GATE CE 2024 Set 1 — Question 9
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Question
In the array shown below, each cell of the first three columns has either a cross (X) or a number, as per the given rule.
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
Correct answer
(C) 2
Solution
Let the cells in the fourth column be from top to bottom. We use the numbers in the third column to establish constraints on these cells.
Substituting into equation (3), we get .
From equation (1), we have .To maximize the total number of crosses in the fourth column ():
1.Cell (1,3) contains 2: Its neighbors are (1,2), (2,2), (2,3), (1,4), and (2,4). From the grid, (2,2) is a cross (X). Thus, one more cross must come from the fourth column neighbors: .
2.Cell (2,3) contains 3: Its neighbors are (1,2), (1,3), (1,4), (2,2), (2,4), (3,2), (3,3), and (3,4). From the grid, (2,2) and (3,2) are crosses. Thus, one more cross must come from the fourth column neighbors: .
3.Cell (3,3) contains 4: Its neighbors are (2,2), (2,3), (2,4), (3,2), (3,4), (4,2), (4,3), and (4,4). From the grid, (2,2), (3,2), and (4,3) are crosses. Thus, one more cross must come from the fourth column neighbors: .
From equations (1) and (2), we find .Substituting into equation (3), we get .
From equation (1), we have .To maximize the total number of crosses in the fourth column ():
- If , then and . Total crosses = .
- If , then and . Total crosses = .
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