The requirement for "suitably shuffled" is that no $k$ cards appear in same order in $D$ and $D'$.
Elementary shuffling operations are defined, very simply, as :
- Interleaving. Split deck in exactly two, parts $D_{top}$, $D_{bottom}$, and place each card of $D_{bottom}$ in between cards of $D_{top}$.
- Cutting. Select a consecutive interval of cards $D_{cut}$, from $c_{start}$ to $c_{end}$ inclusive, (which can be from the middle of the deck) of some length less than |$D$|, and place it either on the top, or the bottom. Defined by 3-tuple ( $c_{start}$, $c_{end}$, $top$ or $bottom$ ) (Aside: there also exists an Identity. One identity operation is to select any cut from the top, and place it back on the top. ( 0, $c_{end}$, $top$ ). Which can be ignored for shuffling.)
How many such elementary shuffling operations are required to ensure the number of cards in the same order for both decks is at most $k$?, with $k$ not less than 8