# Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 12 Cards

## Theorem

Let $D$ be a deck of $12$ cards.

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

Then after $12$ 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:

$2^n \equiv 1 \pmod {13}$
$2^{12} \equiv 1 \pmod {13}$

so we know that $n$ is at most $12$.

But $n$ may be smaller, so it is worth checking the values:

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

 $\ds 2^1$ $\equiv$ $\ds 2$ $\ds \pmod {13}$ $\ds 2^2$ $\equiv$ $\ds 4$ $\ds \pmod {13}$ $\ds 2^3$ $\equiv$ $\ds 8$ $\ds \pmod {13}$ $\ds 2^4$ $\equiv$ $\ds 3$ $\ds \pmod {13}$ $\ds 2^5$ $\equiv$ $\ds 6$ $\ds \pmod {13}$ $\ds 2^6$ $\equiv$ $\ds 12$ $\ds \pmod {13}$ $\ds 2^7$ $\equiv$ $\ds 11$ $\ds \pmod {13}$ $\ds 2^8$ $\equiv$ $\ds 9$ $\ds \pmod {13}$ $\ds 2^9$ $\equiv$ $\ds 5$ $\ds \pmod {13}$ $\ds 2^{10}$ $\equiv$ $\ds 10$ $\ds \pmod {13}$ $\ds 2^{11}$ $\equiv$ $\ds 7$ $\ds \pmod {13}$ $\ds 2^{12}$ $\equiv$ $\ds 1$ $\ds \pmod {13}$

Hence the result.

$\blacksquare$