# Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 12 Cards

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## Theorem

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.

## 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:

- $2^n \equiv 1 \pmod {13}$

From Fermat's Little Theorem:

- $2^{12} \equiv 1 \pmod {13}$

so we know that $n$ is at most $12$.

But $n$ may be smaller, so it is worth checking the values:

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

\(\ds 2^1\) | \(\equiv\) | \(\ds 2\) | \(\ds \pmod {13}\) | |||||||||||

\(\ds 2^2\) | \(\equiv\) | \(\ds 4\) | \(\ds \pmod {13}\) | |||||||||||

\(\ds 2^3\) | \(\equiv\) | \(\ds 8\) | \(\ds \pmod {13}\) | |||||||||||

\(\ds 2^4\) | \(\equiv\) | \(\ds 3\) | \(\ds \pmod {13}\) | |||||||||||

\(\ds 2^5\) | \(\equiv\) | \(\ds 6\) | \(\ds \pmod {13}\) | |||||||||||

\(\ds 2^6\) | \(\equiv\) | \(\ds 12\) | \(\ds \pmod {13}\) | |||||||||||

\(\ds 2^7\) | \(\equiv\) | \(\ds 11\) | \(\ds \pmod {13}\) | |||||||||||

\(\ds 2^8\) | \(\equiv\) | \(\ds 9\) | \(\ds \pmod {13}\) | |||||||||||

\(\ds 2^9\) | \(\equiv\) | \(\ds 5\) | \(\ds \pmod {13}\) | |||||||||||

\(\ds 2^{10}\) | \(\equiv\) | \(\ds 10\) | \(\ds \pmod {13}\) | |||||||||||

\(\ds 2^{11}\) | \(\equiv\) | \(\ds 7\) | \(\ds \pmod {13}\) | |||||||||||

\(\ds 2^{12}\) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod {13}\) |

Hence the result.

$\blacksquare$

## Sources

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