First, let's define isolectronic. Using prefixes, iso- means same or equal, and electronic in this case is referring to electrons. So we are essentially looking for other atoms that have the same number of electrons as argon. Using the periodic table, we should refer to the atomic number since the number of protons in a neutral atom is the same as the number of electrons. So what is Argon's atomic number? 18. This means that it has 18 electrons.
The next aspect to this question relies on your understanding of ions. Remember - what's the definition of an ion? An atom that has lost or gained electrons. Many students get confused on whether the atom will become an ion with a positive or negative charge when it gains or loses electrons. Here's my analogy. Since electrons are negative, think about negativity in life in general. If you lose negativity in your life (get rid of negative thinking, negative friends/influences), then you become a more positive person. If you gain negativity in your life, you only become more and more negative. Thus... If an atom gains electrons, it will become a negatively charged ion. If an atom loses electrons, it will become positively charged.
Look at the elements close to argon such as sulfur, chlorine, potassium, and calcium. Calcium has 20 electrons. In order to get to 18 electrons (like Argon), what would have to happen? Would it have to gain electrons to go from 20 to 18? Or would it have to lose electrons?
It would have to lose 2 electrons. What happens when an atom loses 2 electrons? It becomes a +2 charge. Calcium ions with a +2 are isoelectronic with Argon.
Now let's look at sulfur. Sulfur has 16 electrons. What would have to happen for sulfur to go from having 16 electrons to having 18 electrons like argon? Would it have to lose 2 electrons to go from 16 to 18? Or would it have to gain 2 electrons?
It would have to gain 2 electrons. What happens when an atom gains 2 electrons? It becomes a -2 charge. Sulfur ions with a -2 charge are isoelectronic with Argon.
Now you should have a better understanding to determine which elements as ions with +1 and -1 charges would be isoelectronic with argon. An ion with a +1 charge would have to lose only 1 electron to become isoelectronic with argon. That would have to be potassium. Potassium ions with a +1 charge are isoelectronic with Argon. An ion with a -1 charge would have had to gain only 1 electron to become isoelectronic with argon. That would have to be chlorine. Chlorine ions with a -1 charge are isoelectronic with Argon.
Answer in summary: Ca+2, K+, Cl-, S-2
