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

For a $5 \times 5$ matrix, let us analyze the minimum number of different numerals required.

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Step 1: Understanding the adjacency structure

In a $5 \times 5$ grid, cells have different numbers of neighbors:

• Corner cells (4 total): 3 neighbors each
• Edge cells (12 total): 5 neighbors each
• Interior cells (9 total): 8 neighbors each

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Step 2: Finding the maximum clique

The chromatic number equals the size of the largest clique (set of mutually adjacent cells).

Consider a $2 \times 2$ square of cells:

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

All four cells $\{a,b,c,d\}$ are mutually adjacent (each touches all others either horizontally, vertically, or diagonally).

Thus this forms a clique of size $4$.

For example, cells at positions $(2,2), (2,3), (3,2), (3,3)$:

$$\begin{pmatrix} \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & 7 & 8 & \cdot & \cdot \\ \cdot & 12 & 13 & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot \end{pmatrix}$$

Cells $7,8,12,13$ form a $4$-clique since:

• $7$ is adjacent to $8,12,13$
• $8$ is adjacent to $7,12,13$
• $12$ is adjacent to $7,8,13$
• $13$ is adjacent to $7,8,12$

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Step 3: Proving $4$ is the maximum clique size

Could a $5$-clique exist? That would require $5$ cells where each is adjacent to all $4$ others.

While each cell has up to $8$ neighbors, the grid geometry prevents a fifth cell from being adjacent to all members of a $4$-clique.

Therefore the largest clique is of size $4$.

Thus, the chromatic number is at least $4$.

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Step 4: Construction with 4 numerals

Since the maximum clique size is $4$, we need exactly $4$ different numerals.

A systematic construction (checkerboard-like pattern):

$$\begin{pmatrix} 1 & 2 & 1 & 2 & 1 \\ 3 & 4 & 3 & 4 & 3 \\ 1 & 2 & 1 & 2 & 1 \\ 3 & 4 & 3 & 4 & 3 \\ 1 & 2 & 1 & 2 & 1 \end{pmatrix}$$

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Step 5: Verification

• Horizontally adjacent pairs: $(1,2), (2,1), (3,4), (4,3)$ are all different
• Vertically adjacent pairs: $(1,3), (3,1), (2,4), (4,2)$ are all different
• Diagonally adjacent pairs: $(1,4), (4,1), (2,3), (3,2)$ are all different

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Final Answer:

The minimum number of different numerals required for a $5 \times 5$ matrix is $\boxed{4}$.