Unstable vs. stable nuclei plotted on a graph The enclosed graph shows the number of Protons on the x-axis (Z), and the number of Neutrons on the y-axis for all elements (N). Stable combinations are marked by black squares, whereas unstable ones are marked by grey squares. I understand why there is a grey "line" located above the black "line", as adding electrically positive, mutually repelling, protons would make the nucleus less stable.
However, I do not understand why there is a grey "line" to the right of the black "line": isn't adding neutrons supposed to make a nucleus more stable (as a result of the additional Strong Force and greater distance between the protons)?
Can anyone help out?
Thank you!

from Yoram Kirsh, Fundamentals of Physics B, Tel Aviv, 1998, p. 111.
 A: What your analysis is missing is that the nuclear attraction between a neutron and a proton is somewhat larger than the attraction between two neutrons or two protons. In nuclear physics this difference is called the symmetry energy.  Because of this symmetry interaction the most tightly bound nuclei result from a balanced competition between the attractive symmetry energy and the coulomb repulsion between protons. As you deviate from this balance by having too many or too few neutrons, the resulting nuclei are less stable.
A: Late to the party here, but FWIW…
My recollection from graduate school decades ago — and what I now teach my students, hopefully without doing excessive violence to a more nuanced reality, is that when the ratio of neutrons to protons becomes too large, neutrons have a statistically large enough probability of finding themselves far enough from nearby protons that they can decay (via the weak interaction) into proton-electron pairs, a process called "beta decay". In a stable nucleus, the relative proximity of protons keeps the neutrons from decaying.
And perhaps this answer is actually in agreement with Lewis Miller's, as it could be the "symmetry energy" that makes a neutron near a proton immune to beta decay. It would be energetically favorable for a lone neutron to decay into a proton, electron, and antineutrino, but energetically disfavorable for one close enough to protons to enjoy the symmetry energy benefit.
