International programs have been conducted over the last four decades to quantify the exposure of atom bomb survivors from Hiroshima and Nagasaki. Unfortunately, the quest for accurate gamma-ray and neutron exposure doses of atom bomb survivors has proven illusive. Efforts in the most recent of these programs, designated as Dosimetry System 1986 (DS86), have revealed a serious and persistent discrepancy between neutron transport calculations and thermal neutron activation measurements at the Hiroshima site, which will be called the DS86 neutron dosimetry enigma. It is established that this enigma is a complex puzzle that precludes simple solutions. This conclusion is deduced through the identification of a number of missing pieces to the puzzle. Implications and conclusions that can be inferred from these missing puzzle pieces are advanced.