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:

\hline 16 & 2 & 3 & 13 \\ 5 & 11 & 10 & 8 \\ 9 & 7 & 6 & 12 \\ 4 & 14 & 15 & 1 \\ \hline \end{array}$$ Each row, column, and diagonal sums to 34 in this arrangement. Would you like further explanation on how to construct this or explore other variations? --- 1. What are the properties of a magic square, and how do they apply to different grid sizes? 2. How can the magic constant formula be derived for an $$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.