The noble gas that precedes a given partial electron configuration must <em>itself </em>have an electron configuration that is complete <em>up to </em>the partial electron configuration. The noble gas's electron configuration should, when fully written out right before the partial electron configuration, give us a valid electron configuration for some element.
For the first series, the highest principal energy level has the number 4, so our noble gas should <em>at least </em>be one that is in the third period (numerically, the energy level is the same as the period number). That noble gas would be argon. The partial electron configuration given is not that of a noble gas (note: all noble gases have an electron configuration that contains <em>N</em>p⁶, where <em>N </em>= the highest principal energy level). So, the noble gas that appropriately precedes our first partial electron configuration is [Ar].
Argon's electron configuration is 1s²2s²2p⁶3s²3p⁶. Using the Aufbau Principle, 4s² would correctly follow 3p⁶. [Ar]4s²3d¹⁰4p² is equivalent to writing out 1s²2s²2p⁶3s²3p⁶4s²3d¹⁰4p²; either way, this would happen to be the electron configuration of germanium.
Now that we hopefully have our fundamentals down, we can apply them to figure out the noble gases that precede the remaining partial electron configurations.
[Kr]5s²4d¹⁰5p⁵: This is the electron configuration of iodine.
[He]2s²2p⁵: This is the electron configuration of fluorine.
[Xe]6s²4f¹⁴5d¹⁰6p²: This is the electron configuration of lead.
[Ne]3s²2: This is the electron configuration of magnesium.
