# Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 8 Cards

## Theorem

Let $D$ be a deck of $8$ cards.

Let $D$ be given a sequence of modified perfect faro shuffles.

Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in.

## Proof

From Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order, the cards of $D$ will return to their original order after $n$ such shuffles, where:

$2^n \equiv 1 \pmod 9$

Inspecting $2^n$ for $n$ from $1$:

 $\ds 2^1$ $\equiv$ $\ds 2$ $\ds \pmod 9$ $\ds 2^2$ $\equiv$ $\ds 4$ $\ds \pmod 9$ $\ds 2^3$ $\equiv$ $\ds 8$ $\ds \pmod 9$ $\ds 2^4$ $\equiv$ $\ds 7$ $\ds \pmod 9$ $\ds 2^5$ $\equiv$ $\ds 5$ $\ds \pmod 9$ $\ds 2^6$ $\equiv$ $\ds 1$ $\ds \pmod 9$

Hence the result.

$\blacksquare$