Let's analyze the puzzle.

You are asked to fill in the empty boxes with the correct digits so that the product of the multiplication is accurate. Here's the breakdown of the steps based on the image:

1. The multiplication involves the number 74, and we are multiplying it by a two-digit number where the first digit is missing.
2. The second digit of the multiplicand is 2, and the sum of the products gives us the total result of 4,444.

Steps to solve:

• First, focus on the second row, which represents 74 × 2. The product is: $74 \times 2 = 148$ You can see that the second row shows the number 20_, which should be "148" but represented as 2 blank boxes in the hundreds and tens places.

• This gives us a clue that the first digit to be placed in the hundreds place is 1 and the second digit is 4.

• Now, moving to the next part of the puzzle, let’s deduce the first digit of the missing multiplier. Since the total product (4,444) minus the partial product (2,00) is 2,444, we need to solve: $74 \times ? = 2,444$ Trying different values, we find: $74 \times 3 = 222$

So, the missing digit of the multiplier is 6.

Thus, the missing values are:

• The first missing digit is 6.
• The missing digits in the sum column are 1 and 4.

Final solution:

• The multiplier is 74 × 62 = 4,444.

Would you like to see more details on any part of this solution?

Here are some related questions:

1. What happens if the second digit of the multiplier changes?
2. How would this problem change with different products?
3. How can we check the correctness of multiplication?
4. Why does the algorithm of long multiplication work for all numbers?
5. How can the same puzzle be solved for larger numbers?

Tip: Always work step by step in long multiplication, ensuring that partial sums and carries are handled correctly.