To arrange the ions Li⁺, Mg²⁺, K⁺, and Al³⁺ in order of their increasing radii, we need to consider several factors, primarily the atomic number (i.e., the number of protons in the nucleus), the number of electrons, and the effective nuclear charge experienced by the outer electrons.

Here is a brief overview of each ion:

• Li⁺: This is a lithium ion with one electron removed, leaving it with 2 electrons. The original lithium atom has an atomic number of 3. The loss of one electron increases the effective nuclear charge experienced by the remaining electrons, pulling them closer and reducing the size of the ion compared to the neutral atom.
• Mg²⁺: This is a magnesium ion with two electrons removed, leaving it with 10 electrons. Its atomic number is 12. This ionization results in a smaller ionic radius, as the loss of two electrons greatly increases the effective nuclear charge, causing the electrons to be held more tightly and closer to the nucleus.
• K⁺: This is a potassium ion with one electron removed, leaving it with 18 electrons. It has an atomic number of 19. Like Li⁺, losing one electron increases its effective nuclear charge, but the effect is less pronounced compared to Li⁺ and Mg²⁺ because potassium starts with more electron shells.
• Al³⁺: This is an aluminum ion with three electrons removed, leaving it with 10 electrons. Its atomic number is 13. Removing three electrons substantially increases effective nuclear charge, thereby significantly decreasing the ionic radius.

When arranging ions in order of increasing ionic radius, it's also critical to consider periodic trends:

1. Within a period, as the nuclear charge increases and the number of electrons remains similar (across cations), the radius decreases.
2. Across periods, ions with more electron shells tend to be larger if the number of missing electrons (from being cations) is the same or similar.

Hence, considering that Li⁺, Mg²⁺, and Al³⁺ are found in Period 2, and K⁺ is in Period 4, we can ascertain that K⁺ will generally be larger than those in Period 2. Within Period 2, the increasing removal of electrons (with Al³⁺ losing most) implies that Al³⁺ should be the smallest, followed by Mg²⁺ and then Li⁺.

Thus, the ions in order of increasing radii are:

$$ \text{Al}^{3+} < \text{Mg}^{2+} < \text{Li}^+ < \text{K}^+ $$

This arrangement follows both the periodic trend of decreasing radius with increasing nuclear charge (and ionic charge) in a period and the general increase in size with additional electron shells across periods.