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

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.

Output

Given the complexity of the task, let's first outline a plan and then dive into the implementation: ### Plan 1. **Representation of the Matrix:** - Since the matrix is 8x8 and binary, we can use an `unsigned long` to represent it, where each bit represents a cell in the matrix. The first 8 bits are the first row, the next 8 bits are the second row, and so on. 2. **Checking if a Matrix is Unholy:** - To check if a 1 is isolated, we need to check its immediate neighbors in the north, south, east, and west directions. This can be done using bit-shifting and masking operations. 3. **Dynamic Programming Approach:** - We will use a recursive function that counts the number of unholy matrices by progressively considering more rows. The base case is when we have considered all 8 rows. - To avoid recalculating for the same configurations of the already processed rows, we will store the intermediate results in a cache. 4. **Enumerating the Next Two Rows:** - For each recursion step, we need to consider all possible combinations of the next two rows (2^16 possibilities) and check if adding them keeps the matrix unholy. ### Implementation 

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

#define MAX_ROWS 8
#define CACHE_SIZE (1 << (2*MAX_ROWS)) // 2^16

unsigned long cache[CACHE_SIZE];

// Checks if a given matrix is unholy.
bool is_unholy(unsigned long matrix) {
    for (int row = 0; row < MAX_ROWS; row++) {
        for (int col = 0; col < MAX_ROWS; col++) {
            int bit_position = row * MAX_ROWS + col;
            if (matrix & (1UL << bit_position)) { // If current position is 1
                bool isolated = true;
                // Check North
                if (row > 0 && (matrix & (1UL << (bit_position - MAX_ROWS)))) isolated = false;
                // Check South
                if (row < MAX_ROWS - 1 && (matrix & (1UL << (bit_position + MAX_ROWS)))) isolated = false;
                // Check East
                if (col < MAX_ROWS - 1 && (matrix & (1UL << (bit_position + 1)))) isolated = false;
                // Check West
                if (col > 0 && (matrix & (1UL << (bit_position - 1)))) isolated = false;
                
                if (isolated) return false; // Found an isolated 1
            }
        }
    }
    return true; // No isolated 1s
}

// Dynamic programming function to count unholy matrices.
unsigned long count_unholy(unsigned long matrix, int rows) {
    if (rows == MAX_ROWS) {
        if (is_unholy(matrix)) return 1;
        else return 0;
    }
    
    // Check if result is already cached.
    if (cache[matrix] != -1) {
        return cache[matrix];
    }

    unsigned long count = 0;
    for (unsigned int next_rows = 0; next_rows < (1 << (2*MAX_ROWS)); next_rows++) {
        unsigned long new_matrix = matrix | ((unsigned long)next_rows << (rows * MAX_ROWS));
        count += count_unholy(new_matrix, rows + 2);
    }
    
    // Cache the result before returning.
    cache[matrix] = count;
    return count;
}

int main() {
    memset(cache, -1, sizeof(cache)); // Initialize cache with -1.
    unsigned long total = count_unholy(0, 0);
    printf("Total unholy matrices: %lu\n", total);
    return 0;
}
### Explanation - The `is_unholy` function iterates over every bit in the `unsigned long` matrix representation to check for isolated 1s. - The `count_unholy` function is a recursive function that counts the number of unholy matrices. It uses dynamic programming to cache intermediate results. - The main challenge is efficiently implementing and managing the cache for dynamic programming, as well as correctly handling bit operations to manipulate the matrix representation. - Given the constraints and the exhaustive search space, this program may take a significant amount of time to run, or it may require optimization/refinement to complete in a practical amount of time for larger row counts or different matrix sizes.

Extract Code

I extracted the following code from that output: 

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

#define MAX_ROWS 8
#define CACHE_SIZE (1 << (2*MAX_ROWS)) // 2^16

unsigned long cache[CACHE_SIZE];

// Checks if a given matrix is unholy.
bool is_unholy(unsigned long matrix) {
    for (int row = 0; row < MAX_ROWS; row++) {
        for (int col = 0; col < MAX_ROWS; col++) {
            int bit_position = row * MAX_ROWS + col;
            if (matrix & (1UL << bit_position)) { // If current position is 1
                bool isolated = true;
                // Check North
                if (row > 0 && (matrix & (1UL << (bit_position - MAX_ROWS)))) isolated = false;
                // Check South
                if (row < MAX_ROWS - 1 && (matrix & (1UL << (bit_position + MAX_ROWS)))) isolated = false;
                // Check East
                if (col < MAX_ROWS - 1 && (matrix & (1UL << (bit_position + 1)))) isolated = false;
                // Check West
                if (col > 0 && (matrix & (1UL << (bit_position - 1)))) isolated = false;
                
                if (isolated) return false; // Found an isolated 1
            }
        }
    }
    return true; // No isolated 1s
}

// Dynamic programming function to count unholy matrices.
unsigned long count_unholy(unsigned long matrix, int rows) {
    if (rows == MAX_ROWS) {
        if (is_unholy(matrix)) return 1;
        else return 0;
    }
    
