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

Here's a comprehensive solution to the problem: Program Plan: 1. Data Structures: - Use unsigned long (64 bits) to represent each row - Use dynamic programming with memoization - Create a memo to store intermediate results 2. Key Functions: a. is_isolated(row, above_row): - Check if any 1 in current row is isolated - Use bit manipulation to check cardinal directions b. is_unholy(matrix, num_rows): - Iterate through matrix - Use is_isolated to check each cell - Return false if any isolated 1 found c. count_unholy_matrices(): - Use recursive DP - Memoize results based on current rows - Explore all possible next two rows - Prune impossible configurations Here's the implementation: 

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

#define MATRIX_SIZE 8
#define MEMO_SIZE (1 << 16)

uint64_t memo[MATRIX_SIZE + 1][MEMO_SIZE];
int memo_initialized = 0;

// Check if a 1 is isolated in a specific position
int is_isolated(uint64_t row, uint64_t above_row, int col) {
    // Check if current cell is 1
    if (!(row & (1ULL << col))) return 0;

    // Check cardinal directions
    int left = (col > 0) && (row & (1ULL << (col-1)));
    int right = (col < 7) && (row & (1ULL << (col+1)));
    int up = (above_row & (1ULL << col));

    return !(left || right || up);
}

// Check if matrix configuration is unholy
int is_unholy(uint64_t prev_row, uint64_t current_row) {
    for (int col = 0; col < MATRIX_SIZE; col++) {
        if (is_isolated(current_row, prev_row, col)) {
            return 0;  // Found an isolated 1
        }
    }
    return 1;  // No isolated 1s
}

// Dynamic programming to count unholy matrices
uint64_t count_unholy_matrices_dp(int rows_left, uint64_t prev_rows) {
    // Base case: completed matrix
    if (rows_left == 0) return 1;

    // Check memoization
    if (memo_initialized && memo[rows_left][prev_rows] != 0) {
        return memo[rows_left][prev_rows];
    }

    uint64_t total_count = 0;

    // Try all possible next two rows
    for (uint64_t row1 = 0; row1 < (1ULL << MATRIX_SIZE); row1++) {
        for (uint64_t row2 = 0; row2 < (1ULL << MATRIX_SIZE); row2++) {
            // Check if configuration is valid
            if (is_unholy(prev_rows, row1) && 
                is_unholy(row1, row2)) {
                
                // Recursive call
                total_count += count_unholy_matrices_dp(
                    rows_left - 2, 
                    row2
                );
            }
        }
    }

    // Memoize result
    if (memo_initialized) {
        memo[rows_left][prev_rows] = total_count;
    }

    return total_count;
}

uint64_t count_unholy_matrices() {
    memo_initialized = 1;
    memset(memo, 0, sizeof(memo));
    
    return count_unholy_matrices_dp(MATRIX_SIZE, 0);
}

int main() {
    uint64_t result = count_unholy_matrices();
    printf("Total number of unholy 8x8 matrices: %llu\n", result);
    return 0;
}
Key Implementation Details: 1. Uses 64-bit unsigned long for bit manipulation 2. Recursive dynamic programming approach 3. Memoization to cache intermediate results 4. Checks for isolated 1s using bit shifting and masking 5. Explores all possible row configurations Time Complexity: O(2^16 * rows) Space Complexity: O(2^16 * rows) The program systematically explores all possible matrix configurations, efficiently pruning invalid ones and using memoization to avoid redundant computations. Compilation and Running: 
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gcc -O3 unholy_matrices.c -o unholy_matrices
./unholy_matrices
Note: This is a computationally intensive problem. The program might take some time to compute the result, depending on your machine's performance. Potential Optimizations: 1. Use more aggressive pruning techniques 2. Implement parallel processing 3. Use more compact memoization strategies Would you like me to elaborate on any part of the implementation?

Extract Code

I extracted the following code from that output: 

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

#define MATRIX_SIZE 8
#define MEMO_SIZE (1 << 16)

uint64_t memo[MATRIX_SIZE + 1][MEMO_SIZE];
int memo_initialized = 0;

// Check if a 1 is isolated in a specific position
int is_isolated(uint64_t row, uint64_t above_row, int col) {
    // Check if current cell is 1
    if (!(row & (1ULL << col))) return 0;

    // Check cardinal directions
    int left = (col > 0) && (row & (1ULL << (col-1)));
    int right = (col < 7) && (row & (1ULL << (col+1)));
    int up = (above_row & (1ULL << col));

    return !(left || right || up);
}

// Check if matrix configuration is unholy
int is_unholy(uint64_t prev_row, uint64_t current_row) {
    for (int col = 0; col < MATRIX_SIZE; col++) {
        if (is_isolated(current_row, prev_row, col)) {
            return 0;  // Found an isolated 1
        }
    }
    return 1;  // No isolated 1s
}

// Dynamic programming to count unholy matrices
uint64_t count_unholy_matrices_dp(int rows_left, uint64_t prev_rows) {
    // Base case: completed matrix
    if (rows_left == 0) return 1;

    // Check memoization
    if (memo_initialized && memo[rows_left][prev_rows] != 0) {
        return memo[rows_left][prev_rows];
    }

    uint64_t total_count = 0;

    // Try all possible next two rows
    for (uint64_t row1 = 0; row1 < (1ULL << MATRIX_SIZE); row1++) {
        for (uint64_t row2 = 0; row2 < (1ULL << MATRIX_SIZE); row2++) {
            // Check if configuration is valid
            if (is_unholy(prev_rows, row1) && 
                is_unholy(row1, row2)) {
                
                // Recursive call
                total_count += count_unholy_matrices_dp(
                    rows_left - 2, 
                    row2
                );
            }
        }
    }

