# Card dealing mathematics problem [closed]

I have found the following card trick:

• Take a stack of n cards, with a known order.
• Deal the cards from your hand onto the table like this: Card from the top, then card from the bottom, from the top, from the bottom, etc until the deck is depleted.
• Keep dealing the cards like this. After a certain amount of deals (lets call that d) the original order will be restored.

I have made a table for n=1 through 25. The numbers for d are, consecutively: 1,2,2,3,3,5,6,4,4,9,6,11,10,9,14,5,5,12,18,12,10,7,12,23,21 I'm trying to find a pattern here, but I can't find one. My first thought was that after the same amount of deals as the amount of cards the sequence would repeat. Some seem to conform to n-1=d, but most don't. If you check for n-1=z*d, where z is an integer, most of them conform, but some don't.

Does anyone know what logic is hidden here? What is so special about n=16 and 17 (both just 5 deals required) and n=22 (just 7 deals required)?

• From your table, there is a conjecture: with $2^n$ or $2^n+1$ cards, you get period $n+1$. But please specify your algorithm. The topmost card ends up topmost in the new pile, or at the bottom? For three cards 1-2-3 gives 2-3-1 or 1-3-2? According to your table, it should be 1-3-2. – Sebastian Goette Mar 18 '16 at 15:21
• It's probably this one: oeis.org/A216066 – Wolfgang Mar 18 '16 at 15:42

In mathematical language, you're talking about the order of a certain permutation on $n$ letters, which can be easily computed using the cycle structure of the permutation.