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In the traditional method for cutting a deck of playing cards, the deck is arbitrarily split two parts, which are exchanged beforereassembling the deck. In a more complex cut, called $\text{Scarne’s cut,}$ the deck is broken into three parts and the middle part in placed first in the reassembly. We’ll take $\text{Scarne’s cut}$ as the inspiration for an operation on languages$.$ For a language $A,$ let $\text{CUT(A) = {yxz| xyz ∈ A}}.$                     

  1. Exhibit a language $B$ for which $\text{CUT(B)$\neq$ CUT(CUT(B)).}$
  2. Show that the class of regular languages is closed under $\text{CUT}.$
in Theory of Computation by Veteran
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