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: Number Theory ... (previous) ... (next): $\text {4-3}$ Riffling: $\text {4-3-4}$