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

Let's break this down into a clear plan and then implement it. Plan: 1. Represent 8x8 matrix using an unsigned long (64 bits) 2. Create helper functions: - is_unholy: Checks if a matrix has no isolated 1s - count_unholy: Dynamic programming function to count unholy matrices 3. Main dynamic programming approach: - State: Previous two rows and current height - For each state, try all possible next two rows - Check if new configuration maintains "unholiness" - Use memoization to store results Here's the implementation: 

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

// DP table size: 2^16 states (for two rows) * 4 heights
unsigned long long dp[65536][4];
int calculated[65536][4];

// Check if a specific position has an isolated 1
int is_isolated(unsigned long matrix, int row, int col) {
    if ((matrix & (1UL << (row * 8 + col))) == 0) return 0;
    
    // Check all four directions
    if (row > 0 && (matrix & (1UL << ((row-1) * 8 + col)))) return 0;
    if (row < 7 && (matrix & (1UL << ((row+1) * 8 + col)))) return 0;
    if (col > 0 && (matrix & (1UL << (row * 8 + col-1)))) return 0;
    if (col < 7 && (matrix & (1UL << (row * 8 + col+1)))) return 0;
    
    return 1;
}

// Check if matrix is unholy (no isolated 1s)
int is_unholy(unsigned long matrix) {
    for(int i = 0; i < 8; i++)
        for(int j = 0; j < 8; j++)
            if (is_isolated(matrix, i, j))
                return 0;
    return 1;
}

// Check if current configuration is valid
int is_valid_config(unsigned prev_rows, unsigned new_rows, int height) {
    unsigned long matrix = 0;
    // Construct partial matrix
    matrix |= ((unsigned long)prev_rows) << (8 * (height-2));
    matrix |= ((unsigned long)new_rows) << (8 * height);
    
    // Check only relevant rows for isolation
    for(int i = height-1; i <= height; i++)
        for(int j = 0; j < 8; j++)
            if (is_isolated(matrix, i, j))
                return 0;
    return 1;
}

// Dynamic programming function
unsigned long long count_unholy(unsigned prev_rows, int height) {
    if (height == 4) return 1; // Reached full height
    
    if (calculated[prev_rows][height])
        return dp[prev_rows][height];
    
    calculated[prev_rows][height] = 1;
    unsigned long long result = 0;
    
    // Try all possible next two rows
    for(unsigned new_rows = 0; new_rows < (1<<16); new_rows++) {
        if (is_valid_config(prev_rows, new_rows, height)) {
            result += count_unholy(new_rows, height + 1);
        }
    }
    
    dp[prev_rows][height] = result;
    return result;
}

int main() {
    // Initialize memoization tables
    memset(calculated, 0, sizeof(calculated));
    memset(dp, 0, sizeof(dp));
    
    unsigned long long total = 0;
    
    // Try all possible first two rows
    for(unsigned first_rows = 0; first_rows < (1<<16); first_rows++) {
        total += count_unholy(first_rows, 1);
    }
    
    printf("Total number of unholy 8x8 matrices: %llu\n", total);
    return 0;
}
This implementation: 1. Uses an unsigned long to represent the matrix (64 bits for 8x8) 2. Implements is_isolated and is_unholy to check for isolated 1s 3. Uses dynamic programming with memoization to avoid recalculating states 4. Builds the matrix two rows at a time 5. Maintains a state of previous two rows to ensure proper checking of isolation The program counts all possible unholy 8x8 matrices by: 1. Starting with all possible configurations of first two rows 2. For each valid configuration, recursively trying all possible next two rows 3. Using memoization to store results of previously computed states 4. Combining results to get the total count The state space is manageable because: - We only need to store previous two rows (2^16 possibilities) - We only need to track 4 heights (processing two rows at a time) - The dp table size is 65536 * 4 entries This solution efficiently handles the combinatorial explosion by using dynamic programming and bit operations.

