A 4x4 grid is filled with numbers from 1 to 16. Janet must rearrange the numbers so that every row, column, and diagonal sums to 34.

Janet needs to arrange the numbers from 1 to 16 in a 4x4 grid such that every row, column, and diagonal has a sum of 34. This setup describes a **4x4 magic square**, where each row, column, and diagonal has the same sum.

Here's how to approach this:

### Step 1: Calculate the Magic Constant
For an \( n \times n \) magic square filled with numbers from 1 to \( n^2 \), the magic constant \( M \) (sum of each row, column, and diagonal) is given by:
\[
M = \frac{n(n^2 + 1)}{2}
\]
For a 4x4 grid:
\[
M = \frac{4(16 + 1)}{2} = 34
\]

### Step 2: Arrange the Numbers
One standard 4x4 magic square with these properties is as follows:

\[
\begin{array}{|c|c|c|c|}
\hline
16 &  2 &  3 & 13 \\
5 & 11 & 10 &  8 \\
9 &  7 &  6 & 12 \\
4 & 14 & 15 &  1 \\
\hline
\end{array}
\]

Each row, column, and diagonal sums to 34 in this arrangement.

Would you like further explanation on how to construct this or explore other variations?

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1. What are the properties of a magic square, and how do they apply to different grid sizes?
2. How can the magic constant formula be derived for an \( n \times n \) grid?
3. Are there unique 4x4 magic squares, or can they be generated in multiple ways?
4. How do the diagonals contribute to the balance of a magic square?
5. What are applications of magic squares in real-world contexts?

**Tip:** When constructing magic squares, try starting with the central cell(s) for symmetric balance.

Learn how to arrange numbers 1 through 16 in a 4x4 grid to form a magic square, where each row, column, and diagonal sums to 34. This solution covers concepts like magic squares, the magic constant formula, and their properties. Suitable for high school students in Grades 9-12.

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