# No subject

N. J. A. Sloane njas at research.att.com
Sun Nov 15 02:06:57 CET 1998

```Dear Sequence Fans,
I would like to draw your attention to an old problem
that i think was invented by Richard Guy, and which could
potentially produce several interesting sequences.

Take an infinite deck of cards labeled 1,2,3,4,5,6,...
At step n, pick up the top n cards and interlace them
with the next n cards.  For example, after step 2 we have
3,2,4,1,5,6,7,...
and we pick up 3,2,4 and shuffle them in, getting
1,3,5,2,6,4,7,8,9,...
Do this for ever. The sequence below gives the
sequence of cards that appear on top of the deck.
It is conjectured that eventually every number appears on top of the deck.
Many other sequences suggest themselves-
What about the sequence that tells at which step n appears on top? It begins 0,1,2,8,5,4,78(?),...
Or the sequence that gives the top cards that are new: 1 2 3 6 5 9 4 ...
Or the sequence telling when 1 reaches the top (0,3,7,...)
and so on.
I have not been able to locate any publication by Richard that mentions this.

%I A035485
%S A035485 1,2,3,1,6,5,9,1,4
%N A035485 Card on top of deck at nth stage of Richard Guy's shuffling problem.
%O A035485 0,2
%K A035485 nonn,easy,nice,more
%D A035485 David Gale's column, Mathematical Intelligencer, '91 or '92.
%A A035485 njas, Clark Kimberling (ck6 at cedar.evansville.edu)
%C A035485 At nth step, pick up top n cards and interlace them with the next n.
%C A035485 Here is the deck after steps 0,1,2,3,4,5:
%C A035485 1,2,3,4,5,6,7,...
%C A035485 2,1,3,4,5,6,7,...
%C A035485 3,2,4,1,5,6,7,...
%C A035485 1,3,5,2,6,4,7,8,9,...
%C A035485 6,1,4,3,7,5,8,2,9,10,...
%C A035485 It is conjectured that eventually every number appears on top of the deck.
%C A035485 What about the sequence that tells at which step n appears on top? It begins 0,1,2,8,5,4,78(?),...

Neil Sloane

```