Much of the following explanation comes from the Demonstrations Wichmann-like Rulers [1] and Sparse Rulers [2].

A sparse ruler is a rod of integer length

with a minimal number of marks

so that all distances 1 to

can be measured.

For example, with a rod of length 9, let its five marks be at

with differences between the marks

. The differences 1 to 9 are

,

,

,

,

,

,

,

,

. Therefore, this rod is a sparse ruler.

In [3], Wichmann introduced what is now called the Wichmann ruler with differences (or mark spacing) given by the split form

.

The rule means

,

,

,

,

,

.

For example,

,

gives

as a split form. In differences form, this is

,

.

For a subset of

with

elements, define the excess

. For all

up to at least 300000, a sparse ruler exists with excess

or 1.

In [2], the "show grid" option seen in the third snapshot is known as the excess pattern.

A sparse ruler starting with

ones can be extended by up to

ones with an extra mark at the end.

For example, the sparse ruler

and still be a sparse ruler. This new ruler looks like

.

The new lengths above 68 are handled by differences

,

and

. Note that

is not a sparse ruler since the length 69 cannot be expressed as a difference.

This Demonstration generates a column in the excess pattern by using only sparse rulers made by the first two Wichmann recipes,

and

, and extending these two rulers.

Hover over a value to see the generated sparse rulers for that length.

Red indicates that a sparse ruler cannot be generated by

,

or extending them.

Blue indicates a generated sparse ruler with excess 0.

Green indicates a generated sparse ruler with excess 1.

Column 880 has some red values, one of them for length 257992.