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We enumerate all Hamilton circuits on the dual graph of the dodecahedron, group them according to symmetry, and then discuss which of them may be realized as snake patterns on the Megaminx puzzle. We find that there are eleven shapes that may be obtained without taking the puzzle apart (legal shapes) and ten more that can be achieved if you swap two pieces (illegal). With color taken into account, there are 700 ways to put a snake on the Megaminx, 280 of them legal. Supplemental data for this article is available online at [ABSTRACT FROM AUTHOR]

Motivated by the our recent work in Tan et al., 2016, related to the bi-periodic Fibonacci quaternions, here we introduce the bi-periodic Lucas quaternions that gives the Lucas quaternions as a special case. We give the generating function and the Binet formula for these quaternions. Also, we give the relationships between bi-periodic Fibonacci quaternions and bi-periodic Lucas quaternions. [ABSTRACT FROM AUTHOR]

In this paper, we present a new generalization of the Fibonacci quaternions that are emerged as a generalization of the best known quaternions in the literature, such as classical Fibonacci quaternions, Pell quaternions, k -Fibonacci quaternions. We give the generating function and the Binet formula for these quaternions. By using the Binet formula, we obtain some well-known results. Also, we correct some results in [3] and [4] which have been overlooked that the quaternion multiplication is non commutative. [ABSTRACT FROM AUTHOR]

In this paper, we introduce h ( x ) − Fibonacci quaternion polynomials that generalize the k − Fibonacci quaternion numbers, which in their turn are a generalization of the Fibonacci quaternion numbers. We also present a Binet-style formula, ordinary generating function and some basic identities for the h ( x ) − Fibonacci quaternion polynomial sequences. [ABSTRACT FROM AUTHOR]