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