The given problem presents a larger square divided into nine smaller squares, where each square has a given numerical value except for the center, which is represented by $X$.

We will look for a pattern in the given numbers. One common approach is to check whether the numbers follow an arithmetic or geometric pattern or whether they relate to their positions in some way.

Observing the numbers:

\hline 82 & 7 & 32 \\ \hline 12 & X & 15 \\ \hline 96 & 13 & 16 \\ \hline \end{array}$$ ### Step 1: Checking Row and Column Patterns 1. **Column-wise Sum Check:** - First column: $$82 + 12 + 96 = 190$$ - Second column: $$7 + X + 13$$ - Third column: $$32 + 15 + 16 = 63$$ If all columns sum to the same total, then $$7 + X + 13 = 190$$, which is incorrect since 63 is the sum of another column. 2. **Row-wise Sum Check:** - First row: $$82 + 7 + 32 = 121$$ - Second row: $$12 + X + 15$$ - Third row: $$96 + 13 + 16 = 125$$ If the pattern follows an arithmetic rule where the sum of each row is the same, then: $$12 + X + 15 = 121$$ $$X + 27 = 121$$ $$X = 94$$ Thus, the missing number $$X$$ is **94**. Would you like a different approach or further details? ### Related Questions: 1. What are other possible patterns in the 3×3 grid? 2. How can we generalize the method to any $$n \times n$$ magic square? 3. What is the significance of magic squares in mathematics? 4. Can a geometric approach be used to solve such puzzles? 5. What happens if we replace a number with a variable and create an equation system? ### Tip: For number puzzles, always check sums, differences, or multiplication patterns both row-wise and column-wise to identify hidden structures.