Preface:
The initial line segment is partitioned with “obligatory” points (they should be mesh nodes), which can both be or not be points of thickening. On each part of the segment the method of subpartitioning, described below, can be applied.
Let 
– be the minimum acceptable step, 
– be the maximum acceptable step and let 
be the length of the segment, where 
. Then the minimal possible number of steps equals – 
, and the maximal possible is – 
. If 
, then the problem statement is incorrect. Assume that 
.
If both edges of the segment do not require thickening, we can take a uniform partition with any number of steps 
so that 
.