Arrange 6 coins in two rows of three coins nestled together; what is the minimum number of slide moves to...

geometry/coins.to.hexagon New

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

Problem

Arrange 6 coins in two rows of three coins nestled together; what is the
minimum number of slide moves to produce a circle of coins?

For numbering convenience, the top row extends to the left of the bottom
row.  The top row are coins 1, 2 and 3, left to right, the bottom row
has coins 4, 5 and 6.  All coins are flat on the table.  Notice that
coin 1 touches coins 2 and 4, and coin 2 touches each coin except 6.

Now, make a sequence of slide moves, sliding one coin at a time to a new
position.  Each coin must be slid without lifting it above the other
coins, or causing any of the other coins to move.  Additionally, the
coin slid must stop in a position touching at least two other coins (to
insure accurate placement).

What is the minimum number of such moves to produce a circle of coins?
The circle is equivalent to six coins surrounding a central coin,
without the central coin.

There is not intended to be any trickiness involved with this.  All the
coins are the same size, etc.

The minimum number of moves required is less than 10 and greater than
2.

-- Don Charnley <charnley@interval.com>

[ Editor: the first line joins Charnley's first sentence and his
question. ]

Show solution

> 4 moves.
>
> 1. Move #2 so it touches #3 and #6.
> 2. Move #6 so it touches #3 and #2 above instead of below.
> 3. Move #2 so it touches #3 and #6 on the left instead of the right.
> 4. Move #6 to complete the circle by touching #1 and #2.
>
>   Start     Step 1     Step 2      Step 3     Step 4
>
>                             6         2 6      6 2
>   1 2 3     1   3 2    1   3 2     1   3      1   3
>    4 5 6     4 5 6      4 5         4 5        4 5

3 moves.

1.  1 2 3
       4 5
        6

2.  1 2 3
     4   5
        6

3.    2 3
     4   5
      1 6

-- Dr. Pete Gieser <pete@pog.ufl.edu>

No it isn't [Editor: an original puzzle], Don.  IIRC, it appeared in
Martin Gardner's column in the Scientific American about 30 years ago.
The answer is 3 and there is a very nice trick such that you can show
someone how you do it and replace the coins and they still find it
difficult.  More of that later.

-- Stan Armstrong <Stan@stanleya.demon.co.uk>

This is almost the same as a puzzle in Martin Gardner's Mathematical
Diversions:

Move a pyramid of six pennies into a hexagon in the smallest number
of moves.  A move slides a penny (without disturbing any others) to
a position in which it touches two others.

The answer to MG's problem (below) creates Don's problem on the first
move, so just use the last three for Don's.

Numbered 1,23,456:  1->2,4;  4->5,6;  5->1,2;  1->4,5.

-- Ken Duisenberg <kend@rose.hp.com>

[ Editor: the 4-move answer quoted by Gieser was from an unnamed poster
in the thread. ]