An affine plane is an incidence structure of points and lines satisfying
three axioms. Any two distinct points lie on exactly one
line. Given a point 
not on a line 
, exactly one line through 
is parallel to 
. Finally, there are three noncollinear points.
A finite affine plane has order 
if every line contains 
points. It then has 
points, 
lines, and 
lines through each point.
Its lines split into 
parallel classes, each
containing 
mutually parallel lines that
partition the point set. The figure above shows the four
parallel classes of the affine plane of order 3.
The arithmetic in the labels is modulo 3.
Every finite field 
gives an affine plane whose points
are the pairs in 
and whose lines have equations 
or 
, where 
. Thus an affine plane exists for every prime
power order 
,
although not every affine plane is obtained from a field
in this way.
An affine plane of order 
is a resolvable block
design with parameters (
, 
,
1). Conversely, adjoining one point at infinity
for every parallel class and a line
at infinity containing these new points produces a
projective plane. Consequently, an affine plane
of order 
exists iff a projective plane
of order 
exists.
Interpreting the points as golfers and the lines in one parallel class as the groups in one round
gives a solution of the social golfer problem:
golfers can play in 
groups of 
for 
rounds, with every pair meeting exactly once.