    // Check if result is already cached.
    if (cache[matrix] != -1) {
        return cache[matrix];
    }

    unsigned long count = 0;
    for (unsigned int next_rows = 0; next_rows < (1 << (2*MAX_ROWS)); next_rows++) {
        unsigned long new_matrix = matrix | ((unsigned long)next_rows << (rows * MAX_ROWS));
        count += count_unholy(new_matrix, rows + 2);
    }
    
    // Cache the result before returning.
    cache[matrix] = count;
    return count;
}

int main() {
    memset(cache, -1, sizeof(cache)); // Initialize cache with -1.
    unsigned long total = count_unholy(0, 0);
    printf("Total unholy matrices: %lu\n", total);
    return 0;
}

Run Code Interpreter

Running the following program: 

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

#define MAX_ROWS 8
#define CACHE_SIZE (1 << (2*MAX_ROWS)) // 2^16

unsigned long cache[CACHE_SIZE];

// Checks if a given matrix is unholy.
bool is_unholy(unsigned long matrix) {
    for (int row = 0; row < MAX_ROWS; row++) {
        for (int col = 0; col < MAX_ROWS; col++) {
            int bit_position = row * MAX_ROWS + col;
            if (matrix & (1UL << bit_position)) { // If current position is 1
                bool isolated = true;
                // Check North
                if (row > 0 && (matrix & (1UL << (bit_position - MAX_ROWS)))) isolated = false;
                // Check South
                if (row < MAX_ROWS - 1 && (matrix & (1UL << (bit_position + MAX_ROWS)))) isolated = false;
                // Check East
                if (col < MAX_ROWS - 1 && (matrix & (1UL << (bit_position + 1)))) isolated = false;
                // Check West
                if (col > 0 && (matrix & (1UL << (bit_position - 1)))) isolated = false;
                
                if (isolated) return false; // Found an isolated 1
            }
        }
    }
    return true; // No isolated 1s
}

// Dynamic programming function to count unholy matrices.
unsigned long count_unholy(unsigned long matrix, int rows) {
    if (rows == MAX_ROWS) {
        if (is_unholy(matrix)) return 1;
        else return 0;
    }
    
    // Check if result is already cached.
    if (cache[matrix] != -1) {
        return cache[matrix];
    }

    unsigned long count = 0;
    for (unsigned int next_rows = 0; next_rows < (1 << (2*MAX_ROWS)); next_rows++) {
        unsigned long new_matrix = matrix | ((unsigned long)next_rows << (rows * MAX_ROWS));
        count += count_unholy(new_matrix, rows + 2);
    }
    
    // Cache the result before returning.
    cache[matrix] = count;
    return count;
}

int main() {
    memset(cache, -1, sizeof(cache)); // Initialize cache with -1.
    unsigned long total = count_unholy(0, 0);
    printf("Total unholy matrices: %lu\n", total);
    return 0;
}
And got the output: main.sh: line 2: 16 Segmentation fault (core dumped) ./a.out

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False

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

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.

Output

To tackle this problem, let's break it down into steps following the guidelines provided: ### Plan 1. **Representation of the Matrix**: - Use an `unsigned long` to represent the 8x8 matrix, as each bit can represent a cell in the matrix, allowing for efficient bitwise operations. 2. **Function to Check if a Matrix is Unholy**: - Implement `is_unholy` function that takes an `unsigned long` representing the matrix and returns `1` if the matrix is unholy, and `0` otherwise. - Use bit shifting and masking to check adjacent cells for each 1 in the matrix. 3. **Dynamic Programming Approach**: - Use recursion to count the total number of unholy matrices, based on the number of rows processed. - For each step, enumerate all possible configurations for the next two rows (2^16 possibilities) and check if adding these rows keeps the matrix unholy. - Cache intermediate results to avoid recalculating for the same partial matrix configurations. 4. **Main Algorithm**: - Start with an empty matrix and recursively fill it row by row, checking for the unholy condition as we go. - Keep a global counter to accumulate the total number of unholy matrices found. ### Implementation 

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#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#define MATRIX_SIZE 64
#define ROW_MASK 0xFF // Mask for a single row (8 bits set)

unsigned long long count_unholy_matrices(int rows, unsigned long current_matrix, unsigned long long *memo);

// Checks if the current matrix configuration is unholy
int is_unholy(unsigned long matrix) {
    for (int i = 0; i < MATRIX_SIZE; i++) {
        if ((matrix >> i) & 1) { // If there's a 1 at position i
            // Check cardinal directions (exclude edges)
            if (!((i % 8 == 0 || (matrix >> (i - 1)) & 1) && // Left
                  (i % 8 == 7 || (matrix >> (i + 1)) & 1) && // Right
                  (i < 8 || (matrix >> (i - 8)) & 1) && // Up
                  (i >= 56 || (matrix >> (i + 8)) & 1))) { // Down
                return 0; // Found an isolated 1
            }
        }
    }
    return 1; // No isolated 1s found, matrix is unholy
}

// Recursive function to count unholy matrices
unsigned long long count_unholy_matrices(int rows, unsigned long current_matrix, unsigned long long *memo) {
    if (rows == 8) { // Base case: all rows filled
        return is_unholy(current_matrix);
    }
    
