Complete the following 5 × 5 magic square. Recall that we need to use each of the numbers 1, 2, . . . , 25 exactly once and that the sum of every row, column, and diagonal must be the same. Provide details of your work.
20 136 23 1019
9 17 3
25 7
8 524

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:

\[
\begin{array}{|c|c|c|c|c|}
\hline
20 & 13 & 6 &  & 23 \\
\hline
9 &  & 17 & 3 & \\
\hline
 & 25 & 7 &  & \\
\hline
8 &  &  & 5 & 24 \\
\hline
 & 4 &  & 12 & \\
\hline
\end{array}
\]

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

Complete the following 5 × 5 magic square. Recall that we need to use each of the numbers 1, 2, . . . , 25 exactly once and that the sum of every row, column, and diagonal must be the same. Provide details of your work. Partially filled square: 20 13 6  _ 23 | 9 _ 17 3 _ | _ 25 7 _ _ | 8 _ _ 5 24 | _ 4 _ 12 _

Solve a 5 × 5 magic square by filling in missing numbers so that the sum of each row, column, and diagonal equals 65. This problem demonstrates arithmetic concepts like summation and properties of magic squares, suitable for students in Grades 8-10.

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