Extract Code

I extracted the following code from that output: 

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

// DP table size: 2^16 states (for two rows) * 4 heights
unsigned long long dp[65536][4];
int calculated[65536][4];

// Check if a specific position has an isolated 1
int is_isolated(unsigned long matrix, int row, int col) {
    if ((matrix & (1UL << (row * 8 + col))) == 0) return 0;
    
    // Check all four directions
    if (row > 0 && (matrix & (1UL << ((row-1) * 8 + col)))) return 0;
    if (row < 7 && (matrix & (1UL << ((row+1) * 8 + col)))) return 0;
    if (col > 0 && (matrix & (1UL << (row * 8 + col-1)))) return 0;
    if (col < 7 && (matrix & (1UL << (row * 8 + col+1)))) return 0;
    
    return 1;
}

// Check if matrix is unholy (no isolated 1s)
int is_unholy(unsigned long matrix) {
    for(int i = 0; i < 8; i++)
        for(int j = 0; j < 8; j++)
            if (is_isolated(matrix, i, j))
                return 0;
    return 1;
}

// Check if current configuration is valid
int is_valid_config(unsigned prev_rows, unsigned new_rows, int height) {
    unsigned long matrix = 0;
    // Construct partial matrix
    matrix |= ((unsigned long)prev_rows) << (8 * (height-2));
    matrix |= ((unsigned long)new_rows) << (8 * height);
    
    // Check only relevant rows for isolation
    for(int i = height-1; i <= height; i++)
        for(int j = 0; j < 8; j++)
            if (is_isolated(matrix, i, j))
                return 0;
    return 1;
}

// Dynamic programming function
unsigned long long count_unholy(unsigned prev_rows, int height) {
    if (height == 4) return 1; // Reached full height
    
    if (calculated[prev_rows][height])
        return dp[prev_rows][height];
    
    calculated[prev_rows][height] = 1;
    unsigned long long result = 0;
    
    // Try all possible next two rows
    for(unsigned new_rows = 0; new_rows < (1<<16); new_rows++) {
        if (is_valid_config(prev_rows, new_rows, height)) {
            result += count_unholy(new_rows, height + 1);
        }
    }
    
    dp[prev_rows][height] = result;
    return result;
}

int main() {
    // Initialize memoization tables
    memset(calculated, 0, sizeof(calculated));
    memset(dp, 0, sizeof(dp));
    
    unsigned long long total = 0;
    
    // Try all possible first two rows
    for(unsigned first_rows = 0; first_rows < (1<<16); first_rows++) {
        total += count_unholy(first_rows, 1);
    }
    
    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 <string.h>

// DP table size: 2^16 states (for two rows) * 4 heights
unsigned long long dp[65536][4];
int calculated[65536][4];

// Check if a specific position has an isolated 1
int is_isolated(unsigned long matrix, int row, int col) {
    if ((matrix & (1UL << (row * 8 + col))) == 0) return 0;
    
    // Check all four directions
    if (row > 0 && (matrix & (1UL << ((row-1) * 8 + col)))) return 0;
    if (row < 7 && (matrix & (1UL << ((row+1) * 8 + col)))) return 0;
    if (col > 0 && (matrix & (1UL << (row * 8 + col-1)))) return 0;
    if (col < 7 && (matrix & (1UL << (row * 8 + col+1)))) return 0;
    
    return 1;
}

// Check if matrix is unholy (no isolated 1s)
int is_unholy(unsigned long matrix) {
    for(int i = 0; i < 8; i++)
        for(int j = 0; j < 8; j++)
            if (is_isolated(matrix, i, j))
                return 0;
    return 1;
}

// Check if current configuration is valid
int is_valid_config(unsigned prev_rows, unsigned new_rows, int height) {
    unsigned long matrix = 0;
    // Construct partial matrix
    matrix |= ((unsigned long)prev_rows) << (8 * (height-2));
    matrix |= ((unsigned long)new_rows) << (8 * height);
    
    // Check only relevant rows for isolation
    for(int i = height-1; i <= height; i++)
        for(int j = 0; j < 8; j++)
            if (is_isolated(matrix, i, j))
                return 0;
    return 1;
}

// Dynamic programming function
unsigned long long count_unholy(unsigned prev_rows, int height) {
    if (height == 4) return 1; // Reached full height
    
    if (calculated[prev_rows][height])
        return dp[prev_rows][height];
    
    calculated[prev_rows][height] = 1;
    unsigned long long result = 0;
    
    // Try all possible next two rows
    for(unsigned new_rows = 0; new_rows < (1<<16); new_rows++) {
        if (is_valid_config(prev_rows, new_rows, height)) {
            result += count_unholy(new_rows, height + 1);
        }
    }
    
    dp[prev_rows][height] = result;
    return result;
}

int main() {
    // Initialize memoization tables
    memset(calculated, 0, sizeof(calculated));
    memset(dp, 0, sizeof(dp));
    
    unsigned long long total = 0;
    
    // Try all possible first two rows
    for(unsigned first_rows = 0; first_rows < (1<<16); first_rows++) {
        total += count_unholy(first_rows, 1);
    }
    
