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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. 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?
- a)4
- b)16
- c)9
- d)25
Correct answer is option 'A'. Can you explain this answer?
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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.
4 numbers are required to fill a 5x5 matrix.
It has been given that we are allowed to make 1 mistake - One pair of adjacent cells can contain the same number. In the arrangement given above, we can alter any value along the edge to satisfy this condition. For example, the 2 in the bottom-most row can be changed to 4. Still, the number of numbers required to fill the matrix will be 4.
Another way to approach this problem is as follows: We know that a minimum of 4 numbers are required to fill a 5x5 matrix. If we are allowed to make a mistake, then the number of numbers required should either remain the same or go down. 4 is the smallest value among the given options. Therefore, we can be sure that even if we are allowed to make a mistake, 4 numbers will be required to fill the matrix and hence, option A is the right answer.
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You are given an n×n square matrix to be filled with numerals so that...
Minimum Number of Different Numerals Required to Fill a 5x5 Matrix
To find the minimum number of different numerals required to fill a 5x5 matrix with the given conditions, we can use a simple approach.
Approach:
1. We know that each cell has at most 8 adjacent cells. So, for a cell to have different numerals in all its adjacent cells, we need at least 8 different numerals.
2. Let's consider the cells in the corners of the matrix. Each corner cell has 3 adjacent cells. Therefore, we can assign the same numeral to all the corner cells without violating the condition.
3. Now, let's consider the cells on the edges of the matrix (excluding the corners). Each edge cell has 5 adjacent cells. Since the corner cells already have the same numeral, we can assign a different numeral to each edge cell without violating the condition.
4. Finally, let's consider the cells in the center of the matrix. Each center cell has 8 adjacent cells. Since the corner and edge cells already have numerals assigned to them, we can assign a different numeral to each center cell without violating the condition.
5. Thus, we can assign the same numeral to the corner cells, different numerals to the edge cells, and different numerals to the center cells, satisfying the given condition.
Minimum Number of Numerals Required:
- Corner cells: 1 numeral (same numeral for all 4 corners)
- Edge cells: 4 numerals (different numerals for each edge cell)
- Center cells: 3 numerals (different numerals for each center cell)
Therefore, the minimum number of different numerals required to fill a 5x5 matrix is 1 + 4 + 3 = 8.
Answer:
The minimum number of different numerals required to fill a 5x5 matrix is 8.
To find the minimum number of different numerals required to fill a 5x5 matrix with the given conditions, we can use a simple approach.
Approach:
1. We know that each cell has at most 8 adjacent cells. So, for a cell to have different numerals in all its adjacent cells, we need at least 8 different numerals.
2. Let's consider the cells in the corners of the matrix. Each corner cell has 3 adjacent cells. Therefore, we can assign the same numeral to all the corner cells without violating the condition.
3. Now, let's consider the cells on the edges of the matrix (excluding the corners). Each edge cell has 5 adjacent cells. Since the corner cells already have the same numeral, we can assign a different numeral to each edge cell without violating the condition.
4. Finally, let's consider the cells in the center of the matrix. Each center cell has 8 adjacent cells. Since the corner and edge cells already have numerals assigned to them, we can assign a different numeral to each center cell without violating the condition.
5. Thus, we can assign the same numeral to the corner cells, different numerals to the edge cells, and different numerals to the center cells, satisfying the given condition.
Minimum Number of Numerals Required:
- Corner cells: 1 numeral (same numeral for all 4 corners)
- Edge cells: 4 numerals (different numerals for each edge cell)
- Center cells: 3 numerals (different numerals for each center cell)
Therefore, the minimum number of different numerals required to fill a 5x5 matrix is 1 + 4 + 3 = 8.
Answer:
The minimum number of different numerals required to fill a 5x5 matrix is 8.
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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. 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?a) 4b) 16c) 9d) 25Correct answer is option 'A'. 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. 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?a) 4b) 16c) 9d) 25Correct answer is option 'A'. 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. 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?a) 4b) 16c) 9d) 25Correct answer is option 'A'. 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. 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?a) 4b) 16c) 9d) 25Correct answer is option 'A'. Can you explain this answer?.
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. 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?a) 4b) 16c) 9d) 25Correct answer is option 'A'. 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. 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?a) 4b) 16c) 9d) 25Correct answer is option 'A'. 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. 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?a) 4b) 16c) 9d) 25Correct answer is option 'A'. Can you explain this answer?.
Solutions 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. 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?a) 4b) 16c) 9d) 25Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
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