Problem 14: Two missing numbers

def find_two_missing(numbers: list[int], n: int) -> set[int]: total_sum = n * (n - 1) // 2 arr_sum = sum(numbers) missing_sum = total_sum - arr_sum total_sq_sum = (n - 1) * n * (2 * (n - 1) + 1) // 6 arr_sq_sum = sum(x * x for x in numbers) missing_sq_sum = total_sq_sum - arr_sq_sum # Let the two missing numbers be a and b # a + b = missing_sum # a^2 + b^2 = missing_sq_sum # (a + b)^2 = a^2 + b^2 + 2ab # ab = (missing_sum^2 - missing_sq_sum) / 2 ab = (missing_sum * missing_sum - missing_sq_sum) // 2 # Solve for a and b using quadratic equation derived from: # x^2 - (a + b)x + ab = 0 # x^2 - missing_sum * x + ab = 0 # Discriminant = missing_sum^2 - 4 * ab discriminant = missing_sum * missing_sum - 4 * ab a = (missing_sum + discriminant ** 0.5) // 2 b = missing_sum - a return {int(a), int(b)}