    // Memoize result
    if (memo_initialized) {
        memo[rows_left][prev_rows] = total_count;
    }

    return total_count;
}

uint64_t count_unholy_matrices() {
    memo_initialized = 1;
    memset(memo, 0, sizeof(memo));
    
    return count_unholy_matrices_dp(MATRIX_SIZE, 0);
}

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

Run Code Interpreter

Running the following program: 

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

#define MATRIX_SIZE 8
#define MEMO_SIZE (1 << 16)

uint64_t memo[MATRIX_SIZE + 1][MEMO_SIZE];
int memo_initialized = 0;

// Check if a 1 is isolated in a specific position
int is_isolated(uint64_t row, uint64_t above_row, int col) {
    // Check if current cell is 1
    if (!(row & (1ULL << col))) return 0;

    // Check cardinal directions
    int left = (col > 0) && (row & (1ULL << (col-1)));
    int right = (col < 7) && (row & (1ULL << (col+1)));
    int up = (above_row & (1ULL << col));

    return !(left || right || up);
}

// Check if matrix configuration is unholy
int is_unholy(uint64_t prev_row, uint64_t current_row) {
    for (int col = 0; col < MATRIX_SIZE; col++) {
        if (is_isolated(current_row, prev_row, col)) {
            return 0;  // Found an isolated 1
        }
    }
    return 1;  // No isolated 1s
}

// Dynamic programming to count unholy matrices
uint64_t count_unholy_matrices_dp(int rows_left, uint64_t prev_rows) {
    // Base case: completed matrix
    if (rows_left == 0) return 1;

    // Check memoization
    if (memo_initialized && memo[rows_left][prev_rows] != 0) {
        return memo[rows_left][prev_rows];
    }

    uint64_t total_count = 0;

    // Try all possible next two rows
    for (uint64_t row1 = 0; row1 < (1ULL << MATRIX_SIZE); row1++) {
        for (uint64_t row2 = 0; row2 < (1ULL << MATRIX_SIZE); row2++) {
            // Check if configuration is valid
            if (is_unholy(prev_rows, row1) && 
                is_unholy(row1, row2)) {
                
                // Recursive call
                total_count += count_unholy_matrices_dp(
                    rows_left - 2, 
                    row2
                );
            }
        }
    }

    // Memoize result
    if (memo_initialized) {
        memo[rows_left][prev_rows] = total_count;
    }

    return total_count;
}

uint64_t count_unholy_matrices() {
    memo_initialized = 1;
    memset(memo, 0, sizeof(memo));
    
    return count_unholy_matrices_dp(MATRIX_SIZE, 0);
}

int main() {
    uint64_t result = count_unholy_matrices();
    printf("Total number of unholy 8x8 matrices: %llu\n", result);
    return 0;
}
And got the output: Total number of unholy 8x8 matrices: 53003772538202116 main.c: In function 'main': main.c:80:53: warning: format '%llu' expects argument of type 'long long unsigned int', but argument 2 has type 'uint64_t' {aka 'long unsigned int'} [-Wformat=] 80 | printf("Total number of unholy 8x8 matrices: %llu\n", result); | ~~~^ ~~~~~~ | | | | | uint64_t {aka long unsigned int} | long long unsigned int | %lu

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

Here's a comprehensive plan and implementation for solving the unholy matrix problem: Plan: 1. Define key data structures and constants 2. Implement bit manipulation helper functions 3. Create an is_unholy function 4. Develop a dynamic programming solution with memoization 5. Main function to drive the computation Here's an implementation following this plan: 

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

#define MATRIX_SIZE 8
#define MAX_ROWS (1 << MATRIX_SIZE)

// Memoization cache
uint64_t memo[MATRIX_SIZE + 1][MAX_ROWS][MAX_ROWS];
int memo_initialized = 0;

// Function to check if a specific cell is isolated
int is_isolated(uint64_t matrix, int row, int col) {
    // Check if the bit is set
    if (!(matrix & (1ULL << (row * MATRIX_SIZE + col)))) {
        return 0;
    }

    // Cardinal direction masks
    uint64_t up_mask = (row > 0) ? (1ULL << ((row-1) * MATRIX_SIZE + col)) : 0;
    uint64_t down_mask = (row < MATRIX_SIZE-1) ? (1ULL << ((row+1) * MATRIX_SIZE + col)) : 0;
    uint64_t left_mask = (col > 0) ? (1ULL << (row * MATRIX_SIZE + col - 1)) : 0;
    uint64_t right_mask = (col < MATRIX_SIZE-1) ? (1ULL << (row * MATRIX_SIZE + col + 1)) : 0;

    // Check if no neighboring cells have 1s
    return !(matrix & (up_mask | down_mask | left_mask | right_mask));
}

// Check if entire matrix is unholy
int is_unholy(uint64_t matrix) {
    for (int row = 0; row < MATRIX_SIZE; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            if (is_isolated(matrix, row, col)) {
                return 0;  // Found an isolated 1
            }
        }
    }
    return 1;  // No isolated 1s
}

// Dynamic programming solution
uint64_t count_unholy_matrices(int current_row, uint64_t prev_rows, uint64_t current_rows) {
    // Base case: completed matrix
    if (current_row == MATRIX_SIZE) {
        return 1;
    }

    // Check memoization cache
    if (memo_initialized && memo[current_row][prev_rows][current_rows] != 0) {
        return memo[current_row][prev_rows][current_rows];
    }

    uint64_t total_count = 0;

    // Try all possible configurations for the next two rows
    for (int next_row_config = 0; next_row_config < (1 << (2 * MATRIX_SIZE)); next_row_config++) {
        uint64_t combined_matrix = prev_rows | (current_rows << MATRIX_SIZE) | 
                                   ((uint64_t)next_row_config << (2 * MATRIX_SIZE));

        // Check if current configuration is valid
        if (is_unholy(combined_matrix)) {
            total_count += count_unholy_matrices(
                current_row + 1, 
                current_rows, 
                next_row_config & ((1 << MATRIX_SIZE) - 1)
            );
        }
    }

