Question

What is the minimum number of different numerals needed to ?ll a 3×3 square matrix?

• A. 16
• B. 4
• C. 25
• D. 9
• A. 9
• B. 16
• C. 4
• D. 25

Answer 1

Correct Answer: 4

We are given an \(n \times n\) square matrix to be filled with numerals such that: 

• No two adjacent cells have the same numeral.
• Two cells are called adjacent if they touch each other horizontally, vertically, or diagonally.

Understanding Adjacency

This means each cell is considered to be adjacent to all 8 surrounding cells (if they exist). Therefore, each numeral must differ from all those around it.

Example for a 3 × 3 Matrix

For a \(3 \times 3\) matrix, using the adjacency rule, the minimum number of numerals required can be found by trying to color the grid such that no two adjacent cells share the same number.

One possible arrangement using 4 numerals:

1	2	3
4	1	2
3	4	1

Here, each numeral is surrounded by different numerals in all directions, satisfying the condition.

Conclusion

✅ Minimum number of numerals required: 4

Answer 2

To solve the problem of determining the minimum number of different numerals needed to fill a 5×5 square matrix such that no two adjacent cells have the same numeral, let's analyze the situation step by step. 

1. Definition of "adjacent": A cell is adjacent to another cell if it touches horizontally, vertically, or diagonally, which means each cell has up to 8 possible adjacent cells (except at edges and corners). Therefore, no two adjacent cells can have the same numeral.

2. Strategy: We must color the matrix with the minimum number of different numerals based on the condition of adjacency. This is equivalent to a graph-coloring problem where adjacent vertices (cells) cannot share the same color (numeral).

3. Pattern Consideration: Since this is essentially a graph theory problem on a 2D grid, it is recognized that a checkerboard pattern could minimize the number of colors, achieving a pattern where adjacent cells always have different numerals.

4. Checkerboard Pattern: This pattern uses two colors effectively. One numeral for all "black" cells and another for all "white" cells (using a chessboard analogy). Thus, every cell is different from its adjacent ones:

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

5. Conclusion: Due to the checkerboard configuration, only two numerals are necessary to fill the matrix, satisfying all adjacency conditions. Hence, the minimum number of different numerals required to fill a 5×5 matrix, respecting the adjacency constraint, is 2.

6. Verification against range: The range provided is 4,4. However, the solution identifies that only 2 numerals are sufficient when considering a checkerboard pattern, which is less than the proposed range. This suggests an optimal solution beyond initial expectations.

Answer 3

To solve this problem, we need to fill a 5×5 matrix with numerals such that no two adjacent cells have the same numeral. Here, we are allowed one mistake where one pair of adjacent cells can have the same numeral. The task is to determine the minimum number of different numerals required.
Let's analyze the condition if no mistake was allowed. The optimal strategy to fill the matrix would be similar to a checkerboard pattern using 4 numerals (like a typical 2-color pattern in chess), where no two adjacent cells share the same numeral. 

For instance, consider a pattern using numerals 1, 2, 3, and 4:

1	2	3	4	1
2	3	4	1	2
3	4	1	2	3
4	1	2	3	4
1	2	3	4	1

In this pattern, each cell is surrounded by other cells with different numerals, satisfying the problem's condition without mistakes.

However, we are allowed one mistake. This means one pair of adjacent cells might have the same numeral. Even with this allowance, the pattern using 4 different numerals remains valid as it handles adjacency efficiently.
Thus, the minimum number of different numerals required to fill the 5×5 matrix, considering the one allowed mistake, is 4.

Answer 4

To solve the problem of determining the minimum number of different numerals needed to fill a 5×5 square matrix, where no two adjacent cells can have the same numeral, we essentially aim to create a coloring pattern that ensures this condition.

Step-by-step Solution: 

1. Understand adjacency: Adjacent cells include those touching horizontally, vertically, or diagonally. For a cell in the center, this means 8 adjacent cells. For corner cells, there are 3 adjacent cells, and for edge cells (not corners), there are 5 adjacent cells.

2. Color Graph Theory: In graph theory, this is similar to coloring a graph where each cell is a vertex, and edges exist between adjacent cells. The problem determines the chromatic number (minimum colors) of this graph.

3. Analyze Patterns: For smaller n×n matrices (such as 2×2), it's clear that 4 numerals are needed. Extending to 3×3 requires at least 5 numerals to avoid repeating a numeral in any paths originating from the center.

4. Test Larger Patterns: For the 5×5 matrix, symmetry or periodic coloring patterns suggest that numbers cycle after reaching particular configurations.

5. Validate: Deploy the pattern and check for potential numeral repetition:
Assign numerals in a repetitive L-shaped pattern diagonally, ensuring coverage of all edges.
Using this method in a larger grid ensures all conditions met with minimal numerals.

Conclusion: Following the above logic and ensuring edge compliance, the minimum number of different numerals required is 9, as to fill bigger matrices systematically and satisfy adjacency constraints, larger indices would be unavoidable without covering adjacent possibilities.