Initial Query

A binary 8x8 matrix is 'unholy' if there are no isolated 1s. A 1 is isolated if there isn't another 1 in any of the 4 direct cardinal directions. Write a C program that counts the total number of unholy 8x8 matricies.To solve this question: - Use bit arithmetic and an unsigned long to represent the matrix. - Write a function is_unholy that tests if a matrix is unholy by bit-shifting and masking. Use a dynamic program to count the number of unholy matricies. - Recurse on the number of rows the matrix has. - Enumerate all 2^16 next two rows. This is enough to completely isolate the prior solutions from the next. Write out a plan for the program, and then implement the plan in C.

LLM Generation

Query

Output

### Plan for the Program 1. **Representation of the Matrix**: - Represent each row of the 8x8 matrix as an 8-bit unsigned integer (`uint8_t`). - The entire matrix can be represented as an array of 8 `uint8_t` values. 2. **Check if a Matrix is Unholy**: - Define a function `is_unholy` that takes the matrix as input. - For each cell in the matrix, check if it is a `1` and if it has at least one neighboring `1` in the four cardinal directions (up, down, left, right). - If any `1` is found to be isolated, the matrix is not unholy. 3. **Dynamic Programming Approach**: - Use dynamic programming to count the number of unholy matrices. - Recurse on the number of rows, considering all possible combinations of rows. - Use a memoization table to store intermediate results and avoid redundant calculations. 4. **Enumerate All Possible Rows**: - Since each row is an 8-bit integer, there are `2^8 = 256` possible rows. - Enumerate all possible pairs of rows to ensure that the constraints are satisfied. 5. **Count Unholy Matrices**: - Use the dynamic programming approach to count all possible 8x8 matrices that are unholy. ### Implementation in C

#include <stdio.h>
#include <stdint.h>
#include <stdbool.h>
#include <string.h>

#define ROWS 8
#define COLS 8

// Function to check if a matrix is unholy
bool is_unholy(uint8_t matrix[ROWS]) {
    for (int i = 0; i < ROWS; i++) {
        for (int j = 0; j < COLS; j++) {
            if (matrix[i] & (1 << j)) {
                bool has_neighbor = false;
                // Check left
                if (j > 0 && (matrix[i] & (1 << (j - 1)))) has_neighbor = true;
                // Check right
                if (j < COLS - 1 && (matrix[i] & (1 << (j + 1)))) has_neighbor = true;
                // Check up
                if (i > 0 && (matrix[i - 1] & (1 << j))) has_neighbor = true;
                // Check down
                if (i < ROWS - 1 && (matrix[i + 1] & (1 << j))) has_neighbor = true;
                if (!has_neighbor) return false;
            }
        }
    }
    return true;
}

// Dynamic programming function to count unholy matrices
uint64_t count_unholy_matrices() {
    uint64_t dp[ROWS + 1][1 << COLS];
    memset(dp, 0, sizeof(dp));

    // Base case: empty matrix is considered unholy
    dp[0][0] = 1;

    for (int i = 1; i <= ROWS; i++) {
        for (int prev_mask = 0; prev_mask < (1 << COLS); prev_mask++) {
            for (int curr_mask = 0; curr_mask < (1 << COLS); curr_mask++) {
                uint8_t matrix[ROWS];
                matrix[i - 1] = curr_mask;
                if (i > 1) matrix[i - 2] = prev_mask;

                if (is_unholy(matrix)) {
                    dp[i][curr_mask] += dp[i - 1][prev_mask];
                }
            }
        }
    }

    uint64_t total = 0;
    for (int mask = 0; mask < (1 << COLS); mask++) {
        total += dp[ROWS][mask];
    }

    return total;
}

int main() {
    uint64_t total_unholy = count_unholy_matrices();
    printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy);
    return 0;
}

### Explanation 1. **is_unholy Function**: - This function checks if a given matrix is unholy by iterating through each cell and checking its neighbors. - If any `1` is found to be isolated, the function returns `false`. 2. **count_unholy_matrices Function**: - This function uses dynamic programming to count the number of unholy matrices. - It initializes a DP table where `dp[i][mask]` represents the number of unholy matrices with `i` rows, where the `i-th` row is represented by `mask`. - It iterates through all possible rows and updates the DP table based on whether the current row configuration leads to an unholy matrix. 3. **Main Function**: - The `main` function calls `count_unholy_matrices` and prints the total number of unholy 8x8 matrices. ### Note: - The above implementation is a simplified version and may not be the most optimized solution. The problem is computationally intensive due to the large number of possible matrices (`2^64`), so further optimizations or parallel processing may be required for practical execution.