    // Memoize result
    if (memo_initialized) {
        memo[current_row][prev_rows][current_rows] = total_count;
    }

    return total_count;
}

int main() {
    // Initialize memoization cache
    memo_initialized = 1;
    memset(memo, 0, sizeof(memo));

    // Compute total unholy matrices
    uint64_t total = count_unholy_matrices(0, 0, 0);

    printf("Total number of unholy 8x8 matrices: %llu\n", total);

    return 0;
}
Key Design Choices: 1. Use 64-bit unsigned integers to represent matrices 2. Recursive dynamic programming with memoization 3. Enumerate all possible matrix configurations 4. Use bit manipulation for efficient checks Complexity Considerations: - Time complexity: O(2^(2*8) * 8) - Space complexity: O(8 * 2^8 * 2^8) for memoization - Potential optimization: pruning impossible configurations earlier Potential Improvements: 1. Add more robust error handling 2. Parallelize the computation 3. More advanced pruning techniques The program systematically explores all possible matrix configurations, checking each for the "unholy" property and counting valid arrangements. Would you like me to elaborate on any specific aspect of the solution?

Extract Code

I extracted the following code from that output: 

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

#define MATRIX_SIZE 8
#define MAX_ROWS (1 << MATRIX_SIZE)

// Memoization cache
uint64_t memo[MATRIX_SIZE + 1][MAX_ROWS][MAX_ROWS];
int memo_initialized = 0;

// Function to check if a specific cell is isolated
int is_isolated(uint64_t matrix, int row, int col) {
    // Check if the bit is set
    if (!(matrix & (1ULL << (row * MATRIX_SIZE + col)))) {
        return 0;
    }

    // Cardinal direction masks
    uint64_t up_mask = (row > 0) ? (1ULL << ((row-1) * MATRIX_SIZE + col)) : 0;
    uint64_t down_mask = (row < MATRIX_SIZE-1) ? (1ULL << ((row+1) * MATRIX_SIZE + col)) : 0;
    uint64_t left_mask = (col > 0) ? (1ULL << (row * MATRIX_SIZE + col - 1)) : 0;
    uint64_t right_mask = (col < MATRIX_SIZE-1) ? (1ULL << (row * MATRIX_SIZE + col + 1)) : 0;

    // Check if no neighboring cells have 1s
    return !(matrix & (up_mask | down_mask | left_mask | right_mask));
}

// Check if entire matrix is unholy
int is_unholy(uint64_t matrix) {
    for (int row = 0; row < MATRIX_SIZE; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            if (is_isolated(matrix, row, col)) {
                return 0;  // Found an isolated 1
            }
        }
    }
    return 1;  // No isolated 1s
}

// Dynamic programming solution
uint64_t count_unholy_matrices(int current_row, uint64_t prev_rows, uint64_t current_rows) {
    // Base case: completed matrix
    if (current_row == MATRIX_SIZE) {
        return 1;
    }

    // Check memoization cache
    if (memo_initialized && memo[current_row][prev_rows][current_rows] != 0) {
        return memo[current_row][prev_rows][current_rows];
    }

    uint64_t total_count = 0;

    // Try all possible configurations for the next two rows
    for (int next_row_config = 0; next_row_config < (1 << (2 * MATRIX_SIZE)); next_row_config++) {
        uint64_t combined_matrix = prev_rows | (current_rows << MATRIX_SIZE) | 
                                   ((uint64_t)next_row_config << (2 * MATRIX_SIZE));

        // Check if current configuration is valid
        if (is_unholy(combined_matrix)) {
            total_count += count_unholy_matrices(
                current_row + 1, 
                current_rows, 
                next_row_config & ((1 << MATRIX_SIZE) - 1)
            );
        }
    }

    // Memoize result
    if (memo_initialized) {
        memo[current_row][prev_rows][current_rows] = total_count;
    }

    return total_count;
}

int main() {
    // Initialize memoization cache
    memo_initialized = 1;
    memset(memo, 0, sizeof(memo));

    // Compute total unholy matrices
    uint64_t total = count_unholy_matrices(0, 0, 0);

    printf("Total number of unholy 8x8 matrices: %llu\n", total);

    return 0;
}

Run Code Interpreter

Running the following program: 

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

#define MATRIX_SIZE 8
#define MAX_ROWS (1 << MATRIX_SIZE)

// Memoization cache
uint64_t memo[MATRIX_SIZE + 1][MAX_ROWS][MAX_ROWS];
int memo_initialized = 0;

// Function to check if a specific cell is isolated
int is_isolated(uint64_t matrix, int row, int col) {
    // Check if the bit is set
    if (!(matrix & (1ULL << (row * MATRIX_SIZE + col)))) {
        return 0;
    }

    // Cardinal direction masks
    uint64_t up_mask = (row > 0) ? (1ULL << ((row-1) * MATRIX_SIZE + col)) : 0;
    uint64_t down_mask = (row < MATRIX_SIZE-1) ? (1ULL << ((row+1) * MATRIX_SIZE + col)) : 0;
    uint64_t left_mask = (col > 0) ? (1ULL << (row * MATRIX_SIZE + col - 1)) : 0;
    uint64_t right_mask = (col < MATRIX_SIZE-1) ? (1ULL << (row * MATRIX_SIZE + col + 1)) : 0;

    // Check if no neighboring cells have 1s
    return !(matrix & (up_mask | down_mask | left_mask | right_mask));
}

// Check if entire matrix is unholy
int is_unholy(uint64_t matrix) {
    for (int row = 0; row < MATRIX_SIZE; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            if (is_isolated(matrix, row, col)) {
                return 0;  // Found an isolated 1
            }
        }
    }
    return 1;  // No isolated 1s
}

