A magic square of size $n \times n$ is a square grid filled with distinct integers from $1$ to $n^2$, such that the sum of each row, column, and diagonal is the same. For a $5 \times 5$ magic square, the sum of the integers in each row, column, or diagonal should equal a specific constant called the magic constant.

Step 1: Calculate the Magic Constant

For a magic square, the magic constant $S$ is given by the formula:

$S = \frac{n(n^2 + 1)}{2}$

For $n = 5$:

$S = \frac{5(25 + 1)}{2} = \frac{5 \times 26}{2} = 65$

Thus, the sum of every row, column, and diagonal in the $5 \times 5$ magic square should be 65.

Step 2: Understanding the Partially Completed Magic Square

We are given a partially filled $5 \times 5$ magic square:

\hline 20 & 13 & 6 & & 23 \\ \hline 9 & & 17 & 3 & \\ \hline & 25 & 7 & & \\ \hline 8 & & & 5 & 24 \\ \hline & 4 & & 12 & \\ \hline \end{array}$$ We need to fill in the remaining numbers (1, 2, 10, 11, 14, 15, 16, 18, 19, 21, 22) so that every row, column, and diagonal sums to 65. ### Step 3: Step-by-Step Filling of the Square We'll now solve this step by step, ensuring that every row, column, and diagonal sums to 65. #### Fill the first row: The first row already has the numbers 20, 13, 6, and 23. Their sum is: $$20 + 13 + 6 + 23 = 62$$ So, the missing number in the first row is: $$65 - 62 = 3$$ But the number 3 is already in the second row, so there seems to be an issue with this