    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

Let's break this down into a structured plan and then implement it. Plan: 1. Represent the 8x8 matrix using an unsigned long (64 bits) 2. Create a function to test if a matrix is unholy - Check each 1 bit to ensure it has an adjacent 1 - Use bit operations to check neighbors 3. Create a dynamic programming solution - State: (previous row, current row, number of rows processed) - For each state, try all possible next rows - Use memoization to store results 4. Main program to initialize and run the DP Here's the implementation: 

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

// DP table dimensions
#define ROW_BITS 8
#define MAX_ROWS 8
#define MAX_ROW_VAL (1 << ROW_BITS)

// Store results in DP table
unsigned long long dp[MAX_ROW_VAL][MAX_ROW_VAL][MAX_ROWS];
int computed[MAX_ROW_VAL][MAX_ROW_VAL][MAX_ROWS];

// Check if two consecutive rows make valid unholy pattern
int is_valid_rows(unsigned char prev_row, unsigned char curr_row) {
    // Check if any 1 in prev_row is isolated (no 1s adjacent in curr_row)
    unsigned char mask = prev_row;
    while (mask) {
        int pos = __builtin_ctz(mask);  // Get position of least significant 1
        if ((prev_row & (1 << pos)) &&  // If there's a 1 here
            !(curr_row & (1 << pos))) { // And no 1 below
            // Check left and right in same row
            int has_neighbor = 0;
            if (pos > 0 && (prev_row & (1 << (pos-1)))) has_neighbor = 1;
            if (pos < ROW_BITS-1 && (prev_row & (1 << (pos+1)))) has_neighbor = 1;
            if (!has_neighbor) return 0;
        }
        mask &= (mask - 1);  // Clear least significant 1
    }
    return 1;
}

// Dynamic programming function
unsigned long long solve(unsigned char prev_row, unsigned char curr_row, int row_num) {
    // Base case: reached end of matrix
    if (row_num == MAX_ROWS) {
        // Check if last row is valid
        return is_valid_rows(curr_row, 0);
    }

    // Return memoized result if available
    if (computed[prev_row][curr_row][row_num]) {
        return dp[prev_row][curr_row][row_num];
    }

    unsigned long long result = 0;
    // Try all possible next rows
    for (int next_row = 0; next_row < MAX_ROW_VAL; next_row++) {
        // Check if current configuration is valid
        if (is_valid_rows(prev_row, curr_row) && 
            is_valid_rows(curr_row, next_row)) {
            result += solve(curr_row, next_row, row_num + 1);
        }
    }

    // Memoize and return
    computed[prev_row][curr_row][row_num] = 1;
    dp[prev_row][curr_row][row_num] = result;
    return result;
}

int main() {
    unsigned long long total = 0;
    memset(computed, 0, sizeof(computed));
    memset(dp, 0, sizeof(dp));

    // Try all possible first two rows
    for (int first_row = 0; first_row < MAX_ROW_VAL; first_row++) {
        for (int second_row = 0; second_row < MAX_ROW_VAL; second_row++) {
            if (is_valid_rows(first_row, second_row)) {
                total += solve(first_row, second_row, 2);
            }
        }
    }

    printf("Total number of unholy matrices: %llu\n", total);
    return 0;
}
This program: 1. Uses bit operations to efficiently check if rows are valid 2. Implements dynamic programming with memoization to avoid recalculating states 3. Considers all possible combinations of rows 4. Uses an efficient representation of the matrix using bits The solution works by: 1. Starting with all possible combinations of first two rows 2. For each valid combination, recursively trying all possible next rows 3. Memoizing results to avoid recalculation 4. Using bit operations to efficiently check if configurations are valid Note that this is a computationally intensive problem, and the program might take some time to run depending on your machine. The actual number of unholy matrices will be quite large due to the combinatorial nature of the problem. To optimize further, you could: 1. Use more bit operations instead of loops 2. Implement parallel processing 3. Pre-compute valid row transitions 4. Use a more compact representation of the DP table

Extract Code

I extracted the following code from that output: 

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

// DP table dimensions
#define ROW_BITS 8
#define MAX_ROWS 8
#define MAX_ROW_VAL (1 << ROW_BITS)

// Store results in DP table
unsigned long long dp[MAX_ROW_VAL][MAX_ROW_VAL][MAX_ROWS];
int computed[MAX_ROW_VAL][MAX_ROW_VAL][MAX_ROWS];

