the 44th of the 6₂. The green color in Figure 3 represents the first 16 blocks and the red color represents the 49th to the 64th blocks, and the black color is for the 33th to 47th. That means if we remove the red color and some green and black, then connect the green with black, we will get 4₁. That is exactly what happened.

The DT code of our 6₂ is [4, 8, −12, 2, −14, −6, −10]    (not the best reduction of DT but indeed corresponds to 6₂).

The DT code of the 4₁ is the standard [4, 6, 8, 2]. The three crossings difference between these DT codes are the two red- green crossings and the right and a red-blue crossing in the middle. To summarize, the construction of 6₂ is based on 4₁, going through a hole another time.

Finally consider the knot 6₃. The shortest we found has 64 blocks shown in Figure 6 with its line representation in Figure 5:

[0, 0, 1, 0, 0, 3, 2, 3, 3, 2, 0, 0, 1, 0, 1, 2, 1, 1, 3, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 3, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 3, 0, 0, 1, 3, 0, 0, 1, 0, 0, 1, 1, 0, 2, 3, 0, 3, 0, 3, 3]

Figure 5 The line representation of a Rubik’s snake 6₃ knot with 64 blocks.

Figure 6 A Rubik’s snake 6₃ knot with 64 blocks.

The path is based on the following 3₁ with an extra room   in   a   hole   to   allow   crossing   another   time: [1, 1, 0, 1, 2, 1, 1, 3, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 3, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 0, 1, 2, 0]. Note that the shortest 3₁ we found has 34 blocks but does not have this extra room. This sequence with 36 blocks should be the shortest with such extra room needed for 6₃. Also note that for the last hole if we go across the opposite way it would have been a 5₂ knot instead of a 6₃ knot.

To better understand this construction, we observe that this 3₁ sequence have a common subsequence with the 13th to the 48th block of the 6₃. The green color in Figure 5 represents the first 16 blocks and the red color represents the 49th to the 64th blocks. That means if we ignore the red part and most of the green part starting with the connection point to red, and then reconnect using a short path, we will get a trefoil knot 3₁. This is exactly what happened in that figure.

In addition to two self-crossings that can be easily removed, we could also remove two red-blue crossing because the red line is below the blue line for both crossings above. That gives a DT code of [4, 8, 10, 2, 12, 6] corresponding to 6₃        knot. The corresponding trefoil knot saves 3 crossings (two red-blue and one green-black) and has a DT code of [4, 6, 2].

To summarize, the construction of 6₃ is based on 3₁, going through a hole another time.

Just like the knots with less than 5 crossings, we found many other solutions for 6₁, 6₂ and 6₃ that tied for the shortest as well. The idea is that we could keep some key patterns and let the computer do a local exhaustive search for about 12 blocks.

In,¹² the tube version of the Rubik’s Snake is introduced. The idea is that each block of the Rubik’s Snake corresponds to a tube of the same shape. The connection is smooth with the first order derivative continuous. The roller coasters in parks typically has a path of just a trivial knot. It would be interesting to build a roller coaster with a non- trivial knot, especially with more crossing numbers than just a trefoil knot or a figure-8 knot.

Figure 7 shows the design of a roller coaster using the tube version of Rubik’s snake 6₂ knot with the shortest path we found (62 blocks). The view angle is different from the earlier pictures but the snake sequence remains the same. The reason to choose the shortest path is to save the cost while keeping the complexity. Note that this is under a simplified model and in practice the design does not have to strictly follow the rule of having identical basic blocks.

Figure 7The tube version of Rubik’s snake 6₂ knot with 62 blocks for roller coaster design.

Finding the shortest path for a Rubik’s snake nontrivial knot is challenging. With each joint having 4 choices, the total number of attempts is too many to have an exhaustive search. By using certain key patterns combined with local searches, we found possible shortest Rubik’s Snake 6₁ knot with 64 blocks based on a 4₁ knot with an extra hole that the path could go through, possible shortest 6₂ knot with 62 blocks based on a 4₁ knot with an extra hole that the path could go through, and possible shortest 6₃ knot with 64 blocks based on a 3₁ knot with an extra hole that the path could go through. All the results are verified by the DT code to make sure they are indeed the knots we claimed they are. It would be interesting future work to see if these can be proved to be optimal.

We thank Katie Liu for a discussion about the Roller Coaster design. S. Hou’s research is partially supported by the Walter Koss Endowed Professorship. The title of the professorship is made available through the State of Louisiana Board of Regents Support Funds.

We thank Katie Liu for a discussion about the Roller Coaster design. S. Hou’s research is partially supported by the Walter Koss Endowed Professorship. The title of the professorship is made available through the State of Louisiana Board of Regents Support Funds.

None.

1. Fenyvesi C. Rubik’s snake of infinite possibilities. The Washington Post; 1981.
2. Iguchi K. A toy model for understanding the conceptual framework of protein folding: Rubik’s magic snake model. Mod Phys Lett B. 1998;12(13):499–506.
3. Ding X, Lu S, Yang Y. Configuration transformation theory from a chain-type reconfigurable modular mechanism-rubik’s snake. The 13th World Congress in Mechanism and Machine Science; 2011.
4. Zhang X, Liu J. Prototype design of a rubik snake robot. Mechanisms and Machine Science; 2016:36.
5. Liu J, Zhang X, Zhang K, et al. Configuration analysis of a reconfigurable rubik’s snake robot. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 2019;233(9):3137–3154.
6. Yim M, Roufas K, Duff D, et al. Modular reconfigurable robots in space applications. Autonomous Robots. 2003;14(2-3):225–237.
7. Zhang X, Liu J, Feng J, et al. Effective capture of nongraspable objects for space robots using geometric cage pairs. IEEE/ASME Transactions on Mechatronics. 2020;25(1):95–107.
8. Hull TC, Belcastro SM. Modelling the folding of paper into three dimensions using affine transformations. Linear Algebra and its applications. 2002;348(1-3):273–282.
9. Tachi T. Simulation of rigid origami. Origami. 2009;4(08):175–187.
10. Li Z, Hou S, Bishop T. Computational design and analysis of a magic snake. J Mech Rob. 2020;12(5):054501.
11. Hou S, Chen Y, Li Z. Some mathematical problems related to the rubik’s snake. J Mech Rob. 2021;13(1):014502.
12. Hou S, Atkins S, Chen Y. A rubik’s snake with general rotation angles. SCIREA Journal of Information Science and Systems Science. 2021;5(6):123–135.
13. Hou S, Su J, Chen, Y. Palindromic, periodic and mobius rubik’s snakes. Int Rob Auto J. 2021;7(3):84–88.
14. Adams C. The knot book: An Elementary introduction to the mathematical theory of knots. American Mathematical Society, ISBN. 2004;978-0-8218-3678-1.
15. Hou S, Su J. Shortest paths of trefoil knot designs using rubik’s snakes. Int Rob Auto J. 2022;8(1):18–20.
16. Hou S, Su J. Shortest paths of rubik’s snake prime knots up to 5 crossings. Int Rob Auto J. 2022;8(2):47–50.