What is the maximal value of the expression $\overline{abc}-(a^3+b^3+c^3)?$ Let $\overline{abc}$ be $3$-digits number. What is the maximal value of the expression $\overline{abc}-(a^3+b^3+c^3)?$
Computer gives the answer $396$ for the number $620$ but I have  no idea how to do it by hand.

You can write $\overline{abc} = 100a + 10 b + c$. Then $$\overline{abc} - (a^3 + b^3 + c^3) = a(100-a^2) + b(10 - b^2) + c(1-c^2).$$ You just need to find the digits $a,b,c$ that maximize each term individually. 
