Torsten Sillke, FRA, 1998-10
Double contact moves:

   o            o o
  o o     ->   o . o
 o o o          o o


Solution:

                     o
                    o o
                   o o o

               /       \       \
             /          \        \
           /             \         \

 +---------------------------------------------------------------+ all moves
 |     o o       o         o o       o o       o o           o   | possible
 |    o o       o o o   !>  o    <! o o   !>  o o o       o o    | except for
 |   o o           o o     o o o     o o           o     o o o   |   !>
 +---------------------------------------------------------------+
                                                |
                                               o o
                                              o . o  --> A
                                                 o o
                                                |
                                               o o
                                              o . o
                                               o o
                                        
 If we allow only connected positions we will get at most
 the 82 hexahexes. But the directed graph of double contact moves
 is not connected. Some small components are:

 component1:

   o o o o o o

 component2:
  o
   o o o o o
  o

 component3:
  o
   o o o o
  o
 o

 component4:
 o
  o
   o o o
  o
 o

 component5: 

  o             o o           o o o
   o o o o o --    o o o o --      o o o

 component6: (edges not shown)

  o         o        o o      o o       o         o
   o         o o        o        o o     o o o     o o o o
    o o o o     o o o    o o o      o o       o o         o

 Is there a further small component?
 Determine the big component. Is it strongly connected?

 If we allow unconnected positions we will get infinitely many
 positions. Then we can transform the ring into the 6-bone.

  o   o
 o o o o  = A
    |
      o o
 o o o o
    |
        o
 o o o o o
    |
        o o
 o x x x o     (one of the x is empty)
    |
          o
 o x x x o o   (one of the x is empty)
    |
 o o o o o o

 Moving to A is not reversable. 
 Find a short sequence of moves for the inverse transformation.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Double contact moves:

  o     o        o o o
  o o o o   ->   o   o
  o     o        o o o

 solution (5 moves, [Harry Langman] right angle double contact)

  A     F    A F        A F H      A F H      A F H      A F H
  B D E G -> B D E G -> B D E G -> B D   G -> B   D G -> B   D
  C     H    C     H    C          C E        C E        C E G

 solution (4 moves)

  A     F    A   D F    A   D F    A F D      A F D
  B D E G -> B   E G -> B   E   -> B   E   -> B   E
  C     H    C     H    C   G H    C   G H    C G H

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Double contact moves: in 4 moves

   o           o o
  o o         o   o
 o   o   ->  o     o
o o o o       o o o


References:

- M. Delannoy;
  La Nature, June 1887, p10
  (coin shifting problems: abababab -> bbbbaaaa)

- A. P. Domoryad;
  Mathematical Games and Pastimes,
  Pergamon, 1963
  (p127-128 coin shifting problems: abababab -> bbbbaaaa)

- Martin Gardner;
  The Second Scientific American Book of
  Mathematical Puzzles and Diversions
  Simon & Schuster (1961)
  Chapter 5 Problem 7: The Sliding Pennies 
  triangle <-> circular

- Martin Gardner;
  Mathematical Carnival
  A. Knopf (1975)
  Chapter 7: Penny Puzzles
  rhomboid <-> circular

- Henry Ernest Dudeney;
  Modern Puzzles and How to Solve Them,
  London 1926, p96-97, 182
  rhomboid <-> circular

- Henry Ernest Dudeney;
  536 Puzzles and Curious Problems,
  New York, 1967 (ed. by Martin Gardner)
  No. 383 The six pennies p138, 343

- E R Berlekamp, J H Conway, R Guy;
  Winning Ways,
  for your mathematical play, Vol 2, (Part 4: Diamond)
  Academic Press, London, 1982
  (german: Gewinnen, Strategien f\"ur mathematische Spiele,
   Band 4: Solitairspiele, Vieweg Verlag, Braunschweig, 1985)
  pages after the german translation
  p61-2: Ein Solitaire-\"ahnliches Puzzle und einige M\"unzen-schiebe-Probleme
         Bild 2.24 Ringelreihen-Wechsel
         rhomboid <-> circular
  p113:  Bild 2.64 (L\"osung)

- Pieter van Delft, Jack Botermans;
  Denkspiele der Welt,
  Hugendubel (1977)
  (orig: Creative Puzzles of the World)
  Sektion: Positionpuzzles, Das genaue Hexagon, p178
           triangle <-> circular (4 moves)
  Sektion: Positionpuzzles, Zwei Verwandlungen, p178
           H <-> O ( -> 5 moves, <- 7 moves)

- Harry Langman;
  Scripta Mathematica 19 (Dec. 1953) 242
  H <-> O ( -> 5 moves, <- 7 moves)

- Heinrich Hemme;
  Das Problem des Zw\"olf-Elfs,
  Vandenhoeck & Ruprecht, G\"ottingen, 1998, ISBN 3-525-40736-X
  60. M\"unzschieberei  triangle <-> circular
  61. Eine zweite M\"unzschieberei  H -> O 
  61. Eine dritte M\"unzschieberei  H <- O