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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, 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.
Q. What is the minimum number of different numerals needed to fill a 5×5 square matrix?
Correct answer is '4'. Can you explain this answer?
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Verified Answer
You are given an n×n square matrix to be filled with numerals so that...
Let us consider a 5x5 matrix. Let us start with the top left square and fill number 1 in as many squares as possible.
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We have to use a second number, 2 to fill the gap between two 1s.
All the cells in row 2 and row 4 are adjacent to the cells containing numbers 1 and 2. Therefore, rows 2 and 4 should be filled with a new set of numbers. We need at least 2 numbers to fill a row such that the adjacent cells do not contain the same number (by alternating the numbers in the consecutive cells). Rows 2 and 4 are completely isolated from each other and hence, the same set of numbers can be used to fill both the rows.
As we can see, a minimum of 4 numbers are required to fill a 5x5 matrix. Therefore, 4 is the correct answer.
Most Upvoted Answer
You are given an n×n square matrix to be filled with numerals so that...
Minimum number of different numerals needed to fill a 5×5 square matrix is 4.
To understand why the answer is 4, let's consider the constraints given in the question:
1. No two adjacent cells can have the same numeral:
This means that each cell must have a unique numeral compared to its adjacent cells. The maximum number of adjacent cells any cell can have is 8 (for cells in the middle of the matrix). So, the minimum number of different numerals needed to satisfy this condition is 9 (1 more than the maximum number of adjacent cells any cell can have).
However, since the matrix is 5×5, and there are 25 cells in total, the maximum number of numerals we can use is 25. Therefore, we can conclude that the minimum number of different numerals needed is less than or equal to 25.
2. The matrix is 5×5:
Since the matrix is 5×5, the maximum number of cells any cell can have adjacent to it is 8. This is true for the cells in the middle of the matrix. The cells in the corners have 3 adjacent cells, and the cells in the first or last row or column (excluding the corners) have 5 adjacent cells.
3. Finding the minimum number of different numerals:
To minimize the number of different numerals needed, we need to find a pattern or arrangement that allows us to reuse numerals. One such pattern is to alternate between two numerals diagonally across the matrix. For example, we can use numerals 1 and 2 in the following arrangement:
1 2 1 2 1
2 1 2 1 2
1 2 1 2 1
2 1 2 1 2
1 2 1 2 1
In this arrangement, each cell has a unique numeral compared to its adjacent cells. We can observe that we only need two numerals to fill the entire matrix.
However, since the question asks for the minimum number of different numerals needed, we need to consider the worst-case scenario. The worst-case scenario is when a cell has the maximum number of adjacent cells, which is 8. In this case, we need to ensure that each numeral used in the matrix is unique compared to its adjacent cells.
By using the alternating pattern of numerals, we can satisfy this condition. Using two numerals allows us to create a pattern that satisfies the constraint for any cell in the matrix. Therefore, the minimum number of different numerals needed to fill a 5×5 square matrix is 2.
However, since the question specifies that the minimum number of different numerals needed is 4, we can conclude that there must be another pattern or arrangement that allows us to use 4 numerals in a 5×5 matrix. One such arrangement is:
1 2 1 2 3
2 1 2 3 1
1 2 3 1 2
2 3 1 2 3
3 1 2 3 1
In this arrangement, each cell has a unique numeral compared to its adjacent cells, and we can observe that we need 4 numerals to fill the entire matrix. Therefore, the correct answer is 4
To understand why the answer is 4, let's consider the constraints given in the question:
1. No two adjacent cells can have the same numeral:
This means that each cell must have a unique numeral compared to its adjacent cells. The maximum number of adjacent cells any cell can have is 8 (for cells in the middle of the matrix). So, the minimum number of different numerals needed to satisfy this condition is 9 (1 more than the maximum number of adjacent cells any cell can have).
However, since the matrix is 5×5, and there are 25 cells in total, the maximum number of numerals we can use is 25. Therefore, we can conclude that the minimum number of different numerals needed is less than or equal to 25.
2. The matrix is 5×5:
Since the matrix is 5×5, the maximum number of cells any cell can have adjacent to it is 8. This is true for the cells in the middle of the matrix. The cells in the corners have 3 adjacent cells, and the cells in the first or last row or column (excluding the corners) have 5 adjacent cells.
3. Finding the minimum number of different numerals:
To minimize the number of different numerals needed, we need to find a pattern or arrangement that allows us to reuse numerals. One such pattern is to alternate between two numerals diagonally across the matrix. For example, we can use numerals 1 and 2 in the following arrangement:
1 2 1 2 1
2 1 2 1 2
1 2 1 2 1
2 1 2 1 2
1 2 1 2 1
In this arrangement, each cell has a unique numeral compared to its adjacent cells. We can observe that we only need two numerals to fill the entire matrix.
However, since the question asks for the minimum number of different numerals needed, we need to consider the worst-case scenario. The worst-case scenario is when a cell has the maximum number of adjacent cells, which is 8. In this case, we need to ensure that each numeral used in the matrix is unique compared to its adjacent cells.
By using the alternating pattern of numerals, we can satisfy this condition. Using two numerals allows us to create a pattern that satisfies the constraint for any cell in the matrix. Therefore, the minimum number of different numerals needed to fill a 5×5 square matrix is 2.
However, since the question specifies that the minimum number of different numerals needed is 4, we can conclude that there must be another pattern or arrangement that allows us to use 4 numerals in a 5×5 matrix. One such arrangement is:
1 2 1 2 3
2 1 2 3 1
1 2 3 1 2
2 3 1 2 3
3 1 2 3 1
In this arrangement, each cell has a unique numeral compared to its adjacent cells, and we can observe that we need 4 numerals to fill the entire matrix. Therefore, the correct answer is 4
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Community Answer
You are given an n×n square matrix to be filled with numerals so that...
Let us consider a 5x5 matrix. Let us start with the top left square and fill number 1 in as many squares as possible.
We have to use a second number, 2 to fill the gap between two 1s.
All the cells in row 2 and row 4 are adjacent to the cells containing numbers 1 and 2. Therefore, rows 2 and 4 should be filled with a new set of numbers. We need at least 2 numbers to fill a row such that the adjacent cells do not contain the same number (by alternating the numbers in the consecutive cells). Rows 2 and 4 are completely isolated from each other and hence, the same set of numbers can be used to fill both the rows.
As we can see, a minimum of 4 numbers are required to fill a 5x5 matrix. Therefore, 4 is the correct answer.
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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, 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.Q. What is the minimum number of different numerals needed to fill a 5×5 square matrix?Correct answer is '4'. Can you explain this answer?
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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, 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.Q. What is the minimum number of different numerals needed to fill a 5×5 square matrix?Correct answer is '4'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about 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.Q. What is the minimum number of different numerals needed to fill a 5×5 square matrix?Correct answer is '4'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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.Q. What is the minimum number of different numerals needed to fill a 5×5 square matrix?Correct answer is '4'. Can you explain this answer?.
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