Solves a problem about guessing numbers from sums and products

sums_and_products.py

products = dict()
for i in range(2, 101):
    for j in range(i, 101):
        products[i * j] = products.get(i * j, []) + [(i, j)]
ambiguous_products = set()
for product in products:
    if len(products[product]) > 1:
        ambiguous_products.add(product)
possible_sums = []
for s in range(4, 201):
    all_ambiguous = True
    for i in range(2, s // 2 + 1):
        j = s - i
        product = i * j
        if product not in ambiguous_products:
            all_ambiguous = False
            break
    if all_ambiguous:
        possible_sums.append(s)
new_products = dict()
for s in possible_sums:
    for i in range(2, s // 2 + 1):
        j = s - i
        new_products[i * j] = new_products.get(i * j, []) + [(i, j)]
unique_products = set()
for product in new_products:
    if len(new_products[product]) == 1:
        unique_products.add(product)
sums = dict()
for product in unique_products:
    for i, j in new_products[product]:
        sums[i + j] = sums.get(i + j, []) + [(i, j)]
unique_sums = []
for s in sums:
    if len(sums[s]) == 1:
        unique_sums.append(s)
assert(len(unique_sums) == 1)
real_sum = unique_sums[0]
candidates = []
for i in range(2, real_sum // 2 + 1):
    j = real_sum - i
    if i * j in unique_products:
        candidates.append((i, j))
assert(len(candidates) == 1)
print(candidates[0])