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Dark Satanic Mills on a Cloudy Day


DarkSatanicMillsOnACloudyDay

Plotting the excess values of sparse rulers in batches of maximal Wichmann rulers leads to a pattern called "Dark Satanic Mills on a Cloudy Day" by N. J. A. Sloane (OEIS A289761), illustrated above.

In particular, the maximal length for a Wichmann ruler with k marks is (k^2-(k (mod 6)-3)^2)/3+k, giving the sequence 0, 3, 6, 9, 12, 15, 22, 29, ... (OEIS A289761) for k=0, 1, .... So writing the excesses in blocks of length given by the differences of conseuctive values 3, 3, 3, 3, 7, 7, ... gives

 000 
000 
000 
000 
000 
0000000 
0000000 
0000000 
0000000 
0000000 
1000000 
01000000000

Plotting these values when rotated 90 degrees counterclockwise gives the Dark Satanic Mills on a Cloudy Day pattern. All the windows in the Dark Mill are Wichmann constructions. Due to this pattern, it is believed Wichmann (1963) solved the problem. However, of the black (E=1) excess values in this picture, only the leftmost six are verified (up to length 213), while the rest show the best constructions known. It is therefore an unsolved problem if clouds actually exist.


See also

Sparse Ruler, Wichmann Ruler

Portions of this entry contributed by Ed Pegg, Jr. (author's link)

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References

Sloane, N. J. A. Sequences A289761 and A326499 in "The On-Line Encyclopedia of Integer Sequences."Wichmann, B. "A Note on Restricted Difference Bases." J. Lond. Math. Soc. 38, 465-466, 1963.

Cite this as:

Pegg, Ed Jr. and Weisstein, Eric W. "Dark Satanic Mills on a Cloudy Day." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DarkSatanicMillsonaCloudyDay.html

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