$\underline{A}\ \underline{B}\ \underline{C}$ represent a three-digit base 10 number whose digits are $A$, $B$, and $C$ with $A \ge 1$. The minimum value of$ \underline{A}\ \underline{B}\ \underline{C} - (A^2 + B^2 + C^2) is :
$a=1$, $b=0$ and $c=9$.
Explanation:
Simplified the expression given above as $100a-a^2+10b-b^2+c-c^2$.
This expression contains three quadratic expressions,namely $100a-a^2$,$10b-b^2$,$c-c^2$.
You want to minimize the expression $100a-a^2+10b-b^2+c-c^2,$ which is the sum of three quadratics.
It is minimal when the three quadratics $100a-a^2,\qquad 10b-b^2,\qquad c-c^2,$ are minimal.
These are parabolas with their maxima at $50$, $5$ and $\tfrac{1}{2}$, respectively.
They are minimal when we are as far away from the maximum as possible. This gives us $a=1$, $b=0$ and $c=9$.
