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

Jump to navigation
Jump to search

## Theorem

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.

## 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:

- $2^n \equiv 1 \pmod 9$

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.

$\blacksquare$

## Sources

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