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

a. Exhibit a language B for which CUT(B) 6= CUT(CUT(B)).

b. Show that the class of regular languages is closed under CUT.