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

## Theorem

Let $D$ be a deck of $2 m$ cards.

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

Then the cards of $D$ will return to their original order after $n$ such shuffles, where:

- $2^n \equiv 1 \pmod {2 m + 1}$

## Proof

From Position of Card after n Modified Perfect Faro Shuffles, after $n$ shuffles a card in position $x$ will be in position $2^n x \pmod {m + 1}$.

So for all $2 m$ cards in $D$, we need to find $n$ such that:

- $2^n x \equiv x \pmod {2 m + 1}$

Because $2 m + 1$ is odd, we have:

- $\gcd \set {2, 2 m + 1}$

and so from Cancellability of Congruences:

- $2^n \equiv 1 \pmod {2 m + 1}$

$\blacksquare$

## Examples

### Deck of 6 Cards

Let $D$ be a deck of $6$ cards.

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

Then after $3$ such shuffles, the cards of $D$ will be in the same order they started in.

### Deck of 8 Cards

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.

### Deck of 12 Cards

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.

### Deck of 52 Cards

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.

### Deck of 62 Cards

Let $D$ be a deck of $62$ 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.

## Sources

- 1971: George E. Andrews:
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