Wichmann Columns

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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


and in ruler form,


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

can be extended to

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.


Contributed by: Ed Pegg Jr (January 2020)
Open content licensed under CC BY-NC-SA



[1] E. Pegg Jr. "Wichmann-like Rulers" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/WichmannLikeRulers.

[2] E. Pegg Jr. "Sparse Rulers" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/SparseRulers.

[3] B. Wichmann, "A Note on Restricted Difference Bases," Journal of the London Mathematical Society, s1–38(1), 1963 pp. 465–466. doi:10.1112/jlms/s1-38.1.465.


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