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:
 * $2^n \equiv 1 \pmod {13}$

From Fermat's Little Theorem:
 * $2^{12} \equiv 1 \pmod {13}$

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 $1$:

Hence the result.