sum100.md

The problem is in section Reductio ad absurdum and indirect proof in How to Solve It:

White numbers using each of the the digits exactly once so that the sum of the numbers is exactly 100.

Here is the program to exhaust all sums:

sum_set = set()
numbers = set(range(0, 10))

def append_sum(sum):
    global sum_set, numbers
    l = len(numbers)
    if l == 0:
        sum_set.add(sum)
        return
    i = next(iter(numbers))
    if l == 1:
        sum_set.add(sum + i)
        return
    numbers.remove(i)
    append_sum(sum + i)
    for j in list(numbers):
        numbers.remove(j)
        append_sum(sum + i * 10 + j)
        append_sum(sum + j * 10 + i)
        numbers.add(j)
    numbers.add(i)

append_sum(0)
sum_list = list(sum_set)
sum_list.sort()
print(sum_list)

Actually, all sums are divisible by 9, which can be easily proved by Divisibility Rule.