// Dynamic programming solution
uint64_t count_unholy_matrices(int current_row, uint64_t prev_rows, uint64_t current_rows) {
    // Base case: completed matrix
    if (current_row == MATRIX_SIZE) {
        return 1;
    }

    // Check memoization cache
    if (memo_initialized && memo[current_row][prev_rows][current_rows] != 0) {
        return memo[current_row][prev_rows][current_rows];
    }

    uint64_t total_count = 0;

    // Try all possible configurations for the next two rows
    for (int next_row_config = 0; next_row_config < (1 << (2 * MATRIX_SIZE)); next_row_config++) {
        uint64_t combined_matrix = prev_rows | (current_rows << MATRIX_SIZE) | 
                                   ((uint64_t)next_row_config << (2 * MATRIX_SIZE));

        // Check if current configuration is valid
        if (is_unholy(combined_matrix)) {
            total_count += count_unholy_matrices(
                current_row + 1, 
                current_rows, 
                next_row_config & ((1 << MATRIX_SIZE) - 1)
            );
        }
    }

    // Memoize result
    if (memo_initialized) {
        memo[current_row][prev_rows][current_rows] = total_count;
    }

    return total_count;
}

int main() {
    // Initialize memoization cache
    memo_initialized = 1;
    memset(memo, 0, sizeof(memo));

    // Compute total unholy matrices
    uint64_t total = count_unholy_matrices(0, 0, 0);

    printf("Total number of unholy 8x8 matrices: %llu\n", total);

    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

Here's a comprehensive plan and implementation for solving the unholy matrix problem: Program Plan: 1. Data Structures - Use unsigned long (64-bit) to represent each row - Use dynamic programming with memoization - Create a cache to store intermediate results 2. Core Functions: a) is_unholy function - Check if a 1 is isolated by checking cardinal directions - Use bit manipulation to test adjacency b) count_unholy_matrices function - Recursive with memoization - Base case: when all 8 rows are filled - Recursive case: try all possible next two rows - Memoize results to avoid recomputing 3. Main function - Initialize memoization cache - Call recursive function - Print total count of unholy matrices Here's a C implementation following this plan: 

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

#define SIZE 8
#define CACHE_SIZE (1ULL << 16)

// Memoization cache
uint64_t cache[SIZE + 1][CACHE_SIZE];
int cache_initialized = 0;

// Check if a 1 is isolated in a row
int is_isolated(unsigned long row, int pos) {
    // Check left and right
    int left_check = (pos > 0 && (row & (1UL << (pos-1))));
    int right_check = (pos < SIZE-1 && (row & (1UL << (pos+1))));
    
    return !(left_check || right_check);
}

// Check if a matrix configuration is unholy
int is_unholy(unsigned long prev_row, unsigned long curr_row) {
    for (int i = 0; i < SIZE; i++) {
        if (curr_row & (1UL << i)) {
            // Check if 1 is isolated vertically and horizontally
            if (is_isolated(curr_row, i) &&
                (!prev_row || is_isolated(prev_row, i))) {
                return 0;  // Found an isolated 1
            }
        }
    }
    return 1;  // No isolated 1s
}

// Recursive function with memoization to count unholy matrices
uint64_t count_unholy_matrices(int row, unsigned long prev_rows) {
    // Base case: all rows filled
    if (row == SIZE) return 1;
    
    // Check memoized result
    if (cache_initialized && cache[row][prev_rows] != 0) {
        return cache[row][prev_rows];
    }
    
    uint64_t total = 0;
    
    // Try all possible 2^16 configurations for next two rows
    for (unsigned long next_row1 = 0; next_row1 < (1UL << SIZE); next_row1++) {
        for (unsigned long next_row2 = 0; next_row2 < (1UL << SIZE); next_row2++) {
            // Extract previous rows from state
            unsigned long prev_row1 = (prev_rows >> SIZE) & ((1UL << SIZE) - 1);
            unsigned long prev_row2 = prev_rows & ((1UL << SIZE) - 1);
            
            // Check if current configuration is unholy
            if (is_unholy(prev_row1, next_row1) && 
                is_unholy(next_row1, next_row2)) {
                // Recursive call with updated state
                unsigned long new_state = (next_row1 << SIZE) | next_row2;
                total += count_unholy_matrices(row + 2, new_state);
            }
        }
    }
    
    // Memoize result
    if (cache_initialized) {
        cache[row][prev_rows] = total;
    }
    
    return total;
}

int main() {
    // Initialize cache
    cache_initialized = 1;
    memset(cache, 0, sizeof(cache));
    
    // Count unholy matrices
    uint64_t result = count_unholy_matrices(0, 0);
    
    printf("Total number of unholy 8x8 matrices: %llu\n", result);
    
    return 0;
}
Key Implementation Details: 1. Uses bit manipulation for efficient matrix representation 2. Recursive dynamic programming approach 3. Memoization to cache intermediate results 4. Checks for isolated 1s using bitwise operations 5. Explores all possible row configurations Time Complexity: O(2^16 * 8) Space Complexity: O(2^16) Performance Considerations: - Large state space (2^16) - Memoization helps reduce redundant computations - May require significant memory Potential Optimizations: - Bit-level pruning - Parallel computation - More compact memoization strategy The program systematically explores all possible matrix configurations, identifying and counting those without isolated 1s.

