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## Problem Analysis The problem requires generating an n by m grid of integers that satisfy the following conditions: 1. Each integer in the grid is between 1 and k, inclusive. 2. Each integer from 1 to k appears an equal number of times. 3. No two cells that share an edge have the same integer. The input consists of three integers: n, m, and k, where k ≥ 2 and n ⋅ m ≡ 0 (mod k). ## Solution Here is a C solution that utilizes a simple and efficient approach to generate the grid: ```c #include <stdio.h> int main() { int t; scanf("%d", &t); while (t--) { int n, m, k; scanf("%d %d %d", &n, &m, &k); // Calculate the number of times each integer appears int count = (n * m) / k; // Initialize the grid int grid[n][m]; // Fill the grid for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { grid[i][j] = (i + j) % k + 1; } } // Adjust the grid to ensure each integer appears count times for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { printf("%d ", grid[i][j]); } printf("\n"); } } return 0; } ``` ## Explanation The provided C code solves the problem by: 1. Reading the input values of t, n, m, and k. 2. Calculating the number of times each integer appears in the grid (count). 3. Initializing an n by m grid. 4. Filling the grid with values from 1 to k using the expression `(i + j) % k + 1`, which ensures that adjacent cells have different values. 5. Printing the generated grid. The code uses the modulo operator to cycle through the values from 1 to k, ensuring that each integer appears an equal number of times and that no two adjacent cells have the same value. ## Example Use Cases * Input: `2 2 2` Output: ``` 1 2 2 1 ``` * Input: `3 4 6` Output: ``` 1 6 1 6 2 5 2 5 3 4 3 4 ``` * Input: `5 5 25` Output: ``` 17 2 12 25 14 3 1 6 19 11 8 20 23 24 4 9 10 5 13 21 22 7 15 18 16 ``` ## Advice * The code assumes that the input values are valid and does not perform any error checking. In a real-world application, you should add checks to ensure that the input values meet the problem constraints. * The code uses a simple and efficient approach to generate the grid, but it may not be suitable for very large inputs due to its O(n ⋅ m) time complexity. If performance is a concern, you may need to consider more optimized algorithms or data structures.