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  • You are given an n×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, vertica

You are given an n×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:

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

Option: 1

4


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

We are given an n × n square matrix that needs to be populated with numerals in such a way that no two adjacent cells have the same numeral. Here, the term "adjacent" refers to cells that touch each other either horizontally, vertically, or diagonally. In the context of this problem, we need to consider the various cases of adjacent cells within the matrix.

Consider a 5x5 matrix where we start filling from the top-left square with the number 1. We aim to use as many squares as possible to fill them with the number 1.

1    1    1
         
1   1   1
         
1   1   1

To bridge the gap between two adjacent 1s, we introduce a second number, 2.

1 2 1 2 1
         
1 2 1 2 1
         
1 2 1 2 1

In rows 2 and 4, all the cells are adjacent to the ones containing numbers 1 and 2. Thus, we need to fill rows 2 and 4 with a new set of numbers. To ensure that adjacent cells do not have the same number, we require a minimum of 2 numbers to fill each row by alternating the numbers in consecutive cells. Interestingly, rows 2 and 4 are isolated from each other, allowing us to use the same set of numbers to fill both rows.

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

From these observations, we can conclude that a minimum of 4 numbers are necessary to fill a 5x5 matrix. Therefore, the correct answer is 4.

 

 

 

Posted by

Divya Prakash Singh

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