Question

In the 4 × 4 array shown below, each cell of the first three columns has either a cross (X) or a number, as per the given rule.
\includegraphics[width=0.5\linewidth]{9.png}
Rule: The number in a cell represents the count of crosses around its immediate neighboring cells (left, right, top, bottom, diagonals). As per this rule, the maximum number of crosses possible in the empty column is:

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

For grid problems:
1. Systematically evaluate the constraints row by row or cell by cell.
2. Place elements to satisfy conditions for all neighbors.
3. Verify placements to ensure no violations.

Answer 1

The Correct Option is C

Step 1: Analyze the constraints row by row. The task is to place crosses (\(X\)) in the empty column (column 4) such that the number in each cell of the first three columns matches the total number of crosses in its neighboring cells. We maximize the number of crosses in column 4. Step 2: Analyze each numbered cell and its neighbors. 1. Cell (1,3) with value \(2\): Neighbors are (1,2), (2,2), (2,3), and (1,4). Currently, there is \(1X\) in (2,2). To satisfy the value \(2\), place an \(X\) in (1,4). 2. Cell (2,3) with value \(3\): Neighbors are (1,3), (1,4), (2,2), (2,4), (3,3), and (3,4). There is already \(1X\) in (2,2) and \(1X\) in (1,4). To satisfy \(3\), place an \(X\) in (2,4). 3. Cell (3,3) with value \(4\): Neighbors are (2,3), (2,4), (3,2), (3,4), (4,3), and (4,4). Currently, \(X\) exists in (2,4), \(3,4\), and (3,2). To satisfy \(4\), place an \(X\) in (4,4). 4. Cell (4,3) with value \(1\): Neighbors are (3,3), (3,4), and (4,4). There are already \(X\)'s in (3,4) and (4,4), satisfying the value \(1\). Step 3: Verify the empty column. The crosses placed in column 4 are at positions (1,4), (2,4), and (3,4). This configuration satisfies all the constraints.