Two octahedral complexes of Mn³⁺ (d⁴) have measured magnetic moments of 4.8 BM and 3.0 BM…

Step 1. The first. $\mu^2 = 23.04$, so $n = -1 + \sqrt{24.04} = -1 + 4.90 = 3.90$, which rounds to 4.

Step 2. The second. $\mu^2 = 9.0$, so $n = -1 + \sqrt{10.0} = -1 + 3.16 = 2.16$, which rounds to 2.

Step 3. What the numbers mean. Four unpaired electrons is $t_{2g}^3 e_g^1$, high spin. Two unpaired electrons is $t_{2g}^4$, low spin: the fourth electron paired up in $t_{2g}$, so 4 electrons sit in $t_{2g}$.