Shuffling

[David W. Wilson]

You wrote:

> thanks!  a(54) defeated everyone else who replied.
> I have been updating the data base as new results come in, by the way.

You're welcome, but it's not my result.  I plagiarized it from the followingmessage:

Warut Roonguthai wrote:

> FYI, this card-shuffling topic was discussed in David Gale's column
> "Mathematical Entertainments" on pages 54-56 of "The Mathematical
> Intelligencer" vol. 14, no. 1, 1992.
>
> Here is the number of shuffles required for card n to reach the top
> for the first time according to the 80-hour computation on a NeXT
> workstation mentioned in the article:
>
> card     shuffles
>
>   39        13932
>   43        30452
>   53           30
>   54    252992198
> 3464  21879255397
> 3729  15009146841
> 4546   2263846432
>
> Later,
> Warut (is back!)

The efficient way to calculate this sequence is not to shuffle the deck at
each shuffle, but rather to calculate the position of the relevant card at
each shuffle, which involves a simple integer mapping for each shuffle.
Using this method, a couple hours would probably suffice to confirm the
above value for 54, however, having confirmed the values for 39, 43, and
53, I tend to believe the given value for 54.