Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 8 Cards

Theorem

Let $D$ be a deck of $8$ cards.

Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in.

$2^n \equiv 1 \pmod 9$

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

$\ds 2^1$

$\equiv$

$\ds 2$

$\ds \pmod 9$

$\ds 2^2$

$\equiv$

$\ds 4$

$\ds \pmod 9$

$\ds 2^3$

$\equiv$

$\ds 8$

$\ds \pmod 9$

$\ds 2^4$

$\equiv$

$\ds 7$

$\ds \pmod 9$

$\ds 2^5$

$\equiv$

$\ds 5$

$\ds \pmod 9$

$\ds 2^6$

$\equiv$

$\ds 1$

$\ds \pmod 9$

Hence the result.

$\blacksquare$

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