Mathematical Journeys - Saint Ann's School

58
the
above
case
would
be
described
as:
K2=2
M3=1
M4=1
This
notation
guarantees
that
with
a
given
l
and
given
M
and
K
values,
we
can
compute
p
by
subtracting
from
the
maximum
case
as
described
before.
We
developed
a
Diophantine
equation
(or
rather,
an
equation
template
of
sorts)
to
represent
the
situation.
Here
is
an
example
of
its
application:
C(l,
2)
‐
p
=
k2
C(2,2)
+
k3
C(3,2)
+
...
+
m3
[C(3,2)
‐
1]
+
m4
[C(4,2)
‐
1]
+
...
So
far,
so
good.
But
when
we
got
to
this
point,
we
realized
something
unsettling
that
made
this
problem
even
more
complicated
than
it
already
appeared.
There
can
exist
what
we
termed
“overlaps,”
i.e.
lines
that
are
counted
in
more
than
one
group
of
parallel
or
coincident
lines.
It
would
be
nice
if
the
product
of
all
M
and
K
indices
and
values
equaled
exactly
l,
but
this
is
not
always
the
case:
In
many
ways,
the
pattern
of
our
course
through
this
problem
was
marked
by
two
things:
classification
and
specialization.
We
built
an
entire
notational
system
and
used
it
only
to
solve
a
very
particular
special
case
of
the
original
problem.
But
in
the
process,
I
think
it’s
fair
to
say
that
the
development
of
the
questions
themselves
provided
a
great
deal
better
understanding
of
the
problem
than
I
had
at
outset.
Before
I
get
to
what
the
special
case
was,
I
need
to
introduce
the
“valance”
system
I
designed
to
deal
with
overlaps.
Here's
a
way
to
visualize
our
classification
system
for
configurations
of
lines,
while
accounting
for
overlaps
between
multiple
M/K
groups:
Draw
a
group
of
dots
to
represent
each
M/K
group.
Connect
these
dots,
or
valances,
with
“lines”
to
illustrate
the
overlaps.
However,
our
definition
of
“line”
is
not
what
one
might
expect:
consider
the
case
where
more
than
two
M/K
groups
share
a
line.
This
is
represented
in
the
new
system
as
lines
emanating
from
all
of
the
groups
involved,
and
converging,
becoming
a
single
“line.”
Each
valance
in
the
new
system
represents
a
line,
unless
it
is
connected
to
one
or
more
other
valances,
in
which
case
it
represents
the
same
line
as
do
those.
K2=1
M3=1
Here,
the
topmost
horizontal
line
belongs
to
both
a
pair
of
parallel
lines
and
a
group
of
3
coinciding
lines.