Explanation:

Let's analyze the given pattern column by column. The numbers in the second row are perfect cubes. Let's find their cube roots: $\sqrt[3]{216} = 6$, $\sqrt[3]{512} = 8$, $\sqrt[3]{729} = 9$. Now, let's observe the relationship between the numbers in the first row, the third row, and these cube roots for each column. For Column 1: First row number is $2$, third row number is $4$. Their sum is $2 + 4 = 6$. The cube of this sum is $6^3 = 216$, which matches the second row. For Column 2: First row number is $1$, third row number is $7$. Their sum is $1 + 7 = 8$. The cube of this sum is $8^3 = 512$, which matches the second row. For Column 3: First row number is $3$, third row number is $?$ (let's call it $x$). The second row number is $729$. Following the pattern, $(3 + x)^3 = 729$. Taking the cube root of $729$, we get $3 + x = 9$. Solving for $x$, we find $x = 9 - 3 = 6$. Therefore, the number that replaces the question mark is $6$. Option (1) $6$ is the correct answer. Options (2) $5$, (3) $8$, and (4) $9$ would not satisfy the established relationship of the second row number being the cube of the sum of the first and third row numbers in their respective columns.