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In discrete mathematics, a riffle shuffle is a shuffling of cards in which the top half is placed in one hand the the other half lies in the opposite hand. The cards are then alternatively interlaced with one another. There are two different types of riffle shuffles that will be the topic of discussion: the out-shuffle and the in-shuffle. An out-shuffle keeps the top card on top and the bottom card on bottom. When the top card is, instead, placed in the second position we have an in-shuffle.  \vspace{2mm}  Consider a typical $52$ card deck. If we name and order the cards $1,2,3,4...$, $0,1,2,3,4...$,  then after an out-shuffle we receive the order $1,27,2,28,3,29...$. $0,26,1,27,2,28...$ with the exception that $51$ lies in position $51$.  After an in-shuffle, the cards appear as $27,1,28,2,29,3...$. $26,0,27,1,28,2...$. Note that the first card is in the $0$ position.  To return to the original order, we must make $8$ out-shuffles. Interestingly enough, however, it takes $52$ in-shuffles.