A Wichmann ruler is a ruler obtained from one of the Wichmann (1963) constructions below. Its marks lie at integer positions and measure every integer distance up to its length. Such a ruler need not use the minimum possible number of marks, as required of a sparse ruler. Leech (1956) and Wichmann (1963) gave bounds and constructions. Modern computer analyses include Robison (2014) and Pegg (2020), but exact minimum mark counts remain unknown for many lengths.
An example sparse ruler can be given by the marks 
0, 1, 2, 3, 27, 32, 36, 40, 44, 48, 52,
55, 58
,
with consecutive differences between marks 
1, 1, 1, 24, 5, 4, 4, 4, 4, 4, 3, 3
. A difference representation with repeated gaps has a shortened
form written as 
.
This notation leads to the major 
and minor 
Wichmann constructions, where 
and 
are nonnegative integers
and powers denote repetition of consecutive gaps:
 |
 |
The maximal length for a Wichmann ruler with 
segments (
marks) is 
for 
. For 
, 3, ..., these have lengths 3, 6, 9, 12, 15, 22, 29, 36,
43, 50, 57, 68, 79, ... (OEIS A289761). An
optimal sparse ruler has the greatest length for
a given number of marks. Among known greatest lengths, the exceptions to attainment
by Wichmann rulers are 1, 13, 17, 23, and 58 (OEIS A349978).
This is an existence assertion and does not imply that every optimal sparse
ruler is a Wichmann ruler. It is conjectured that for every number of marks greater
than 13, a Wichmann ruler attains the greatest possible length (Saarela and Vanhatalo
2024, p. 10).
by Differences." J. London Math. Soc. 31,
160-169, 1956. 
Pegg, E. Jr. "Wichmann-Like Rulers." Wolfram Demonstrations
Project. 2019.