WOLFRAM NOTEBOOK

WOLFRAM|DEMONSTRATIONS PROJECT

Sparse Rulers

​
pick length
58
sort by
length
start
random seed
show ruler
length
58
marks
13
excess
0
shortform
1
3
24
1
5
1
4
5
3
2
excessfraction
1
12
A sparse ruler is a rod of integer length
n
with a minimal number of marks so that all distances 1 to
n
can be measured. Many lengths, such as 36, have unique rulers with a twin by subtracting marks from
n
, as seen here:
36-{0,1,3,6,13,20,27,31,35,36}={36,35,33,30,23,16,9,5,1,0}
.
This Demonstration has many but not all sparse rulers. Up to length 198, there are
106520
sparse rulers, with
31980
of them for length 59 alone. Many of the longest known sparse rulers for a particular number of marks are Wichmann rulers. A Wichmann ruler generator is given in the Initialization.
In a Golomb ruler, distances can be missing but none can be repeated.
In a sparse ruler, distances can be repeated but none can be missing.
In a difference set, modular distances cannot be missing or repeated.
For a minimal sparse ruler of length
n
with
m
marks, let the excess
e
be
e=m-round(sqrt(3n+9/4))
. The excess is always 0 or 1.
The "show ruler" box is a toggle for "show grid" of excess to length 2103.
Wolfram Cloud

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.


I understand and wish to continue anyway »

You are using a browser not supported by the Wolfram Cloud. Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.