A sparse ruler is described by a finite sequence g=(g1,…,gn) of positive integers called segment lengths or gaps. Its marks are the partial sums m0=0<m1<⋯<mn=L where mi=g1+⋯+gi, and L=g1+⋯+gn is the length of the ruler. A distance k∈{0,1,…,L} is measurable if k=mj−mi for some marks mi,mj.
A sparse ruler is complete if every distance k∈{0,1,…,L} is measurable. A complete ruler is minimal if no complete ruler of the same length has fewer segments. A complete ruler is maximal if no complete ruler with the same number of segments has greater length. A ruler is optimal if it is both minimal and maximal.
For natural numbers r and s, the Wichmann ruler W(r,s) is the sparse ruler with segment-length sequence
1r,(r+1),(2r+1)r,(4r+3)s,(2r+2)r+1,1r,
where ab denotes b consecutive segments of length a. This ruler has 4r+s+2 segments and length 4r(r+s+2)+3(s+1).
Prove that if g is an optimal sparse ruler with more than 13 segments, then there exist natural numbers r and s such that g equals W(r,s) or the reverse of W(r,s).