    // Check if result is already memoized
    if (memo[(rows << 16) | current_matrix] != -1) {
        return memo[(rows << 16) | current_matrix];
    }

    unsigned long long total = 0;

    // Enumerate all possibilities for the next row (2^8)
    for (unsigned int next_row = 0; next_row <= 0xFF; next_row++) {
        unsigned long new_matrix = (current_matrix << 8) | next_row; // Add next row to current matrix
        total += count_unholy_matrices(rows + 1, new_matrix, memo); // Recurse
    }

    // Memoize and return result
    memo[(rows << 16) | current_matrix] = total;
    return total;
}

int main() {
    unsigned long long memo[8 * (1 << 16)];
    memset(memo, -1, sizeof(memo)); // Initialize memo array with -1

    unsigned long long total_unholy_matrices = count_unholy_matrices(0, 0, memo);

    printf("Total Unholy Matrices: %llu\n", total_unholy_matrices);

    return 0;
}
### Explanation - **Matrix Representation**: The matrix is represented by an `unsigned long`, where each bit corresponds to a cell in the 8x8 matrix. - **`is_unholy` Function**: This function checks for isolated `1`s by iterating through each bit/cell of the matrix and checking the 4 cardinal directions for adjacent `1`s, considering edge cases. - **Dynamic Programming**: The `count_unholy_matrices` function is the core of the dynamic programming approach, which recursively fills the matrix row by row and counts configurations that are unholy. A memoization array (`memo`) is used to cache results for partial configurations to avoid redundant computations. - **Main Algorithm**: The program initializes the memoization array and calls `count_unholy_matrices` starting with an empty matrix (0 rows filled) and accumulates the count of unholy matrices. This approach ensures all possible 8x8 binary matrices are considered, and only those that meet the "unholy" criteria (no isolated `1`s) are counted, leveraging efficient bit manipulation and dynamic programming techniques.

Extract Code

I extracted the following code from that output: 

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#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#define MATRIX_SIZE 64
#define ROW_MASK 0xFF // Mask for a single row (8 bits set)

unsigned long long count_unholy_matrices(int rows, unsigned long current_matrix, unsigned long long *memo);

// Checks if the current matrix configuration is unholy
int is_unholy(unsigned long matrix) {
    for (int i = 0; i < MATRIX_SIZE; i++) {
        if ((matrix >> i) & 1) { // If there's a 1 at position i
            // Check cardinal directions (exclude edges)
            if (!((i % 8 == 0 || (matrix >> (i - 1)) & 1) && // Left
                  (i % 8 == 7 || (matrix >> (i + 1)) & 1) && // Right
                  (i < 8 || (matrix >> (i - 8)) & 1) && // Up
                  (i >= 56 || (matrix >> (i + 8)) & 1))) { // Down
                return 0; // Found an isolated 1
            }
        }
    }
    return 1; // No isolated 1s found, matrix is unholy
}

// Recursive function to count unholy matrices
unsigned long long count_unholy_matrices(int rows, unsigned long current_matrix, unsigned long long *memo) {
    if (rows == 8) { // Base case: all rows filled
        return is_unholy(current_matrix);
    }
    
    // Check if result is already memoized
    if (memo[(rows << 16) | current_matrix] != -1) {
        return memo[(rows << 16) | current_matrix];
    }

    unsigned long long total = 0;

    // Enumerate all possibilities for the next row (2^8)
    for (unsigned int next_row = 0; next_row <= 0xFF; next_row++) {
        unsigned long new_matrix = (current_matrix << 8) | next_row; // Add next row to current matrix
        total += count_unholy_matrices(rows + 1, new_matrix, memo); // Recurse
    }

    // Memoize and return result
    memo[(rows << 16) | current_matrix] = total;
    return total;
}

int main() {
    unsigned long long memo[8 * (1 << 16)];
    memset(memo, -1, sizeof(memo)); // Initialize memo array with -1

    unsigned long long total_unholy_matrices = count_unholy_matrices(0, 0, memo);

    printf("Total Unholy Matrices: %llu\n", total_unholy_matrices);

    return 0;
}

Run Code Interpreter

Running the following program: 

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#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#define MATRIX_SIZE 64
#define ROW_MASK 0xFF // Mask for a single row (8 bits set)

unsigned long long count_unholy_matrices(int rows, unsigned long current_matrix, unsigned long long *memo);

// Checks if the current matrix configuration is unholy
int is_unholy(unsigned long matrix) {
    for (int i = 0; i < MATRIX_SIZE; i++) {
        if ((matrix >> i) & 1) { // If there's a 1 at position i
            // Check cardinal directions (exclude edges)
            if (!((i % 8 == 0 || (matrix >> (i - 1)) & 1) && // Left
                  (i % 8 == 7 || (matrix >> (i + 1)) & 1) && // Right
                  (i < 8 || (matrix >> (i - 8)) & 1) && // Up
                  (i >= 56 || (matrix >> (i + 8)) & 1))) { // Down
                return 0; // Found an isolated 1
            }
        }
    }
    return 1; // No isolated 1s found, matrix is unholy
}

