smc

We start by noting that there are $10$ letters, meaning there are $10$ digits in total. Listing them all out, we have $0,1,2,3,4,5,6,7,8,9$. Clearly, the most restrictive condition is the consecutive odd digits, so we create casework based on that. Case 1: $G$, $H$, $I$, and $J$ are $7$, $5$, $3$, and $1$ respectively. A cursory glance allows us to deduce that the smallest possible sum of $A+B+C$ is $11$ when $D$, $E$, and $F$ are $8$, $6$, and $4$ respectively, so this is out of the question. Case 2: $G$, $H$, $I$, and $J$ are $3$, $5$, $7$, and $9$ respectively. A cursory glance allows us to deduce the answer. Clearly, when $D$, $E$, and $F$ are $6$, $4$, and $2$ respectively, $A+B+C$ is $9$ when $A$, $B$, and $C$ are $8$, $1$, and $0$ respectively, giving us a final answer of $\boxed{8}$