Whisper Words of Wisdom

To minimize the function, we must be careful with where we put the "big" values. In this problem it is very simple: we can eliminate them by letting $GHIJ = 0 \times 9 \times 8 \times 7 = 0$ .

This leaves us with minimizing $A+BC+DEF$ , with numbers $1 \rightarrow 6$ .

Because $6$ taking part in $DEF$ would make it really big, we'll allow $A=6$ . To minimize the three-digit product, $DEF$ , let $F =1$ . We are left now with $BC+DE$ and the numbers $2 \rightarrow 5$ .

The minimum value for $BC+DE$ is held when the "edge" numbers (first with last, second with penult, third with antepenult, etc...) are combined. The order of the numbers will thus be $\; (2,5,3,4)$ .

We finally get that the minimum value of $A+BC+DEF+GHIJ$ is $\; \boxed{28.}$

PS: Go check a very similar problem , submitted by Chung Kevin .