The 288-residue structure that Wrinch proposed was attractive in part because it fit with a principle that other researchers believed might underlie the structure of all proteins. The case had been put forward by Max Bergmann, an expatriate German researcher and respected biochemist, along with the same Carl Niemann who helped Pauling with his paper. The Bergmann-Niemann hypothesis seemed to show how a field as puzzling as protein structure might be explained with simple mathematical equations. They postulated that all proteins were formed from a certain number of amino acid residues expressed as powers of the integers two and three. This gave rise to the 288 amino acids in a cyclol cage, and its multiples, which were observed in fibrin (576 amino acids) and silk (2,592 amino acids). The simple formula 2ⁿ x 3m might, they thought, help unlock the secrets of proteins. "Everyone who is familiar with the history of protein chemistry may feel somewhat amazed on being confronted with a simple stoichiometry of the protein molecule," Bergmann wrote.

Pauling was not amazed. He was skeptical. He could see no good chemical reason for the Bergmann-Niemann formula, or for the 288-residue basic unit; while some proteins fell into the pattern, others, it seemed to him, did not. In any case, he distrusted the whole idea of "magic numbers" designed to explain natural phenomena. His work on crystal structures showed, if anything, how many ways nature could find to put things together. There was a sort of philosophical attraction in linking simple equations to natural phenomena, and in some cases these approaches worked, but Pauling did not see any chemical reason why it should work for proteins.

Then, in 1939, an associate of Bergmann's named William Stein showed definitively that some protein data could not be accommodated by the Bergmann-Niemann principle. Faced with the new information, even Bergmann was forced to abandon his idea. From that point on, protein researchers were freed to look for new answers that did not rely on magic numbers. And Pauling would lead the way.