A perfect ruler of length
n
is a ruler with a subset of the integer markings
{
0
,
a
2
,
⋯
,
a
n
}
⊂
{
0
,
1
,
2
,
…
,
n
}
that appear on a regular ruler. The defining criterion of this subset is that there exists an
m
such that any positive integer
k
≤
m
can be expressed uniquely as a difference
k
=
a
i
−
a
j
for some
i
,
j
. This is referred to as an
m
-perfect ruler.
A 4-perfect ruler of length
7
is given by
{
0
,
1
,
3
,
7
}
. To verify this, we need to show that every number
1
,
2
,
3
,
4
can be expressed as a difference of two numbers in the above set:
1
=
1
−
0
2
=
3
−
1
3
=
3
−
0
4
=
7
−
3
An optimal perfect ruler is one where for a fixed value of
n
the value of
a
n
is minimized.