Extract Code

I extracted the following code from that output: 

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

#define SIZE 8
#define CACHE_SIZE (1ULL << 16)

// Memoization cache
uint64_t cache[SIZE + 1][CACHE_SIZE];
int cache_initialized = 0;

// Check if a 1 is isolated in a row
int is_isolated(unsigned long row, int pos) {
    // Check left and right
    int left_check = (pos > 0 && (row & (1UL << (pos-1))));
    int right_check = (pos < SIZE-1 && (row & (1UL << (pos+1))));
    
    return !(left_check || right_check);
}

// Check if a matrix configuration is unholy
int is_unholy(unsigned long prev_row, unsigned long curr_row) {
    for (int i = 0; i < SIZE; i++) {
        if (curr_row & (1UL << i)) {
            // Check if 1 is isolated vertically and horizontally
            if (is_isolated(curr_row, i) &&
                (!prev_row || is_isolated(prev_row, i))) {
                return 0;  // Found an isolated 1
            }
        }
    }
    return 1;  // No isolated 1s
}

// Recursive function with memoization to count unholy matrices
uint64_t count_unholy_matrices(int row, unsigned long prev_rows) {
    // Base case: all rows filled
    if (row == SIZE) return 1;
    
    // Check memoized result
    if (cache_initialized && cache[row][prev_rows] != 0) {
        return cache[row][prev_rows];
    }
    
    uint64_t total = 0;
    
    // Try all possible 2^16 configurations for next two rows
    for (unsigned long next_row1 = 0; next_row1 < (1UL << SIZE); next_row1++) {
        for (unsigned long next_row2 = 0; next_row2 < (1UL << SIZE); next_row2++) {
            // Extract previous rows from state
            unsigned long prev_row1 = (prev_rows >> SIZE) & ((1UL << SIZE) - 1);
            unsigned long prev_row2 = prev_rows & ((1UL << SIZE) - 1);
            
            // Check if current configuration is unholy
            if (is_unholy(prev_row1, next_row1) && 
                is_unholy(next_row1, next_row2)) {
                // Recursive call with updated state
                unsigned long new_state = (next_row1 << SIZE) | next_row2;
                total += count_unholy_matrices(row + 2, new_state);
            }
        }
    }
    
    // Memoize result
    if (cache_initialized) {
        cache[row][prev_rows] = total;
    }
    
    return total;
}

int main() {
    // Initialize cache
    cache_initialized = 1;
    memset(cache, 0, sizeof(cache));
    
    // Count unholy matrices
    uint64_t result = count_unholy_matrices(0, 0);
    
    printf("Total number of unholy 8x8 matrices: %llu\n", result);
    
    return 0;
}

Run Code Interpreter

Running the following program: 

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

#define SIZE 8
#define CACHE_SIZE (1ULL << 16)

// Memoization cache
uint64_t cache[SIZE + 1][CACHE_SIZE];
int cache_initialized = 0;

// Check if a 1 is isolated in a row
int is_isolated(unsigned long row, int pos) {
    // Check left and right
    int left_check = (pos > 0 && (row & (1UL << (pos-1))));
    int right_check = (pos < SIZE-1 && (row & (1UL << (pos+1))));
    
    return !(left_check || right_check);
}

// Check if a matrix configuration is unholy
int is_unholy(unsigned long prev_row, unsigned long curr_row) {
    for (int i = 0; i < SIZE; i++) {
        if (curr_row & (1UL << i)) {
            // Check if 1 is isolated vertically and horizontally
            if (is_isolated(curr_row, i) &&
                (!prev_row || is_isolated(prev_row, i))) {
                return 0;  // Found an isolated 1
            }
        }
    }
    return 1;  // No isolated 1s
}

// Recursive function with memoization to count unholy matrices
uint64_t count_unholy_matrices(int row, unsigned long prev_rows) {
    // Base case: all rows filled
    if (row == SIZE) return 1;
    
    // Check memoized result
    if (cache_initialized && cache[row][prev_rows] != 0) {
        return cache[row][prev_rows];
    }
    
    uint64_t total = 0;
    
    // Try all possible 2^16 configurations for next two rows
    for (unsigned long next_row1 = 0; next_row1 < (1UL << SIZE); next_row1++) {
        for (unsigned long next_row2 = 0; next_row2 < (1UL << SIZE); next_row2++) {
            // Extract previous rows from state
            unsigned long prev_row1 = (prev_rows >> SIZE) & ((1UL << SIZE) - 1);
            unsigned long prev_row2 = prev_rows & ((1UL << SIZE) - 1);
            
            // Check if current configuration is unholy
            if (is_unholy(prev_row1, next_row1) && 
                is_unholy(next_row1, next_row2)) {
                // Recursive call with updated state
                unsigned long new_state = (next_row1 << SIZE) | next_row2;
                total += count_unholy_matrices(row + 2, new_state);
            }
        }
    }
    
    // Memoize result
    if (cache_initialized) {
        cache[row][prev_rows] = total;
    }
    
    return total;
}

int main() {
    // Initialize cache
    cache_initialized = 1;
    memset(cache, 0, sizeof(cache));
    
    // Count unholy matrices
    uint64_t result = count_unholy_matrices(0, 0);
    
    printf("Total number of unholy 8x8 matrices: %llu\n", result);
    
    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

Here's a comprehensive solution to the problem: Plan: 1. Define a matrix representation using unsigned long (64 bits) 2. Create an is_isolated function to check if a single cell is isolated 3. Create an is_unholy function that checks the entire matrix 4. Implement a dynamic programming solution with memoization 5. Main function to calculate and print the total number of unholy matrices Here's the implementation: 

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

#define MATRIX_SIZE 8
#define MAX_MEMO_SIZE (1 << 16)

// Check if a specific bit is 1
#define IS_BIT_SET(val, pos) ((val) & (1ULL << (pos)))

// Memoization table
uint64_t memo[MATRIX_SIZE + 1][MAX_MEMO_SIZE];
int memo_init[MATRIX_SIZE + 1][MAX_MEMO_SIZE];

// Check if a single cell is isolated
int is_isolated(unsigned long matrix, int row, int col) {
    // Check if current cell is 0
    if (!IS_BIT_SET(matrix, row * MATRIX_SIZE + col)) 
        return 0;

    // Check left
    if (col > 0 && IS_BIT_SET(matrix, row * MATRIX_SIZE + col - 1))
        return 0;
    
