Take three white poker chips and three red poker chips, line them up thusly:

combinatorics/sliding.pairs New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

Take three white poker chips and three red poker chips, line them up
thusly:

   WWWRRR

Now, ALWAYS moving two adjacent poker chips (color doesn't matter), in
three moves convert the above into this:

   RWRWRW

(Of course, you can play this with pennies and dimes, or any other
easily distinguishable and easily slid objects.)

The same thing can be done with four of each color in four moves, five
of each color in five moves, and so on.

-- Everett Kaser <everett@cv.hp.com>

Show solution

 Start:  WWWRRR
Move 1:    WRRRWW
Move 2:    WRR  WRW
Move 3:      RWRWRW

I have solutions at home for up through 11 of each color, I think.  I
know I've solved at least up through 9 of each.  There appears to be a
pattern after a while, but there's always a slight "kink" in the pattern
as you progress to the next level.

-- Everett Kaser <everett@cv.hp.com> (a puzzle he learned from his
   father)

[ Editor: the solutions for more chips were not posted. ]

[ Editor: the solutions for more chips, posted by Everett Kaser in
October 1994, and a reply that preceded them. ]

Wei-Hwa Huang (whuang@cco.caltech.edu) wrote:
:   AABB  ->  AA   ->  A
:             BB       B
:                      A
:                      B
: (No one said the pieces had to stay on a line)

Yes, pretty cute! :-)

Here's my solutions to the poker chip game, levels 3 through 10:

Place X white poker chips in a row followed by X red poker chips.  So,
for X=3, it would look like this:

 WWWRRR

In X moves (in this case, in 3 moves) convert the above into
alternating red and white:

 RWRWRW

The chips will not necessarily end up in the same "places".  They may
be slid a little bit left or right as a result of the moves.

A "move" consists of placing two fingers on two adjacent chips and
sliding them together towards you (out of the line), then sliding them
left or right until there is a "two-chip space" into which they can be
slid.  In the case above, there are five possible initial pairs and two
possible destinations for each pair (the left end or the right end),
giving 10 possible initial moves.

As the number of chips grow, there may be multiple solutions that are
not simple mirrors of each other, I don't know.  But, one solution for
each where 3 <= X <= 10 are:

    WWWRRR
  RRWWWR
RWR  WWR
RWRWRW

  WWWWRRRR
RRWWWWR  R
RRWW  RWWR
R  WRWRWWR
RWRWRWRW

  WWWWWRRRRR
RRWWWWWRR  R
RRW  WWRRWWR
RRWWRW  RWWR
R  WRWRWRWWR
RWRWRWRWRW

  WWWWWWRRRRRR
RRWWWWWWRRR  R
RRWWW  WRRRWWR
RRWWWRRWR  WWR
RRWW  RWRWRWWR
R WRWRWRWRWWR
RWRWRWRWRWRW

  WWWWWWWRRRRRRR
RRWWWWWWWRRRR  R
RRWW  WWWRRRRWWR
RRWWRRWWWR  RWWR
RRWWR  WWRRWRWWR
RRWWRWRW  RWRWWR
R  WRWRWRWRWRWWR
RWRWRWRWRWRWRW

  WWWWWWWWRRRRRRRR
RRWWWWWWWWRRRRR  R
RRWWW  WWWRRRRRWWR
RRWWWRRWWWRR  RWWR
RRWWWRRW  RRWWRWWR
RRWWWRRWRWR  WRWWR
RRWW  RWRWRWRWRWWR
R  WRWRWRWRWRWRWWR
RWRWRWRWRWRWRWRW

  WWWWWWWWWRRRRRRRRR
RRWWWWWWWWWRRRRRR  R
RRWWWW  WWWRRRRRRWWR
RRWWWWRRWWWRR  RRWWR
RRW  WRRWWWRRWWRRWWR
RRWWRWRRWWWRRW  RWWR
RRWWRWR  WWRRWRWRWWR
RRWWRWRWRW  RWRWRWWR
R  WRWRWRWRWRWRWRWWR
RWRWRWRWRWRWRWRWRW

  WWWWWWWWWWRRRRRRRRRR
RRWWWWWWWWWWRRRRRRR  R
RRWWW  WWWWWRRRRRRRWWR
RRWWWRRWWWWWRR  RRRWWR
RRWWWRRWW  WRRWWRRRWWR
RRWWWRRWWRRWRRWWR  WWR
RRWWWRRW  RWRRWWRWRWWR
RRWWWRRWRWRWR  WRWRWWR
RRWW  RWRWRWRWRWRWRWWR
R  WRWRWRWRWRWRWRWRWWR
RWRWRWRWRWRWRWRWRWRW

-- Everett Kaser <everett@cv.hp.com>