4x4 Magic Square Solution¶
Here's a solution for the 4x4 magic square where the sum of each row, column, and diagonal is 34.
Given Partially Filled Matrix¶
\[
\begin{matrix}
1 & {\color{red} ?} & {\color{red} ?} & 4 \\
{\color{red} ?} & 6 & {\color{red} ?} & {\color{red} ?} \\
{\color{red} ?} & {\color{red} ?} & 11 & {\color{red} ?} \\
13 & {\color{red} ?} & {\color{red} ?} & 16
\end{matrix}
\]
Solution¶
To solve the puzzle while preserving the given digits and ensuring that each digit from 1 to 16 is used exactly once, follow these steps:
- Determine Missing Values: Calculate the missing numbers.
- Fill in the Matrix: Place the numbers in such a way that each row, column, and diagonal sums to 34.
A 4x4 magic square is a 4x4 grid where the sum of each row, column, and both main diagonals are equal. Here's a 4x4 magic square matrix:
| 1 | 15 | 14 | 4 |
| 12 | 6 | 7 | 9 |
| 8 | 10 | 11 | 5 |
| 13 | 3 | 2 | 16 |
Solution Verification:
To confirm it's a magic square, check the sum of rows, columns, and diagonals:
- Rows:
- \(1 + 15 + 14 + 4 = 34\)
- \(12 + 6 + 7 + 9 = 34\)
- \(8 + 10 + 11 + 5 = 34\)
-
\(13 + 3 + 2 + 16 = 34\)
-
Columns:
- \(1 + 12 + 8 + 13 = 34\)
- \(15 + 6 + 10 + 3 = 34\)
- \(14 + 7 + 11 + 2 = 34\)
-
\(4 + 9 + 5 + 16 = 34\)
-
Diagonals:
- \(1 + 6 + 11 + 16 = 34\)
- \(4 + 7 + 10 + 13 = 34\)
All sums are equal, so this is a valid 4x4 magic square!