Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 12 Cards
Jump to navigation
Jump to search
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 $1$