// Recursive function to count unholy matrices
unsigned long long count_unholy_matrices(int rows, unsigned long current_matrix, unsigned long long *memo) {
    if (rows == 8) { // Base case: all rows filled
        return is_unholy(current_matrix);
    }
    
    // Check if result is already memoized
    if (memo[(rows << 16) | current_matrix] != -1) {
        return memo[(rows << 16) | current_matrix];
    }

    unsigned long long total = 0;

    // Enumerate all possibilities for the next row (2^8)
    for (unsigned int next_row = 0; next_row <= 0xFF; next_row++) {
        unsigned long new_matrix = (current_matrix << 8) | next_row; // Add next row to current matrix
        total += count_unholy_matrices(rows + 1, new_matrix, memo); // Recurse
    }

    // Memoize and return result
    memo[(rows << 16) | current_matrix] = total;
    return total;
}

int main() {
    unsigned long long memo[8 * (1 << 16)];
    memset(memo, -1, sizeof(memo)); // Initialize memo array with -1

    unsigned long long total_unholy_matrices = count_unholy_matrices(0, 0, memo);

    printf("Total Unholy Matrices: %llu\n", total_unholy_matrices);

    return 0;
}
And got the output: main.sh: line 2: 16 Segmentation fault (core dumped) ./a.out

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False

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

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.

Output

Given the complexity and constraints of this problem, I'll start by outlining a plan for the program and then proceed to implement it according to the plan. ### Plan for the Program 1. **Represent the Matrix:** - Use an `unsigned long` to represent the 8x8 matrix, where each bit corresponds to a cell in the matrix. 2. **Function `is_unholy`:** - Write a function `is_unholy` that accepts an `unsigned long` representing the matrix and returns if the matrix is unholy. Use bit-shifting and masking to identify isolated 1s. 3. **Dynamic Programming Approach:** - Use dynamic programming to count the number of unholy matrices. The state of the DP will be defined by how many rows have been processed so far. 4. **Enumerate Next Two Rows:** - For each state, enumerate all possible configurations of the next two rows (2^16 possibilities) and check if they lead to an unholy matrix when combined with the state representing the already processed rows. 5. **Recursion with Memoization:** - Implement recursion with memoization to avoid recalculating the count for states that have already been processed. 6. **Main Logic:** - The main function will initialize the dynamic programming and start the recursion from the state representing 0 processed rows. It will output the total count of unholy matrices. ### C Program Implementation 

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

#define MAX_ROWS 8
#define MAX_STATES (1 << 16) // 2^16 possible states for two rows

// Memoization array
unsigned long long dp[MAX_ROWS + 1][MAX_STATES];

// Checks if the given matrix configuration is unholy
bool is_unholy(unsigned long matrix) {
    for (int i = 0; i < 64; i++) { // Check all bits
        if (matrix & (1ULL << i)) { // If bit is 1
            // Check if isolated
            if (!((i % 8 > 0 && (matrix & (1ULL << (i - 1)))) || // Left
                  (i % 8 < 7 && (matrix & (1ULL << (i + 1)))) || // Right
                  (i > 7 && (matrix & (1ULL << (i - 8)))) || // Up
                  (i < 56 && (matrix & (1ULL << (i + 8))))) // Down
               ) {
                return false; // Found an isolated 1
            }
        }
    }
    return true; // No isolated 1s, matrix is unholy
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int processedRows, unsigned long prevRows) {
    if (processedRows == MAX_ROWS)
        return is_unholy(prevRows);

    // Check memoization
    if(dp[processedRows][prevRows] != -1)
        return dp[processedRows][prevRows];

    unsigned long long count = 0;
    for (int i = 0; i < MAX_STATES; i++) { // Enumerate next two rows
        unsigned long nextRows = (prevRows << 16) | i; // Combine with previous state
        if (is_unholy(nextRows)) {
            count += count_unholy(processedRows + 2, nextRows >> (8 * (processedRows >= 6 ? 2 : 0)));
        }
    }
    dp[processedRows][prevRows] = count;
    return count;
}

int main() {
    memset(dp, -1, sizeof(dp)); // Initialize memoization array
    unsigned long long totalCount = count_unholy(0, 0); // Start recursion
    printf("Total number of unholy 8x8 matrices: %llu\n", totalCount);
    return 0;
}
**Note:** This implementation outlines a direct approach to solving the problem. However, due to the exponential nature of the enumeration (2^16 for each step), the actual execution might be impractical without further optimizations or approximations, especially on limited hardware. This code serves as a conceptual demonstration and might require optimization or different techniques (such as parallel processing or heuristic pruning) for practical use cases.