    // Check right
    if (col < MATRIX_SIZE - 1 && IS_BIT_SET(matrix, row * MATRIX_SIZE + col + 1))
        return 0;
    
    // Check up
    if (row > 0 && IS_BIT_SET(matrix, (row - 1) * MATRIX_SIZE + col))
        return 0;
    
    // Check down
    if (row < MATRIX_SIZE - 1 && IS_BIT_SET(matrix, (row + 1) * MATRIX_SIZE + col))
        return 0;
    
    return 1;
}

// Check if the entire matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int row = 0; row < MATRIX_SIZE; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            if (is_isolated(matrix, row, col))
                return 0;
        }
    }
    return 1;
}

// Dynamic programming to count unholy matrices
uint64_t count_unholy_matrices(int rows_left, unsigned long prev_rows) {
    // Base case: if we've filled all rows
    if (rows_left == 0)
        return 1;
    
    // Check memoization
    if (memo_init[rows_left][prev_rows])
        return memo[rows_left][prev_rows];
    
    uint64_t total_count = 0;
    
    // Try all possible next two rows
    for (unsigned long next_rows = 0; next_rows < (1ULL << (2 * MATRIX_SIZE)); next_rows++) {
        // Combine previous rows and new rows
        unsigned long current_matrix = (prev_rows << (2 * MATRIX_SIZE)) | next_rows;
        
        // Check if this configuration is unholy
        if (is_unholy(current_matrix)) {
            total_count += count_unholy_matrices(rows_left - 2, next_rows);
        }
    }
    
    // Memoize and return
    memo[rows_left][prev_rows] = total_count;
    memo_init[rows_left][prev_rows] = 1;
    
    return total_count;
}

int main() {
    // Clear memoization table
    memset(memo, 0, sizeof(memo));
    memset(memo_init, 0, sizeof(memo_init));
    
    // Start with 0 previous rows, 8 rows to fill
    uint64_t result = count_unholy_matrices(MATRIX_SIZE, 0);
    
    printf("Total number of unholy 8x8 matrices: %llu\n", result);
    
    return 0;
}
Key points about the solution: 1. Matrix Representation: - Uses an unsigned long (64-bit) to represent the entire matrix - Bits are laid out row by row - Efficient bitwise operations for checking cells 2. Isolation Check: - `is_isolated()` checks a single cell's isolation by looking at 4 cardinal directions - `is_unholy()` checks the entire matrix for isolated 1s 3. Dynamic Programming: - Recursive function `count_unholy_matrices()` - Memoization to cache and reuse subproblem solutions - Enumerates all possible 2-row configurations - Builds matrices row by row 4. Complexity: - Time: O(8 * 2^(2*8)) ≈ O(2^16) - Space: O(2^16) The program will output the total number of "unholy" 8x8 matrices where no 1 is isolated. Note: Due to computational complexity, this might take a while to run and could require significant memory. Potential Improvements: - Could use more efficient memoization techniques - Parallel processing for faster computation - Pruning unnecessary matrix explorations

Extract Code

I extracted the following code from that output: 

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

#define MATRIX_SIZE 8
#define MAX_MEMO_SIZE (1 << 16)

// Check if a specific bit is 1
#define IS_BIT_SET(val, pos) ((val) & (1ULL << (pos)))

// Memoization table
uint64_t memo[MATRIX_SIZE + 1][MAX_MEMO_SIZE];
int memo_init[MATRIX_SIZE + 1][MAX_MEMO_SIZE];

// Check if a single cell is isolated
int is_isolated(unsigned long matrix, int row, int col) {
    // Check if current cell is 0
    if (!IS_BIT_SET(matrix, row * MATRIX_SIZE + col)) 
        return 0;

    // Check left
    if (col > 0 && IS_BIT_SET(matrix, row * MATRIX_SIZE + col - 1))
        return 0;
    
    // Check right
    if (col < MATRIX_SIZE - 1 && IS_BIT_SET(matrix, row * MATRIX_SIZE + col + 1))
        return 0;
    
    // Check up
    if (row > 0 && IS_BIT_SET(matrix, (row - 1) * MATRIX_SIZE + col))
        return 0;
    
    // Check down
    if (row < MATRIX_SIZE - 1 && IS_BIT_SET(matrix, (row + 1) * MATRIX_SIZE + col))
        return 0;
    
    return 1;
}

// Check if the entire matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int row = 0; row < MATRIX_SIZE; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            if (is_isolated(matrix, row, col))
                return 0;
        }
    }
    return 1;
}

// Dynamic programming to count unholy matrices
uint64_t count_unholy_matrices(int rows_left, unsigned long prev_rows) {
    // Base case: if we've filled all rows
    if (rows_left == 0)
        return 1;
    
    // Check memoization
    if (memo_init[rows_left][prev_rows])
        return memo[rows_left][prev_rows];
    
    uint64_t total_count = 0;
    
    // Try all possible next two rows
    for (unsigned long next_rows = 0; next_rows < (1ULL << (2 * MATRIX_SIZE)); next_rows++) {
        // Combine previous rows and new rows
        unsigned long current_matrix = (prev_rows << (2 * MATRIX_SIZE)) | next_rows;
        
        // Check if this configuration is unholy
        if (is_unholy(current_matrix)) {
            total_count += count_unholy_matrices(rows_left - 2, next_rows);
        }
    }
    
    // Memoize and return
    memo[rows_left][prev_rows] = total_count;
    memo_init[rows_left][prev_rows] = 1;
    
    return total_count;
}

int main() {
    // Clear memoization table
    memset(memo, 0, sizeof(memo));
    memset(memo_init, 0, sizeof(memo_init));
    
    // Start with 0 previous rows, 8 rows to fill
    uint64_t result = count_unholy_matrices(MATRIX_SIZE, 0);
    
    printf("Total number of unholy 8x8 matrices: %llu\n", result);
    
    return 0;
}

Run Code Interpreter

Running the following program: 

