Deck of 52 Cards returns to Original Order after 52 Modified Perfect Faro Shuffles/Proof 1

Theorem

Let $D$ be a deck of $52$ cards.

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

Then after $52$ such shuffles, the cards of $D$ will be in the same order they started in.

Proof

From Position of Card after n Modified Perfect Faro Shuffles, after $n$ shuffles a card in position $x$ will be in position $2^n x \pmod {53}$.

So for all $52$ cards in $D$, we need to find $n$ such that:

$2^n x \equiv x \pmod {53}$

We have that $53$ is a prime number, so $x$ can be cancelled from either side:

$2^n \equiv 1 \pmod {53}$

By Fermat's Little Theorem:

$2^{52} \equiv 1 \pmod {53}$

So the cards of $D$ will return to their original order after $52$ modified perfect faro shuffles.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-3}$ Riffling