The following problem is from the very beginning of the book and it even has a solution with explanation. However, I keep banging my head against the wall unable to understand the reasoning behind it. I would be grateful if you could explain his reasoning in a very detailed way since I cannot understand the reasoning behind every of his assumptions.
I have uploaded the problem here http://tinypic.com/r/o018id/6.
When you add two numbers, the most you can carry is $1$. This is because the maximum of each is $9$ (and here we know they are different, so one is no more than $8$) plus you might carry $1$ in from the place to the right. $9+9+1=19 \lt 20$