Extract Code

I extracted the following code from that output: 

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

#define MAX_ROWS 8
#define MAX_STATES (1 << 16) // 2^16 possible states for two rows

// Memoization array
unsigned long long dp[MAX_ROWS + 1][MAX_STATES];

// Checks if the given matrix configuration is unholy
bool is_unholy(unsigned long matrix) {
    for (int i = 0; i < 64; i++) { // Check all bits
        if (matrix & (1ULL << i)) { // If bit is 1
            // Check if isolated
            if (!((i % 8 > 0 && (matrix & (1ULL << (i - 1)))) || // Left
                  (i % 8 < 7 && (matrix & (1ULL << (i + 1)))) || // Right
                  (i > 7 && (matrix & (1ULL << (i - 8)))) || // Up
                  (i < 56 && (matrix & (1ULL << (i + 8))))) // Down
               ) {
                return false; // Found an isolated 1
            }
        }
    }
    return true; // No isolated 1s, matrix is unholy
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int processedRows, unsigned long prevRows) {
    if (processedRows == MAX_ROWS)
        return is_unholy(prevRows);

    // Check memoization
    if(dp[processedRows][prevRows] != -1)
        return dp[processedRows][prevRows];

    unsigned long long count = 0;
    for (int i = 0; i < MAX_STATES; i++) { // Enumerate next two rows
        unsigned long nextRows = (prevRows << 16) | i; // Combine with previous state
        if (is_unholy(nextRows)) {
            count += count_unholy(processedRows + 2, nextRows >> (8 * (processedRows >= 6 ? 2 : 0)));
        }
    }
    dp[processedRows][prevRows] = count;
    return count;
}

int main() {
    memset(dp, -1, sizeof(dp)); // Initialize memoization array
    unsigned long long totalCount = count_unholy(0, 0); // Start recursion
    printf("Total number of unholy 8x8 matrices: %llu\n", totalCount);
    return 0;
}

Run Code Interpreter

Running the following program: 

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

#define MAX_ROWS 8
#define MAX_STATES (1 << 16) // 2^16 possible states for two rows

// Memoization array
unsigned long long dp[MAX_ROWS + 1][MAX_STATES];

// Checks if the given matrix configuration is unholy
bool is_unholy(unsigned long matrix) {
    for (int i = 0; i < 64; i++) { // Check all bits
        if (matrix & (1ULL << i)) { // If bit is 1
            // Check if isolated
            if (!((i % 8 > 0 && (matrix & (1ULL << (i - 1)))) || // Left
                  (i % 8 < 7 && (matrix & (1ULL << (i + 1)))) || // Right
                  (i > 7 && (matrix & (1ULL << (i - 8)))) || // Up
                  (i < 56 && (matrix & (1ULL << (i + 8))))) // Down
               ) {
                return false; // Found an isolated 1
            }
        }
    }
    return true; // No isolated 1s, matrix is unholy
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int processedRows, unsigned long prevRows) {
    if (processedRows == MAX_ROWS)
        return is_unholy(prevRows);

    // Check memoization
    if(dp[processedRows][prevRows] != -1)
        return dp[processedRows][prevRows];

    unsigned long long count = 0;
    for (int i = 0; i < MAX_STATES; i++) { // Enumerate next two rows
        unsigned long nextRows = (prevRows << 16) | i; // Combine with previous state
        if (is_unholy(nextRows)) {
            count += count_unholy(processedRows + 2, nextRows >> (8 * (processedRows >= 6 ? 2 : 0)));
        }
    }
    dp[processedRows][prevRows] = count;
    return count;
}

int main() {
    memset(dp, -1, sizeof(dp)); // Initialize memoization array
    unsigned long long totalCount = count_unholy(0, 0); // Start recursion
    printf("Total number of unholy 8x8 matrices: %llu\n", totalCount);
    return 0;
}
And got the output: Timeout: function took too long to complete

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False

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

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.

Output

To tackle this problem, we'll break it down into several steps as outlined in the plan below, and then proceed to implement it in C. ### Plan 1. **Representation**: Use an `unsigned long` to represent an 8x8 binary matrix, where each bit represents a cell in the matrix. 2. **is_unholy Function**: Implement a function to check if a given matrix is unholy. This involves checking each 1 bit to see if it has another 1 bit in any of the 4 cardinal directions. 3. **Dynamic Programming Approach**: Use dynamic programming to count the number of unholy matrices, considering one or two rows at a time and building up to the full 8x8 matrix. 4. **Recursion with Memoization**: Implement recursion to explore all possible matrices row by row, using memoization to avoid recalculating the number of unholy matrices for any given partial configuration. 5. **Bit Arithmetic for Next Rows**: For each partial matrix, enumerate all 2^16 possible configurations of the next two rows, ensuring these do not make the matrix holy (i.e., ensure that no 1s become isolated). ### Implementation 

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#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#define MATRIX_SIZE 8
#define ROW_MASK 0xFF // Mask for one row
#define FULL_MASK 0xFFFFFFFFFFFFFFFF // Mask for the full 8x8 matrix

unsigned long long dp[1 << 16]; // DP array for memoization, large enough for all 2-row states

