Mathematical_Recreations-Kraitchik-2e

CHAPTER TEN
THE PROBLEM OF THE QUEENS
MATHEMATICS owes many interesting problems to the game
of chess. Indeed the game itself is a single enormously complicated
mathematical problem that. has never been - and
probably never will be - completely solved. In this chapter
and the next we shall deal with certain problems suggested
by the game, but we hasten to assure any of our
readers who do not play chess that each of these problems
will be restated in nontechnical language, and that no knowledge
of chess is involved in their solution.
The problem from which this chapter derives its name is
that of placing eight queens on a chessboard so that no one
of them can take any other in a single move. This is a particular
case of the more general problem: On a square array
of n 2 cells place n objects, one on each of n different cells, in
such a way that no two of them lie on the same row, column,
or diagonal.
Since the problem is again one of arranging objects on the
cells of a square array, as was the problem of magic squares,
we shall use much of the terminology and many of the ideas
of Chapter Seven. In particular, n is the order of the problem,
and rows and columns will be referred to indifferently
as orthogonals. However, the term diagonal will usually be
restricted to mean a main diagonal or one of the two connected
pieces of a broken diagonal. The diagonals which go
upward from left to right will be called upward diagonals,
the others downward diagonals. The array of cells on which
the objects are to be placed will be called a chessboard of
order n.
~38