A game played on a board of a given shape consisting of a number of holes of which all but one are initially filled with pegs. The goal is to remove all pegs but one by jumping pegs from one side of an occupied peg hole to an empty space, removing the peg which was jumped over.
One of the most common configurations is a cross-shaped board with 33 holes. All holes but the middle one are initially filled with pegs. Strategies and symmetries
are discussed by Gosper et al. (1972). Berlekamp et al. (1982) give
a complete solution of the puzzle.
There is also triangular variant with 15 holes (where 15 is the 5th triangular number)
and 14 pegs (Beeler 1972). Numbering hole 1 at the apex of the triangle and thereafter
from left to right on the next lower row, etc., the following table gives possible
ending holes for a single peg removed (Beeler 1972). Because of symmetry, only the
first five pegs need be considered. Also because of symmetry, removing peg 2 is equivalent
to removing peg 3 and flipping the board horizontally. Bell (2008) surveys this problem,
which dates back to Smith (1891). Bell gives necessary and sufficient conditions
for this problem to be solvable and a simple solution algorithm.
remove
possible ending pegs
1
1, , 13
2
2,
6, 11, 14
4
, 4, 9, 15
5
13
Kraitchik (1942) considered a board with one additional hole placed at the vertices of the central right angles.
)
and 14 pegs (Beeler 1972). Numbering hole 1 at the apex of the triangle and thereafter
from left to right on the next lower row, etc., the following table gives possible
ending holes for a single peg removed (Beeler 1972). Because of symmetry, only the
first five pegs need be considered. Also because of symmetry, removing peg 2 is equivalent
to removing peg 3 and flipping the board horizontally. Bell (2008) surveys this problem,
which dates back to Smith (1891). Bell gives necessary and sufficient conditions
for this problem to be solvable and a simple solution algorithm.
, 13
, 4, 9, 15