Some Mathematical Problems Related to the Rubik’s Snake

Abstract

We proved several theorems about the Rubik's snake toy, which has a chain-like structure with applications in robot design. These mathematical theories can guide people to design beautiful shapes using the Rubik's snake. We presented an example for shape design to demonstrate the effectiveness of our approach.

A Rubik’s Snake with General Rotation Angles

“…Calculating the sum of the degrees for our example sequence repeated n times, we obtain S = We would make a remark that it can be seen from the proof that the period of a closed loop with general angle rotations can be any positive integer. This is different from the integer degree case in [15] where the period is proved to be no more than 4. 3．Tube Representation of a Rubik's Snake…”

“…Let θ be the angle of rotation between two blocks. As in [14] and [15], we define d = θ/90°to be the degree of this rotation. In a normal Rubik's Snake the degree of rotation is fixed to be 125 either 0, 1, 2, or 3, corresponding to a rotation of 0°, 90°, 180°, or 270°, respectively.…”

“…In [15], we proved that the sum of degrees of a Rubik's Snake closed loop with integer degrees of rotation must be an even number. In other words it is 0 or 2 in the mod 4 sense.…”

“…Some ideas in the study of Rubik's Snake such as the use of rotation matrix is also used in rigid Origami folding [12,13]. In previous papers, strategies have been given for the design of a Rubik's Snake [14], and some mathematical problems concerning a Rubik's Snake have been studied [15]. Rotations that are not 0°(or 360°), 90°, 180°, or 270°are mentioned in [14] but not much theoretical work is presented.…”

“…Rotations that are not 0°(or 360°), 90°, 180°, or 270°are mentioned in [14] but not much theoretical work is presented. On the other hand, [15] has quite some theoretical work but is only concerned with integer multiple of 90 degree rotations. Also, in [16], theorems about palindromic, periodic and Möbius Rubik's snakes were proved but most of them were concerned with integer multiple of 90 degree rotations.…”