// Check if a given matrix (up to row) is unholy
int is_unholy(unsigned long matrix, int rows) {
    for (int row = 0; row < rows; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            // If this is a 1, check for neighbors
            if ((matrix >> (row * MATRIX_SIZE + col)) & 1) {
                int hasNeighbor = 0;
                // Check above
                if (row > 0 && (matrix >> ((row - 1) * MATRIX_SIZE + col)) & 1) hasNeighbor = 1;
                // Check below
                if (row < rows - 1 && (matrix >> ((row + 1) * MATRIX_SIZE + col)) & 1) hasNeighbor = 1;
                // Check left
                if (col > 0 && (matrix >> (row * MATRIX_SIZE + col - 1)) & 1) hasNeighbor = 1;
                // Check right
                if (col < MATRIX_SIZE - 1 && (matrix >> (row * MATRIX_SIZE + col + 1)) & 1) hasNeighbor = 1;

                if (!hasNeighbor) return 0; // Found an isolated 1
            }
        }
    }
    return 1; // No isolated 1s found
}

// Recursive function to count unholy matrices, given a partial matrix represented by 'state'
unsigned long long count_unholy(unsigned long state, int rows) {
    // Base case: if we've filled all rows and the matrix is unholy
    if (rows == MATRIX_SIZE) {
        return is_unholy(state, rows);
    }

    // Check memoization
    if (dp[state] != -1) return dp[state];

    unsigned long long count = 0;
    // Try all possible next rows
    for (unsigned int nextRow = 0; nextRow < (1 << MATRIX_SIZE); nextRow++) {
        unsigned long newState = (state << MATRIX_SIZE) | nextRow;
        count += count_unholy(newState, rows + 1);
    }

    // Memoize and return the result
    dp[state] = count;
    return count;
}

int main() {
    memset(dp, -1, sizeof(dp)); // Initialize DP array with -1
    unsigned long long total_unholy = count_unholy(0, 0); // Start with an empty matrix
    printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy);
    return 0;
}
### Note This solution outlines a general approach and a basic implementation. Given the constraints, the actual number of matrices to be considered is extremely large (the recursion explores all possible matrices), and the 2^16 next-row states enumeration may be computationally intensive. In practice, optimizations or approximations might be necessary to handle the large search space efficiently. This code aims to demonstrate the algorithmic approach and may require adaptations for practical use.

Extract Code

I extracted the following code from that output: 

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#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#define MATRIX_SIZE 8
#define ROW_MASK 0xFF // Mask for one row
#define FULL_MASK 0xFFFFFFFFFFFFFFFF // Mask for the full 8x8 matrix

unsigned long long dp[1 << 16]; // DP array for memoization, large enough for all 2-row states

// Check if a given matrix (up to row) is unholy
int is_unholy(unsigned long matrix, int rows) {
    for (int row = 0; row < rows; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            // If this is a 1, check for neighbors
            if ((matrix >> (row * MATRIX_SIZE + col)) & 1) {
                int hasNeighbor = 0;
                // Check above
                if (row > 0 && (matrix >> ((row - 1) * MATRIX_SIZE + col)) & 1) hasNeighbor = 1;
                // Check below
                if (row < rows - 1 && (matrix >> ((row + 1) * MATRIX_SIZE + col)) & 1) hasNeighbor = 1;
                // Check left
                if (col > 0 && (matrix >> (row * MATRIX_SIZE + col - 1)) & 1) hasNeighbor = 1;
                // Check right
                if (col < MATRIX_SIZE - 1 && (matrix >> (row * MATRIX_SIZE + col + 1)) & 1) hasNeighbor = 1;

                if (!hasNeighbor) return 0; // Found an isolated 1
            }
        }
    }
    return 1; // No isolated 1s found
}

// Recursive function to count unholy matrices, given a partial matrix represented by 'state'
unsigned long long count_unholy(unsigned long state, int rows) {
    // Base case: if we've filled all rows and the matrix is unholy
    if (rows == MATRIX_SIZE) {
        return is_unholy(state, rows);
    }

    // Check memoization
    if (dp[state] != -1) return dp[state];

    unsigned long long count = 0;
    // Try all possible next rows
    for (unsigned int nextRow = 0; nextRow < (1 << MATRIX_SIZE); nextRow++) {
        unsigned long newState = (state << MATRIX_SIZE) | nextRow;
        count += count_unholy(newState, rows + 1);
    }

    // Memoize and return the result
    dp[state] = count;
    return count;
}

int main() {
    memset(dp, -1, sizeof(dp)); // Initialize DP array with -1
    unsigned long long total_unholy = count_unholy(0, 0); // Start with an empty matrix
    printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy);
    return 0;
}

Run Code Interpreter

Running the following program: 

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#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#define MATRIX_SIZE 8
#define ROW_MASK 0xFF // Mask for one row
#define FULL_MASK 0xFFFFFFFFFFFFFFFF // Mask for the full 8x8 matrix

unsigned long long dp[1 << 16]; // DP array for memoization, large enough for all 2-row states

// Check if a given matrix (up to row) is unholy
int is_unholy(unsigned long matrix, int rows) {
    for (int row = 0; row < rows; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            // If this is a 1, check for neighbors
            if ((matrix >> (row * MATRIX_SIZE + col)) & 1) {
                int hasNeighbor = 0;
                // Check above
                if (row > 0 && (matrix >> ((row - 1) * MATRIX_SIZE + col)) & 1) hasNeighbor = 1;
                // Check below
                if (row < rows - 1 && (matrix >> ((row + 1) * MATRIX_SIZE + col)) & 1) hasNeighbor = 1;
                // Check left
                if (col > 0 && (matrix >> (row * MATRIX_SIZE + col - 1)) & 1) hasNeighbor = 1;
                // Check right
                if (col < MATRIX_SIZE - 1 && (matrix >> (row * MATRIX_SIZE + col + 1)) & 1) hasNeighbor = 1;

                if (!hasNeighbor) return 0; // Found an isolated 1
            }
        }
    }
    return 1; // No isolated 1s found
}

