Your dearest friend approaches, asking for a favor. They say it is imperative that you find all sets of consecutive positive integers whose sum is 100. How many such sets are there?
Solution
Solution by exhaustion:
Are there sets with two numbers? No, 49 + 50 and 50 + 51 won't work.
Are there sets with three numbers grouped around 33? No
Four? No
Five? Yes, 18 + 19 + 20 + 21 + 22 = 100.
Six? No
Seven? No
Eight? Yes, 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 = 100.
Etc., up to 14, where the numbers would be grouped around 7.
There can't be any strings longer than 14 numbers, because the string would be so long that the smallest number would be non-positive.
So there are two strings of consecutive positive integers whose sum is 100: one of five numbers, the other of eight, as shown above.
Whew, you did it. You tell your friend. They sigh in relief. They ask, how many minutes did it take you to solve that problem? You answer. They respond, saying that around 830 puppies are born every minute. This is a special fact because while you were solving this problem, a bunch of good dogs came into existence. You are alive at the same time as they are. By extension, you are part of something good.