There Are No Further Counterexamples to S. Piccard's Theorem

In 1977, G. S. Bloom, in the J. Combinatorial Theory, showed that Sophie Piccard's “theorem” had counterexamples for six-mark rulers. Subsequent research into finding additional counterexamples has focused on a variety of computer algorithms, such as searching the space of rulers with relatively few marks in an attempt to find another counterexample. Recent analytic effort has made use of Golomb's “Polynomial Method,” which made strides in eliminating specific types of rulers which cannot contain counterexamples. The question as to whether other larger length ruler counterexamples exist, however, was left unanswered. In this correspondence, a geometric manipulation of the “Polynomial Method” is used to demonstrate that no additional counterexamples are possible.