// Recursive function to count unholy matrices, given a partial matrix represented by 'state'
unsigned long long count_unholy(unsigned long state, int rows) {
    // Base case: if we've filled all rows and the matrix is unholy
    if (rows == MATRIX_SIZE) {
        return is_unholy(state, rows);
    }

    // Check memoization
    if (dp[state] != -1) return dp[state];

    unsigned long long count = 0;
    // Try all possible next rows
    for (unsigned int nextRow = 0; nextRow < (1 << MATRIX_SIZE); nextRow++) {
        unsigned long newState = (state << MATRIX_SIZE) | nextRow;
        count += count_unholy(newState, rows + 1);
    }

    // Memoize and return the result
    dp[state] = count;
    return count;
}

int main() {
    memset(dp, -1, sizeof(dp)); // Initialize DP array with -1
    unsigned long long total_unholy = count_unholy(0, 0); // Start with an empty matrix
    printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy);
    return 0;
}
And got the output: Total number of unholy 8x8 matrices: 146680

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False

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

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.

Output

To tackle this problem, we'll first outline a plan based on the requirements and constraints you've provided, and then we will implement it in C. ### Plan 1. **Matrix Representation**: - Use an `unsigned long` (assuming it's 64-bit) to represent the 8x8 matrix, where each bit corresponds to a cell in the matrix. 2. **is_unholy Function**: - This function will take an `unsigned long` representing the matrix and determine if it is "unholy" by checking that each `1` bit is not isolated. - It will use bit-shifting and masking to inspect the neighbors of each bit. 3. **Dynamic Programming Approach**: - Use a recursive function that takes the number of rows processed so far and the current state of the matrix as arguments. - At each step, enumerate all possible configurations for the next two rows (2^16 possibilities) and check if they can be added without violating the "unholy" property. - Use memoization to avoid recomputing states. 4. **Base Case and Recursion**: - The base case occurs when all 8 rows have been processed. If we reach this point, we have found a valid "unholy" matrix, so we return 1. - Otherwise, for each recursive call, enumerate all possible states for the next two rows, and keep a running total of the counts of "unholy" matrices. 5. **Main Function**: - Initialize the dynamic programming array or use a hash table for memoization. - Call the recursive function with 0 rows processed initially and an empty matrix. - Print the final count of "unholy" matrices. ### Implementation Let's now translate this plan into C code. 

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

// Assuming 64-bit unsigned long
#define MATRIX_SIZE 64
#define ROWS 8
#define COLS 8

// Use a simple hash map for memoization (for demonstration purposes; a real implementation may need a more efficient approach)
unsigned long long dp[1 << (ROWS * 2)]; // Enough space for a simple demonstration
int dp_filled[1 << (ROWS * 2)] = {0}; // To check if a dp value has been filled

// Function to check if a matrix configuration is unholy
int is_unholy(unsigned long matrix) {
    for (int row = 0; row < ROWS; row++) {
        for (int col = 0; col < COLS; col++) {
            int bit_pos = row * COLS + col;
            unsigned long bit = (matrix >> bit_pos) & 1UL;
            if (bit == 1) { // If current bit is 1, check for isolation
                unsigned long neighbors = 0;
                if (row > 0) neighbors |= (matrix >> ((row-1) * COLS + col)) & 1UL; // Up
                if (row < ROWS-1) neighbors |= (matrix >> ((row+1) * COLS + col)) & 1UL; // Down
                if (col > 0) neighbors |= (matrix >> (row * COLS + (col-1))) & 1UL; // Left
                if (col < COLS-1) neighbors |= (matrix >> (row * COLS + (col+1))) & 1UL; // Right
                if (neighbors == 0) return 0; // Isolated 1 found
            }
        }
    }
    return 1; // Not isolated, matrix is unholy
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int rows_processed, unsigned long matrix) {
    if (rows_processed == ROWS) return is_unholy(matrix);
    
    int idx = (rows_processed << 16) | (matrix & ((1UL << 16) - 1)); // Simple hash
    if (dp_filled[idx]) return dp[idx];
    
    unsigned long long count = 0;
    for (unsigned int next_rows = 0; next_rows < (1 << (COLS * 2)); next_rows++) {
        unsigned long new_matrix = (matrix << (COLS * 2)) | next_rows;
        if (is_unholy(new_matrix)) {
            count += count_unholy(rows_processed + 2, new_matrix);
        }
    }
    
    dp[idx] = count;
    dp_filled[idx] = 1;
    return count;
}

int main() {
    unsigned long long total_unholy = count_unholy(0, 0UL);
    printf("Total Unholy Matrices: %llu\n", total_unholy);
    return 0;
}
### Notes - This implementation is a straightforward translation of the plan into code. However, it might not be efficient enough due to the simplistic memoization approach and might need optimizations or a more sophisticated memoization/hash table implementation for practical use. - The `is_unholy` function and the recursive approach should work conceptually, but due to the nature of the problem (enumerating a very large number of possibilities), running this code as-is might not be feasible without further optimizations, especially in memory usage and hash table efficiency.

