The single-digit solution can also be fond using the same iterative process. that's the way I did it, letting Excel do the dirty work. But it takes a bit of luck to find the right starting point. Most led to a situation where the iteration was cycling between two incorrect answers.
But there is an analytical way of getting to the answer with no iteration--just checking a very few possibilities.
Start with a few observations that you can easily see, but need to see how to use them.
1) the numbers in the blanks must add to 20.
2) the numbers in the blanks must all be at least 1
2a) Said another way, the numbers in the blanks can be stated as 1,1+a,1+b,1+c,1+d,1+e,1+f,1+g,1+h,1+i
3) There is a relationship between a and the number of letters that are non-zero.
And while I've omitted one important observation, I've probably said too much :-)