Extract Code

I extracted the following code from that output:

#include <stdio.h>
#include <stdint.h>
#include <stdbool.h>
#include <string.h>

#define ROWS 8
#define COLS 8

// Function to check if a matrix is unholy
bool is_unholy(uint8_t matrix[ROWS]) {
    for (int i = 0; i < ROWS; i++) {
        for (int j = 0; j < COLS; j++) {
            if (matrix[i] & (1 << j)) {
                bool has_neighbor = false;
                // Check left
                if (j > 0 && (matrix[i] & (1 << (j - 1)))) has_neighbor = true;
                // Check right
                if (j < COLS - 1 && (matrix[i] & (1 << (j + 1)))) has_neighbor = true;
                // Check up
                if (i > 0 && (matrix[i - 1] & (1 << j))) has_neighbor = true;
                // Check down
                if (i < ROWS - 1 && (matrix[i + 1] & (1 << j))) has_neighbor = true;
                if (!has_neighbor) return false;
            }
        }
    }
    return true;
}

// Dynamic programming function to count unholy matrices
uint64_t count_unholy_matrices() {
    uint64_t dp[ROWS + 1][1 << COLS];
    memset(dp, 0, sizeof(dp));

    // Base case: empty matrix is considered unholy
    dp[0][0] = 1;

    for (int i = 1; i <= ROWS; i++) {
        for (int prev_mask = 0; prev_mask < (1 << COLS); prev_mask++) {
            for (int curr_mask = 0; curr_mask < (1 << COLS); curr_mask++) {
                uint8_t matrix[ROWS];
                matrix[i - 1] = curr_mask;
                if (i > 1) matrix[i - 2] = prev_mask;

                if (is_unholy(matrix)) {
                    dp[i][curr_mask] += dp[i - 1][prev_mask];
                }
            }
        }
    }

    uint64_t total = 0;
    for (int mask = 0; mask < (1 << COLS); mask++) {
        total += dp[ROWS][mask];
    }

    return total;
}

int main() {
    uint64_t total_unholy = count_unholy_matrices();
    printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy);
    return 0;
}

Run Code Interpreter

Running the following program:

#include <stdio.h>
#include <stdint.h>
#include <stdbool.h>
#include <string.h>

#define ROWS 8
#define COLS 8

// Function to check if a matrix is unholy
bool is_unholy(uint8_t matrix[ROWS]) {
    for (int i = 0; i < ROWS; i++) {
        for (int j = 0; j < COLS; j++) {
            if (matrix[i] & (1 << j)) {
                bool has_neighbor = false;
                // Check left
                if (j > 0 && (matrix[i] & (1 << (j - 1)))) has_neighbor = true;
                // Check right
                if (j < COLS - 1 && (matrix[i] & (1 << (j + 1)))) has_neighbor = true;
                // Check up
                if (i > 0 && (matrix[i - 1] & (1 << j))) has_neighbor = true;
                // Check down
                if (i < ROWS - 1 && (matrix[i + 1] & (1 << j))) has_neighbor = true;
                if (!has_neighbor) return false;
            }
        }
    }
    return true;
}

// Dynamic programming function to count unholy matrices
uint64_t count_unholy_matrices() {
    uint64_t dp[ROWS + 1][1 << COLS];
    memset(dp, 0, sizeof(dp));

    // Base case: empty matrix is considered unholy
    dp[0][0] = 1;

    for (int i = 1; i <= ROWS; i++) {
        for (int prev_mask = 0; prev_mask < (1 << COLS); prev_mask++) {
            for (int curr_mask = 0; curr_mask < (1 << COLS); curr_mask++) {
                uint8_t matrix[ROWS];
                matrix[i - 1] = curr_mask;
                if (i > 1) matrix[i - 2] = prev_mask;

                if (is_unholy(matrix)) {
                    dp[i][curr_mask] += dp[i - 1][prev_mask];
                }
            }
        }
    }

    uint64_t total = 0;
    for (int mask = 0; mask < (1 << COLS); mask++) {
        total += dp[ROWS][mask];
    }

    return total;
}

int main() {
    uint64_t total_unholy = count_unholy_matrices();
    printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy);
    return 0;
}

And got the output:

Total number of unholy 8x8 matrices: 53003772538202116
main.c: In function 'main':
main.c:63:53: warning: format '%llu' expects argument of type 'long long unsigned int', but argument 2 has type 'uint64_t' {aka 'long unsigned int'} [-Wformat=]
   63 |     printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy);
      |                                                  ~~~^     ~~~~~~~~~~~~
      |                                                     |     |
      |                                                     |     uint64_t {aka long unsigned int}
      |                                                     long long unsigned int
      |                                                  %lu

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False