N x N Square Matrix Numeral Filling

CAT 2018 Slot 1 · DILR · Medium · Grids

All CAT 2018 Slot 1 questionsMore Grids questionsAll CAT DILR

Passage / data set

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.

Question 1 of 4

What is the minimum number of different numerals needed to fill a 3×33 \times 3 square matrix?

Answer

4

Explanation

Any 2×22 \times 2 subgrid has 4 mutually adjacent cells, so at least 4 numerals are required.

Question 2 of 4

What is the minimum number of different numerals needed to fill a 5×55 \times 5 square matrix?

Answer

4

Explanation

Repeating a 2×22 \times 2 block pattern of 4 numerals across the 5×55 \times 5 grid satisfies all adjacency requirements.

Question 3 of 4

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 5×55 \times 5 matrix?

  1. A.

    4

  2. B.

    16

  3. C.

    9

  4. D.

    25

Answer

A

Explanation

Since 4 numerals can fill the matrix with 0 mistakes, allowing 1 mistake still requires 4 numerals.

Question 4 of 4

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.

    9

  2. B.

    25

  3. C.

    16

  4. D.

    4

Answer

A

Explanation

For an interior cell, it has 8 distinct adjacent cells, all of which must be distinct from each other and from the center cell, necessitating at least 9 different numerals.

Practise this under exam conditions

Sign in to solve it with a live timer, the on-screen CAT calculator, and streak and accuracy tracking across every question you attempt.

Solve in the workspace