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CAT 2018 Slot 1 DILR · Set 3 of 8Logical ReasoningEasy DifficultyQ9–Q12 · 2 MCQ + 2 TITA

CAT 2018 Slot 1 DILR Set 3: Grids (4 Questions) with Solutions

The Grids set from the official CAT 2018 Slot 1 DILR paper: the caselet, all 4 questions, the official answer key and a step-by-step solution for each. 100% free, no sign-up.

The Set: Read the Caselet First

CAT 2018 Slot 1 DILR · Questions 9–12
You are given an n×nn \times n square matrix to be filled with numerals so that no two adjacent cells have the same numeral. Two cells are called adjacent if they touch each other horizontally, vertically or diagonally. So a cell in one of the four corners has three cells adjacent to it, and a cell in the first or last row or column which is not in the corner has five cells adjacent to it. Any other cell has eight cells adjacent to it.

In the exam this set competed with 7 others for 60 minutes. Budget about 8 minutes if you attempt it.

Q9. Grids

TITA · +3 / 0
What is the minimum number of different numerals needed to fill a 3×33\times 3 square matrix?
Correct Answer
4
Step-by-step solution
📌 Core Concept
Cells in any 2×22\times 2 block all touch each other (horizontally, vertically or diagonally), so they need pairwise different numerals. This gives a lower bound, which a repeating pattern then achieves.
🔢 Step-by-Step Solution
1
Take the top-left 2×22\times 2 block. Each of its four cells is adjacent to the other three, so four different numerals are needed. Hence at least 4 numerals are required.
2
Show that 4 is enough by repeating the block, with numerals 1, 2, 3, 4:
121343121\begin{matrix}1&2&1\\3&4&3\\1&2&1\end{matrix}
3
Check the pattern. Horizontal neighbours are 1 and 2, or 3 and 4. Vertical neighbours are 1 and 3, or 2 and 4. Diagonal neighbours are 1 and 4, or 2 and 3. No two touching cells match.
Quick Check
The lower bound is 4 and the pattern uses exactly 4, so the minimum is 4.
Correct Answer: 4

Q10. Grids

TITA · +3 / 0
What is the minimum number of different numerals needed to fill a 5×55 \times 5 square matrix?
Correct Answer
4
Step-by-step solution
Clue Breakdown:
The grid is 5x5.
Adjacent cells include horizontal, vertical, and diagonal neighbors.
Each cell must have a unique numeral compared to all adjacent cells.
Goal: Minimum numerals required.
Step-by-Step Deduction Grid:
1
Understanding Adjacency: Each cell has up to 8 neighbors.
2
Graph Coloring Insight: This is a graph coloring problem where each cell is a node, and edges connect adjacent cells. The minimum number of colors needed is the chromatic number.
3
Smaller Grid Analysis: For a 2x2 grid, 4 colors are needed as each cell is adjacent to all others.
4
Pattern Repetition: Use a 2x2 tile with 4 colors and repeat it across the 5x5 grid.
5
Color Arrangement:
`` 1 2 1 2 1 3 4 3 4 3 1 2 1 2 1 3 4 3 4 3 1 2 1 2 1 ``
6
Verification: Each cell's diagonal neighbors have different colors, satisfying the condition.
Question Resolution: The 5x5 grid can be successfully filled using a 4-color repeating pattern, ensuring no two adjacent cells (including diagonally) share the same numeral.
Final Answer Confirmation: Correct Answer: 4

Q11. Grids

MCQ · +3 / −1
Suppose you are allowed to make one mistake, that is, one pair of adjacent cells can have the same numeral. What is the minimum number of different numerals required to fill a 5×5 matrix?
  1. A) 4
  2. B) 16
  3. C) 9
  4. D) 25
Correct Answer
A) 4
Step-by-step solution
📌 Core Concept
Any 2×22\times 2 block needs 4 different numerals. A 5×5 grid has several disjoint 2×22\times 2 blocks, so one allowed mistake cannot lower the requirement below 4. Four numerals in a repeating pattern are enough.
🔢 Step-by-Step Solution
1
Four disjoint 2×22\times 2 blocks fit in a 5×5 grid, for example in rows 1-2 and 4-5 and columns 1-2 and 4-5.
2
The single allowed mistake is one pair of adjacent cells. Both cells of that pair lie in at most one of these disjoint blocks. So at least 3 blocks have no repeat and each needs 4 distinct numerals. Hence at least 4 numerals are needed.
3
Four numerals suffice even with no mistake. Use numeral 1 for odd row and odd column, 2 for odd row and even column, 3 for even row and odd column, and 4 for even row and even column. Touching cells always differ in row parity, column parity or both, so they get different numerals.
4
The mistake is not needed, so the minimum is 4.
Why Not the Others
B) 16: far more than the 4 that the pattern already achieves.
C) 9: also more than needed; 9 is the answer only when cells sharing a common neighbour must differ.
D) 25: this would give every cell its own numeral, which is not required.
Correct Answer: A

Q12. Grids

MCQ · +3 / −1
Suppose that all the cells adjacent to any particular cell must have different numerals. What is the minimum number of different numerals needed to fill a 5×55 \times 5 square matrix?
  1. A) 25
  2. B) 4
  3. C) 16
  4. D) 9
Correct Answer
D) 9
Step-by-step solution
📌 Core Concept
The condition now says that any two cells touching a common cell must differ. Together with the original adjacency rule, every cell in a 3×33\times 3 block must then have a different numeral.
🔢 Step-by-Step Solution
1
Take the centre cell of the 5×5 grid. It has 8 adjacent cells, and these must all differ from each other.
2
Each of the 8 also differs from the centre cell, since it is adjacent to it. So the 3×33\times 3 block around the centre needs 9 distinct numerals. Hence at least 9 are required.
3
Show that 9 is enough. Give the cell in row ii, column jj the numeral given by the pair (i mod 3, j mod 3)(i \bmod 3,\ j \bmod 3), which has 9 possible values.
4
Two cells that touch a common cell differ by at most 2 in row and at most 2 in column, and are not the same cell. So the row difference or the column difference is 1 or 2, which is not a multiple of 3. Their pairs therefore differ, and their numerals differ.
5
The same argument covers cells that touch each other directly, so all conditions hold with 9 numerals.
Why Not the Others
A) 25: every cell distinct is not required.
B) 4: this only handles direct adjacency, not cells sharing a common neighbour.
C) 16: more than the 9 that work.
Correct Answer: D
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CAT 2018 Slot 1 DILR at a Glance

Sets in this section8
Questions in this section32
Section time60 minutes
Convenor IIMIIM Calcutta

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