In general, examining the characteristics of the boards, for anything to happen
there must be a "reactive" set of pegs, on which a valid move can be performed,
and which consists of two or more adjacent pegs, and not three or seven in a
row such that none of the pegs can move, as in
  111        000
  000        000
0000000    1111111
0000000 or 0000000
0000000    0000000
  000        000
  000        000

Other single pegs may become involved as the moves performed bring other pegs
adjacent to them.

There cannot be more than 17 pegs in a final configuration, as below.  If there
are any more pegs than that, some will be adjacent, and unless all the holes
are filled, there will be a hole with two adjacent pegs next to it.
  101
  010
1010101
0101010
1010101
  010
  101

If there are only two pegs on the board, in order for the board to "reduce" to
one peg, the two pegs must be adjacent.  If they not adjacent, the board cannot
be reduced, which means that there is no win.  In order for a peg to be at
least one "hole" away, i.e. there is an empty hole between them, and for the
board to be reducible, there must be at least three pegs in the initial
configuration.  Here, holes are counted by the minimum number of empty holes
that are between two pegs, following the valid move directions.
If there are three pegs in an initial configuration, in order for the board to
be reducible, the pegs must be

  000        000
  000        000
0000000    0000000
0000100 or 0000000
0000100    0000000
  000        110
  001        001

in relation to each other.  In order for a peg to at least two holes from every
other one, and the board to be reducible, there must be at least four pegs, as
in
  000
  000
0000000
0000100
0011000
  000
  001
This is the "least" board, starting from a board with a peg in the right bottom
hole, that is reducible and has a peg two away from any others.  If there are
fewer than four pegs on the board and a peg is more than one hole away from any
other, the board is not reducible.  If there are fewer than 6 pegs on the board
and one is more than two holes away from any other, the board is not reducible.
  000
  001
0001100
0001000
0010000
  000
  001
If there are fewer than nine pegs on a board and one peg is more than three
holes away from any other, the board is not reducible.
  000
  001
0001011
0110000
1100000
  000
  001