Mathematical_Recreations-Kraitchik-2e

~40 Mathematical Recreations
the columns of a chessboard of order n from left to right, and
its rows from bottom to top, every cell will correspond to
just one pair of numbers ex, y), the one which gives the number
x of the column and the number y of the row in which it
lies. Thus each permutation of n objects corresponds to a
set of n cells of our chessboard, so selected that no two cells
lie in the same orthogonal. The graphical representation
may be completed by marking the selected cells on a set of
chessboards, one for each permutation. Figure 112 gives the
representation of all permutations of three things, namely
123,132,213,231,312,321.
2. THE PROBLEM OF THE ROOKS
As we have pointed out, the graphical representation of
the permutations of n different things constitutes a set of
solutions - and the complete set - of the problem of arranging
n objects on a chessboard of order n so that no two
of them lie in the same orthogonal. While this is not quite
the problem of the queens, it is precisely the corresponding
problem for rooks - to place n rooks on an nth-order chessboard
so that no one of them can take any other in a single
move. Before going on to the problem of the queens let us
examine the solutions of this problem in more detail, particularly
in respect to their geometric properties.
In dealing with magic squares we agreed (p. 144) to consider
as equivalent any solutions which could be derived from
each other by certain simple geometrical transformations of
the square - namely, rotation in its own plane through a
right angle in the counterclockwise direction, reflection in a
mirror, and repetitions and combinations of these operations.
We make the same agreement here. By means of these
transformations and their effect on the solutions we are able
to make a very simple and useful classification of the solu-