Hitler and Goering

Add Hitler to Goering and get—what? Letters substitute for digits in the sum below. The same letter stands for the same digit wherever it appears, and different letters stand for different digits.

The slightly inappropriate addition puzzle

Find the digits the letters represent.

(Source: Brain Puzzler's Delight by E.R. Emmet)

The key to getting started is focusing on the leftmost column, where the result of the addition is an H. This can only be a one carried over from the column to the right, so H is equal to one. Then we turn our attention to second leftmost column. Recalling that this column carries over into the last, the only possibility with only one number in the column is for G to be nine with a carryover from the third leftmost column, adding to ten, making T zero. In turn, that carryover from the third leftmost column is only possible if—remembering that each letter stands for a different digit—there was a carryover from the fourth leftmost column and if O is equal to eight. Four letters are already knocked out!

Now look at the rightmost column. This contains two known columns and one unknown that can be solved. G (nine) adds with R, resulting in H (one) and a carry digit, meaning that G plus R equals 11. This means that R is equal to two. The middle column that contains R can then be figured out. T (zero) adds with R (two), resulting in L. Again, because each of the letters stands for a different digit, the only way this is possible is if there was a carryover from the column to the right, with the result that L is equal to three.

The last three letters fall into place pretty quickly now. The least significant digit of L (three) plus I is H (one), meaning that they added up to 11. Eight has already been used up by O, so I is equal to seven, and there was a carryover from the column to the right. In turn, the least significant digit of I (seven) plus E is L (three), meaning they added up to 13, which is only possible if E is equal to six. Finally, the least significant digit of E (six) plus N is H (one) again, meaning they added up to 11. Recall that there is a carry digit from the rightmost column, so N is equal to four.

Final answer: 1,170,362 plus 9,862,749 equals 10,033,111.