A Markov Chain with a finite number of states is said to be regular if there exists a non negative integer n such that for any i, j ∈ S, P(n)i, j > 0 for any n >_ n.
(a) Prove that a regular Markov Chain is irreducible.
(b) Prove that a regular Markov Chain is aperiodic.

The pyraminx is a rubik's cube-type toy in the shape of a tetrahedron. the pyraminx shown below has edges 15\,\text{cm}15cm long and vertical height h=12.2\,\text{cm}h=12.2cm. the triangle drawn with dashed lines is a right triangle. what is the distance rr? round your answer to the nearest tenth.