Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 6 Cards

Theorem

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.

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:

$2^n \equiv 1 \pmod 7$

Inspecting $2^n$ for $n$ from $1$:

\(\ds 2^1\)

\(\equiv\)

\(\ds 2\)

\(\ds \pmod 7\)

\(\ds 2^2\)

\(\equiv\)

\(\ds 4\)

\(\ds \pmod 7\)

\(\ds 2^3\)

\(\equiv\)

\(\ds 1\)

\(\ds \pmod 7\)

Hence the result.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-3}$ Riffling: Exercise $1$