# Can two nuclei with the same number of protons and neutrons be different?

In school, the nucleus is often portrayed with this caricature

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My question is: can two nuclei with the same number of protons and neutrons be different? In other words, can the protons and neutrons be "arranged" differently? Is there a rule(s) on how a stable nucleus should arrange its protons and neutrons? Or is there no meaning on arrangement of nucleus?

Each energy level can be occupied by two particles of the same type and so when the bottom energy level (shell) is filled with 2 neutrons and 2 protons a very stable nucleus is formed $(^4_2 \rm He)$. When other shells are filled the nucleus tends to be very stable as per the filling of electron shells with the result of the very stable (chemically almost inert) noble gases.