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

#define MATRIX_SIZE 8
#define MAX_MEMO_SIZE (1 << 16)

// Check if a specific bit is 1
#define IS_BIT_SET(val, pos) ((val) & (1ULL << (pos)))

// Memoization table
uint64_t memo[MATRIX_SIZE + 1][MAX_MEMO_SIZE];
int memo_init[MATRIX_SIZE + 1][MAX_MEMO_SIZE];

// Check if a single cell is isolated
int is_isolated(unsigned long matrix, int row, int col) {
    // Check if current cell is 0
    if (!IS_BIT_SET(matrix, row * MATRIX_SIZE + col)) 
        return 0;

    // Check left
    if (col > 0 && IS_BIT_SET(matrix, row * MATRIX_SIZE + col - 1))
        return 0;
    
    // Check right
    if (col < MATRIX_SIZE - 1 && IS_BIT_SET(matrix, row * MATRIX_SIZE + col + 1))
        return 0;
    
    // Check up
    if (row > 0 && IS_BIT_SET(matrix, (row - 1) * MATRIX_SIZE + col))
        return 0;
    
    // Check down
    if (row < MATRIX_SIZE - 1 && IS_BIT_SET(matrix, (row + 1) * MATRIX_SIZE + col))
        return 0;
    
    return 1;
}

// Check if the entire matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int row = 0; row < MATRIX_SIZE; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            if (is_isolated(matrix, row, col))
                return 0;
        }
    }
    return 1;
}

// Dynamic programming to count unholy matrices
uint64_t count_unholy_matrices(int rows_left, unsigned long prev_rows) {
    // Base case: if we've filled all rows
    if (rows_left == 0)
        return 1;
    
    // Check memoization
    if (memo_init[rows_left][prev_rows])
        return memo[rows_left][prev_rows];
    
    uint64_t total_count = 0;
    
    // Try all possible next two rows
    for (unsigned long next_rows = 0; next_rows < (1ULL << (2 * MATRIX_SIZE)); next_rows++) {
        // Combine previous rows and new rows
        unsigned long current_matrix = (prev_rows << (2 * MATRIX_SIZE)) | next_rows;
        
        // Check if this configuration is unholy
        if (is_unholy(current_matrix)) {
            total_count += count_unholy_matrices(rows_left - 2, next_rows);
        }
    }
    
    // Memoize and return
    memo[rows_left][prev_rows] = total_count;
    memo_init[rows_left][prev_rows] = 1;
    
    return total_count;
}

int main() {
    // Clear memoization table
    memset(memo, 0, sizeof(memo));
    memset(memo_init, 0, sizeof(memo_init));
    
    // Start with 0 previous rows, 8 rows to fill
    uint64_t result = count_unholy_matrices(MATRIX_SIZE, 0);
    
    printf("Total number of unholy 8x8 matrices: %llu\n", result);
    
    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

Here's a comprehensive solution to the problem: Program Plan: 1. Define data structures and constants - Use unsigned long (64-bit) to represent 2 rows of 8 bits each - Create memoization cache for dynamic programming - Define constants for matrix size 2. Implement bit manipulation helper functions - is_isolated(): Check if a 1 is isolated - is_unholy(): Check if entire matrix is unholy 3. Implement dynamic programming solution - Recursive function with memoization - Base cases for small matrices - Enumerate all possible next two rows - Use memoization to cache intermediate results 4. Main function to drive computation and print result Here's the implementation: 

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

#define ROWS 8
#define COLS 8
#define MEMO_SIZE (1ULL << 16)

// Memoization cache
uint64_t memo[ROWS + 1][MEMO_SIZE];
int memo_initialized = 0;

// Check if a specific bit is isolated
int is_isolated(uint64_t matrix, int row, int col) {
    // Check 4 cardinal directions
    int left = (col > 0 && (matrix & (1ULL << (row * COLS + col - 1))));
    int right = (col < COLS - 1 && (matrix & (1ULL << (row * COLS + col + 1))));
    int up = (row > 0 && (matrix & (1ULL << ((row - 1) * COLS + col))));
    int down = (row < ROWS - 1 && (matrix & (1ULL << ((row + 1) * COLS + col))));

    return !(left || right || up || down);
}

// Check if entire matrix is unholy (no isolated 1s)
int is_unholy(uint64_t matrix) {
    for (int row = 0; row < ROWS; row++) {
        for (int col = 0; col < COLS; col++) {
            if (matrix & (1ULL << (row * COLS + col))) {
                if (is_isolated(matrix, row, col)) {
                    return 0;  // Found isolated 1
                }
            }
        }
    }
    return 1;  // No isolated 1s
}

// Dynamic programming solution with memoization
uint64_t count_unholy_matrices(int rows_left, uint64_t prev_rows) {
    // Base case: completed matrix
    if (rows_left == 0) {
        return is_unholy(prev_rows) ? 1 : 0;
    }

    // Check memoization cache
    if (memo_initialized && memo[rows_left][prev_rows] != 0) {
        return memo[rows_left][prev_rows];
    }

    uint64_t total_count = 0;

    // Enumerate all possible next two rows (2^16 possibilities)
    for (uint64_t next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        // Combine previous rows and next rows
        uint64_t current_matrix = prev_rows | (next_rows << (COLS * 2));

        // Check if current matrix is valid so far
        if (is_unholy(current_matrix)) {
            // Recursively count remaining matrices
            total_count += count_unholy_matrices(rows_left - 2, next_rows);
        }
    }

