The problem of finding a sparse ruler of a given length was solved partially by Leech (1956) and mostly by Wichmann (1963). However, the power of Wichmann's solution wasn't realized until a modern computer analysis by Robinson (2014) and Pegg (2020).
An example sparse ruler can be given by the marks 
0, 1, 2, 3, 27, 32, 36, 40, 44, 48, 52,
55, 58
,
with differences between marks 
1, 1, 1, 24, 5, 4, 4, 4, 4, 4, 3, 3
. A ruler like this having repeated values has a shortened
form written as 
.
This notation leads to the major 
and minor 
Wichmann constructions:
 |
 |
The maximal length for a Wichmann ruler with 
marks is 
. For 
, 2, ..., these have lengths 3, 6, 9, 12, 15, 22, 29, 36,
43, 50, 57, 68, 79, ... (A289761). An optimal
sparse ruler has the greatest length for a given
number of marks. Except for lengths 1, 13, 17, 23 and 58 (A349978),
all known optimal sparse rulers are Wichmann constructions.
The optimal ruler conjecture posits that all optimal sparse rulers other than these
exceptions are Wichmann constructions.
by Differences." J. London Math. Soc. 31,
160-169, 1956.Pegg, E. "Hitting All the Marks: Exploring New Bounds
for Sparse Rulers and a Wolfram Language Proof." 2020.