ADE plus ODO

In the addition below, each letter stands for a digit, and different letters stand for different digits. Find them. $$\mathrm{ADE} + \mathrm{ODO} = \mathrm{DEED}.$$

Solution

Two three-digit numbers add to the four-digit $\mathrm{DEED}$, and the largest two three-digit numbers can make is under $2000$, so the leading digit is $1$: thus $D = 1$.

Now read the columns from the right. In the tens column the digits are $D$ and $D$, so $D + D + c_1 = 2 + c_1$, where $c_1$ is whatever carried out of the units. This has to end in the tens digit of the answer, which is $E$, with no carry onward (since $2 + c_1$ is at most $3$). So $E = 2 + c_1$. The units column reads $E + O$, ending in $D = 1$; that forces a carry, $c_1 = 1$, and $E + O = 11$. Then $E = 2 + 1 = 3$, so $O = 8$. Finally the hundreds column $A + O$ must give $E$ with a carry of $1$ into the thousands (which supplies the leading $D = 1$), so $A + O = E + 10 = 13$, giving $A = 5$. The unique solution is $$513 + 818 = 1331.$$