Wolfram Function Repository
Instantuse addon functions for the Wolfram Language
Function Repository Resource:
Find a nearminimal set of integer marks so that all integers up to a given length are represented as differences
ResourceFunction["Excess01Ruler"][length] finds a set of integers representing all differences up to length using a compact form for the result. 

ResourceFunction["Excess01Ruler"][length,"Full"] finds a set of integers representing all differences up to length. 
Find the marks for a sparse ruler of length 9:
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Check that all differences are covered:
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Find the marks for a sparse ruler of length 57:
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Check that the ruler is complete (i.e. all differences are covered):
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Show the intervals between marks in the full form of a length396 ruler:
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Split the differences:
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Make a series of lists from the compact form of a length396 ruler:
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Show the compact form:
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Verify that the compact form from the previous result has length 396:
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Show the full form for a length396 ruler:
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Show the marks for the first 51 rulers:
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Show the excess values for the first 51 rulers:
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Count the number of marks for a length58 ruler in full form or compact form:
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The excess of a ruler with length L and M marks is :
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An excess01 ruler has an excess of 0 or 1:
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A list of some rulers with a single mark less than those generated by this function:
Show the length999 ruler with one less mark:
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Show how the Leech upper bound for the number of marks in a complete ruler compares to the number of marks in functiongenerated rulers:
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Generate rulers of length n! and plot the number of marks:
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Show the default shortened form for a lengthgoogol ruler:
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Show the number of marks in a lengthgoogol ruler:
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Showing the full form of a googollength ruler is not recommended.
Generate reasonable rulers for various integer powers and calculate their excess:
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All rulers to length 213 are minimal. Show a pixel representation of these sparse rulers:
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The maximal length for a given number of marks is usually a Wichmann value:
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Arrange lengths in columns ending in Wichmann values and bold the lengths with excess 1:
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