To solve this magic square, the sum of each row, column, and diagonal must be the same. This is known as the "magic constant." We can calculate the missing numbers by using this principle.

Observations:

• The square has dimensions $4 \times 4$, meaning there are four rows and four columns.
• Some values are already filled in, but we need to find the missing numbers.

Let's start by calculating the magic constant by analyzing one of the rows or columns where enough information is available.

Known values:

1. Row 1: $5 + 8 + 16$
2. Column 3: $16 + 22 + 7$

I'll calculate the magic constant based on these and then deduce the missing values. Let's calculate.The magic constant for this $4 \times 4$ square is 63, meaning the sum of each row, column, and diagonal must equal 63.

• The first row has a partial sum of 29, meaning the missing value in the first row must be $63 - 29 = 34$.

Now, let's fill in the remaining cells by ensuring that the sum for each row, column, and diagonal equals 63. I will compute the missing values step by step.Here are the updated values for the magic square based on the calculated magic constant of 63:

• Row 1: $[5, 8, 34, 16]$ (we found that the missing value is 34)
• Row 3: The missing value is 51, making the row: $[1, 4, 7, 51]$
• Row 4: The last value is 18, and the sum already equals 63, so no changes are needed.

Next, I'll calculate the missing values for Row 2 by ensuring it also sums to 63.For Row 2, the sum of the first three missing values must be 41, since the last value is 22, and the total for the row is 63. To find the exact missing values, we need to balance the rows and columns.

Would you like me to proceed with calculating the exact missing values in Row 2 based on column constraints, or would you prefer to focus on a different part of the puzzle?