Deck of 52 Cards returns to Original Order after 52 Modified Perfect Faro Shuffles/Proof 2
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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 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:
- $2^n \equiv 1 \pmod {53}$
- $2^{52} \equiv 1 \pmod {53}$
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