Extract Code

I extracted the following code from that output: 

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

// Assuming 64-bit unsigned long
#define MATRIX_SIZE 64
#define ROWS 8
#define COLS 8

// Use a simple hash map for memoization (for demonstration purposes; a real implementation may need a more efficient approach)
unsigned long long dp[1 << (ROWS * 2)]; // Enough space for a simple demonstration
int dp_filled[1 << (ROWS * 2)] = {0}; // To check if a dp value has been filled

// Function to check if a matrix configuration is unholy
int is_unholy(unsigned long matrix) {
    for (int row = 0; row < ROWS; row++) {
        for (int col = 0; col < COLS; col++) {
            int bit_pos = row * COLS + col;
            unsigned long bit = (matrix >> bit_pos) & 1UL;
            if (bit == 1) { // If current bit is 1, check for isolation
                unsigned long neighbors = 0;
                if (row > 0) neighbors |= (matrix >> ((row-1) * COLS + col)) & 1UL; // Up
                if (row < ROWS-1) neighbors |= (matrix >> ((row+1) * COLS + col)) & 1UL; // Down
                if (col > 0) neighbors |= (matrix >> (row * COLS + (col-1))) & 1UL; // Left
                if (col < COLS-1) neighbors |= (matrix >> (row * COLS + (col+1))) & 1UL; // Right
                if (neighbors == 0) return 0; // Isolated 1 found
            }
        }
    }
    return 1; // Not isolated, matrix is unholy
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int rows_processed, unsigned long matrix) {
    if (rows_processed == ROWS) return is_unholy(matrix);
    
    int idx = (rows_processed << 16) | (matrix & ((1UL << 16) - 1)); // Simple hash
    if (dp_filled[idx]) return dp[idx];
    
    unsigned long long count = 0;
    for (unsigned int next_rows = 0; next_rows < (1 << (COLS * 2)); next_rows++) {
        unsigned long new_matrix = (matrix << (COLS * 2)) | next_rows;
        if (is_unholy(new_matrix)) {
            count += count_unholy(rows_processed + 2, new_matrix);
        }
    }
    
    dp[idx] = count;
    dp_filled[idx] = 1;
    return count;
}

int main() {
    unsigned long long total_unholy = count_unholy(0, 0UL);
    printf("Total Unholy Matrices: %llu\n", total_unholy);
    return 0;
}

Run Code Interpreter

Running the following program: 

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

// Assuming 64-bit unsigned long
#define MATRIX_SIZE 64
#define ROWS 8
#define COLS 8

// Use a simple hash map for memoization (for demonstration purposes; a real implementation may need a more efficient approach)
unsigned long long dp[1 << (ROWS * 2)]; // Enough space for a simple demonstration
int dp_filled[1 << (ROWS * 2)] = {0}; // To check if a dp value has been filled

// Function to check if a matrix configuration is unholy
int is_unholy(unsigned long matrix) {
    for (int row = 0; row < ROWS; row++) {
        for (int col = 0; col < COLS; col++) {
            int bit_pos = row * COLS + col;
            unsigned long bit = (matrix >> bit_pos) & 1UL;
            if (bit == 1) { // If current bit is 1, check for isolation
                unsigned long neighbors = 0;
                if (row > 0) neighbors |= (matrix >> ((row-1) * COLS + col)) & 1UL; // Up
                if (row < ROWS-1) neighbors |= (matrix >> ((row+1) * COLS + col)) & 1UL; // Down
                if (col > 0) neighbors |= (matrix >> (row * COLS + (col-1))) & 1UL; // Left
                if (col < COLS-1) neighbors |= (matrix >> (row * COLS + (col+1))) & 1UL; // Right
                if (neighbors == 0) return 0; // Isolated 1 found
            }
        }
    }
    return 1; // Not isolated, matrix is unholy
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int rows_processed, unsigned long matrix) {
    if (rows_processed == ROWS) return is_unholy(matrix);
    
    int idx = (rows_processed << 16) | (matrix & ((1UL << 16) - 1)); // Simple hash
    if (dp_filled[idx]) return dp[idx];
    
    unsigned long long count = 0;
    for (unsigned int next_rows = 0; next_rows < (1 << (COLS * 2)); next_rows++) {
        unsigned long new_matrix = (matrix << (COLS * 2)) | next_rows;
        if (is_unholy(new_matrix)) {
            count += count_unholy(rows_processed + 2, new_matrix);
        }
    }
    
    dp[idx] = count;
    dp_filled[idx] = 1;
    return count;
}

int main() {
    unsigned long long total_unholy = count_unholy(0, 0UL);
    printf("Total Unholy Matrices: %llu\n", total_unholy);
    return 0;
}
And got the output: main.sh: line 2: 16 Segmentation fault (core dumped) ./a.out

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False