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GATE 2024 XH2 (English) – Question 9

General Aptitude · Analytical Aptitude: Logical Reasoning and Deduction · 2 marks · Multiple choice

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.

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 4 by 4 grid whose last column is empty. Rows from the top: 1, 1, 2, blank; 2, X, 3, blank; 2, X, 4, blank; 1, 2, X, blank.
  1. 0
  2. 1
  3. 2
  4. 3

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Correct answer: (C) 2

Explanation

Let the empty column from the top be $a, b, c, d$ (1 for a cross). The top 2 in column 3 has one known cross (below-left), so $a + b = 1$. The 3 below it has two known crosses, so $a + b + c = 1$, which gives $c = 0$. The 4 has three known crosses, so $b + c + d = 1$, which gives $b + d = 1$. The total is $a + b + c + d = 1 + 0 + d$ with $d = 1 - b$. It is largest at $b = 0$: $a = 1, d = 1$, a total of 2.