2. In the addition shown, each of the letters $D, O, G, C, A$ and $T$ represents a different digit.
$$
\begin{array}{r}
D O G \\
+C A T \\
\hline 1000
\end{array}
$$

What is the value of $D+O+G+C+A+T$ ?
Solution
We first note that when we add two different single digits, the result is between $0+1$ and $8+9$, that is, the sum is between 1 and 17 .

Starting with the 'ones' column of the addition in the question, $G$ and $T$ add to give a result that is between 1 and 17 and with 0 as its 'ones' digit, so their total is 10 , that is,
$$
G+T=10 .
$$

Moving to the 'tens' column, we have to include the 1 carried from the 'ones' column so that
$$
O+A+1=10,
$$
that is,
$$
O+A=9
$$
and therefore 1 is carried forward.
Similarly, in the 'hundreds' column, we include the 1 carried forward and have
$$
D+C+1=10
$$
so that
$$
D+C=9 .
$$

Summing the three equations (3) to (1), we obtain $D+O+G+C+A+T=28$. Is a solution possible?
$$
\begin{array}{r}
321 \\
+679 \\
\hline 1000
\end{array}
$$