    // Memoize result
    if (!memo_initialized) {
        memset(memo, 0, sizeof(memo));
        memo_initialized = 1;
    }
    memo[rows_left][prev_rows] = total_count;

    return total_count;
}

int main() {
    uint64_t total_unholy_matrices = count_unholy_matrices(ROWS, 0);
    printf("Total unholy matrices: %llu\n", total_unholy_matrices);
    return 0;
}
Key Components: 1. `is_isolated()`: Checks if a specific 1 is isolated 2. `is_unholy()`: Checks if an entire matrix has no isolated 1s 3. `count_unholy_matrices()`: Recursive DP solution with: - Base case when all rows are filled - Memoization to cache intermediate results - Enumerate all possible next two rows - Recursively count valid matrices Optimization Strategies: - Use bit manipulation for efficient representation - Memoization to avoid recomputing subproblems - Process two rows at a time to ensure isolation checks Complexity: - Time: O(rows * 2^(2*cols)) - Space: O(rows * 2^(2*cols)) for memoization Limitations: - Large matrices become computationally infeasible - 8x8 is near the limit of practical computation The program efficiently counts the number of "unholy" matrices by systematically exploring all possible configurations while pruning invalid arrangements.

Extract Code

I extracted the following code from that output: 

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

#define ROWS 8
#define COLS 8
#define MEMO_SIZE (1ULL << 16)

// Memoization cache
uint64_t memo[ROWS + 1][MEMO_SIZE];
int memo_initialized = 0;

// Check if a specific bit is isolated
int is_isolated(uint64_t matrix, int row, int col) {
    // Check 4 cardinal directions
    int left = (col > 0 && (matrix & (1ULL << (row * COLS + col - 1))));
    int right = (col < COLS - 1 && (matrix & (1ULL << (row * COLS + col + 1))));
    int up = (row > 0 && (matrix & (1ULL << ((row - 1) * COLS + col))));
    int down = (row < ROWS - 1 && (matrix & (1ULL << ((row + 1) * COLS + col))));

    return !(left || right || up || down);
}

// Check if entire matrix is unholy (no isolated 1s)
int is_unholy(uint64_t matrix) {
    for (int row = 0; row < ROWS; row++) {
        for (int col = 0; col < COLS; col++) {
            if (matrix & (1ULL << (row * COLS + col))) {
                if (is_isolated(matrix, row, col)) {
                    return 0;  // Found isolated 1
                }
            }
        }
    }
    return 1;  // No isolated 1s
}

// Dynamic programming solution with memoization
uint64_t count_unholy_matrices(int rows_left, uint64_t prev_rows) {
    // Base case: completed matrix
    if (rows_left == 0) {
        return is_unholy(prev_rows) ? 1 : 0;
    }

    // Check memoization cache
    if (memo_initialized && memo[rows_left][prev_rows] != 0) {
        return memo[rows_left][prev_rows];
    }

    uint64_t total_count = 0;

    // Enumerate all possible next two rows (2^16 possibilities)
    for (uint64_t next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        // Combine previous rows and next rows
        uint64_t current_matrix = prev_rows | (next_rows << (COLS * 2));

        // Check if current matrix is valid so far
        if (is_unholy(current_matrix)) {
            // Recursively count remaining matrices
            total_count += count_unholy_matrices(rows_left - 2, next_rows);
        }
    }

    // Memoize result
    if (!memo_initialized) {
        memset(memo, 0, sizeof(memo));
        memo_initialized = 1;
    }
    memo[rows_left][prev_rows] = total_count;

    return total_count;
}

int main() {
    uint64_t total_unholy_matrices = count_unholy_matrices(ROWS, 0);
    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 <stdint.h>
#include <string.h>

#define ROWS 8
#define COLS 8
#define MEMO_SIZE (1ULL << 16)

// Memoization cache
uint64_t memo[ROWS + 1][MEMO_SIZE];
int memo_initialized = 0;

// Check if a specific bit is isolated
int is_isolated(uint64_t matrix, int row, int col) {
    // Check 4 cardinal directions
    int left = (col > 0 && (matrix & (1ULL << (row * COLS + col - 1))));
    int right = (col < COLS - 1 && (matrix & (1ULL << (row * COLS + col + 1))));
    int up = (row > 0 && (matrix & (1ULL << ((row - 1) * COLS + col))));
    int down = (row < ROWS - 1 && (matrix & (1ULL << ((row + 1) * COLS + col))));

    return !(left || right || up || down);
}

// Check if entire matrix is unholy (no isolated 1s)
int is_unholy(uint64_t matrix) {
    for (int row = 0; row < ROWS; row++) {
        for (int col = 0; col < COLS; col++) {
            if (matrix & (1ULL << (row * COLS + col))) {
                if (is_isolated(matrix, row, col)) {
                    return 0;  // Found isolated 1
                }
            }
        }
    }
    return 1;  // No isolated 1s
}

// Dynamic programming solution with memoization
uint64_t count_unholy_matrices(int rows_left, uint64_t prev_rows) {
    // Base case: completed matrix
    if (rows_left == 0) {
        return is_unholy(prev_rows) ? 1 : 0;
    }

    // Check memoization cache
    if (memo_initialized && memo[rows_left][prev_rows] != 0) {
        return memo[rows_left][prev_rows];
    }

    uint64_t total_count = 0;

    // Enumerate all possible next two rows (2^16 possibilities)
    for (uint64_t next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        // Combine previous rows and next rows
        uint64_t current_matrix = prev_rows | (next_rows << (COLS * 2));

        // Check if current matrix is valid so far
        if (is_unholy(current_matrix)) {
            // Recursively count remaining matrices
            total_count += count_unholy_matrices(rows_left - 2, next_rows);
        }
    }

    // Memoize result
    if (!memo_initialized) {
        memset(memo, 0, sizeof(memo));
        memo_initialized = 1;
    }
    memo[rows_left][prev_rows] = total_count;

    return total_count;
}

int main() {
    uint64_t total_unholy_matrices = count_unholy_matrices(ROWS, 0);
    printf("Total unholy matrices: %llu\n", total_unholy_matrices);
    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