I've been trying to understand the overview given for problem 9 on Project Euler and it mentions that the upper bound for iterating through possible values of a is (s - 3)/3 and for b it is (s - a)/2. I don't understand how they got these values. In googling for possible explanations, I found several places where they use s/3 as an upper bound for a and s/2 for b.
Now, s/3 for a I understand because a + b + c = s and since a < b < c. How do you get the others?
$a, b, c$ are integers, so that $a < b < c$ implies $b \ge a +1 $ and $ c \ge b+1 \ge a+2$. Therefore $$ s = a+b+c \ge a + (a+1) + (a+2) = 3a+3 $$ gives the first estimate. Similarly, $$ s -a = b+c \ge b + (b+1) = 2b +1 $$ gives the second estimate.