Find all sets of consecutive positive integers whose sum is 100. How many such sets are there?

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.