GATE 2024 CH – Question 9
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.
| 1 | 1 | 2 | |
|---|---|---|---|
| 2 | X | 3 | |
| 2 | X | 4 | |
| 1 | 2 | X |
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
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Correct answer: (C) 2
Explanation
Call the empty cells $c_1, \ldots, c_4$ from the top. The 2 in row 1, column 3 has the cross at row 2, column 2 as a neighbour, and its other neighbours in the empty column are $c_1, c_2$, so exactly one of $c_1, c_2$ is a cross. The 3 in row 2, column 3 has two crosses already (row 2 column 2 and row 3 column 2), so exactly one of $c_1, c_2, c_3$ is a cross. The 4 in row 3, column 3 has three crosses already (row 2 column 2, row 3 column 2 and row 4 column 3), so exactly one of $c_2, c_3, c_4$ is a cross. The first two conditions give that $c_3$ is not a cross. Then either $c_1$ and $c_4$ are crosses (2 crosses), or only $c_2$ is a cross (1 cross). The maximum is 2.