Mathematical_Recreations-Kraitchik-2e

The Problem oj the Queens 241
tions. First let us examine the transformations themselves
more closely.
Denote a rotation through 90° by R, a reflection by S, and
the result of performing two transformations in succession
by the product (discussed below) of their symbols in the same
order. Then RR represents a rotation through 180° and may
be denoted by R2. Both R2R and RR2 correspond to a rotation
through 270°, so they may be denoted by Ra. In the
same way Rh-lR and RRh-l both represent rotations through
h right angles, so we shall denote them by Rh. However, R4,
rotation through 4 right angles, has no effect on the chessboard,
so we shall denote it by I, representing the identical
transformation; and higher powers may consequently be
reduced modulo 4. S may also be thought of as being the
result of a rotation through 180° in a plane vertical to that
of the board and about the middle vertical line of the board
as axis. SR, SR2, and SR3 are then seen to be the results of
similar rotations or reflections, with the upward main diagonal,
the middle horizontal and the downward main diagonal
as axes, respectively. One can readily verify that SR = Ras,
SR2 = R2S, and SRa = RS; and that each of these reflections
yields I when repeated. From the three relations
R4 = I, S2 = I, SR = RaS one can easily show that even unlimited
repetitions and combinations of Rand S yield only
the 8 distinct transformations: I, R, R2, R3, S, SR, SR2, SR3.
The symbols of these transformations have been combined
as by multiplication, and we shall call the combinations
(both of the symbols and the transformations they denote)
products. This mUltiplication is just like ordinary multiplication
except in one important respect: a product may not
be independent of the order of the factors. Thus R2R = RR2,
but RS '¢ SR. The set has the further important property
that the product of any two members of it lies in the set.
Such a set is called in mathematics a group - a concept