// Check if two consecutive rows make valid unholy pattern
int is_valid_rows(unsigned char prev_row, unsigned char curr_row) {
    // Check if any 1 in prev_row is isolated (no 1s adjacent in curr_row)
    unsigned char mask = prev_row;
    while (mask) {
        int pos = __builtin_ctz(mask);  // Get position of least significant 1
        if ((prev_row & (1 << pos)) &&  // If there's a 1 here
            !(curr_row & (1 << pos))) { // And no 1 below
            // Check left and right in same row
            int has_neighbor = 0;
            if (pos > 0 && (prev_row & (1 << (pos-1)))) has_neighbor = 1;
            if (pos < ROW_BITS-1 && (prev_row & (1 << (pos+1)))) has_neighbor = 1;
            if (!has_neighbor) return 0;
        }
        mask &= (mask - 1);  // Clear least significant 1
    }
    return 1;
}

// Dynamic programming function
unsigned long long solve(unsigned char prev_row, unsigned char curr_row, int row_num) {
    // Base case: reached end of matrix
    if (row_num == MAX_ROWS) {
        // Check if last row is valid
        return is_valid_rows(curr_row, 0);
    }

    // Return memoized result if available
    if (computed[prev_row][curr_row][row_num]) {
        return dp[prev_row][curr_row][row_num];
    }

    unsigned long long result = 0;
    // Try all possible next rows
    for (int next_row = 0; next_row < MAX_ROW_VAL; next_row++) {
        // Check if current configuration is valid
        if (is_valid_rows(prev_row, curr_row) && 
            is_valid_rows(curr_row, next_row)) {
            result += solve(curr_row, next_row, row_num + 1);
        }
    }

    // Memoize and return
    computed[prev_row][curr_row][row_num] = 1;
    dp[prev_row][curr_row][row_num] = result;
    return result;
}

int main() {
    unsigned long long total = 0;
    memset(computed, 0, sizeof(computed));
    memset(dp, 0, sizeof(dp));

    // Try all possible first two rows
    for (int first_row = 0; first_row < MAX_ROW_VAL; first_row++) {
        for (int second_row = 0; second_row < MAX_ROW_VAL; second_row++) {
            if (is_valid_rows(first_row, second_row)) {
                total += solve(first_row, second_row, 2);
            }
        }
    }

    printf("Total number of unholy matrices: %llu\n", total);
    return 0;
}

Run Code Interpreter

Running the following program: 

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

// DP table dimensions
#define ROW_BITS 8
#define MAX_ROWS 8
#define MAX_ROW_VAL (1 << ROW_BITS)

// Store results in DP table
unsigned long long dp[MAX_ROW_VAL][MAX_ROW_VAL][MAX_ROWS];
int computed[MAX_ROW_VAL][MAX_ROW_VAL][MAX_ROWS];

// Check if two consecutive rows make valid unholy pattern
int is_valid_rows(unsigned char prev_row, unsigned char curr_row) {
    // Check if any 1 in prev_row is isolated (no 1s adjacent in curr_row)
    unsigned char mask = prev_row;
    while (mask) {
        int pos = __builtin_ctz(mask);  // Get position of least significant 1
        if ((prev_row & (1 << pos)) &&  // If there's a 1 here
            !(curr_row & (1 << pos))) { // And no 1 below
            // Check left and right in same row
            int has_neighbor = 0;
            if (pos > 0 && (prev_row & (1 << (pos-1)))) has_neighbor = 1;
            if (pos < ROW_BITS-1 && (prev_row & (1 << (pos+1)))) has_neighbor = 1;
            if (!has_neighbor) return 0;
        }
        mask &= (mask - 1);  // Clear least significant 1
    }
    return 1;
}

// Dynamic programming function
unsigned long long solve(unsigned char prev_row, unsigned char curr_row, int row_num) {
    // Base case: reached end of matrix
    if (row_num == MAX_ROWS) {
        // Check if last row is valid
        return is_valid_rows(curr_row, 0);
    }

    // Return memoized result if available
    if (computed[prev_row][curr_row][row_num]) {
        return dp[prev_row][curr_row][row_num];
    }

    unsigned long long result = 0;
    // Try all possible next rows
    for (int next_row = 0; next_row < MAX_ROW_VAL; next_row++) {
        // Check if current configuration is valid
        if (is_valid_rows(prev_row, curr_row) && 
            is_valid_rows(curr_row, next_row)) {
            result += solve(curr_row, next_row, row_num + 1);
        }
    }

    // Memoize and return
    computed[prev_row][curr_row][row_num] = 1;
    dp[prev_row][curr_row][row_num] = result;
    return result;
}

int main() {
    unsigned long long total = 0;
    memset(computed, 0, sizeof(computed));
    memset(dp, 0, sizeof(dp));

    // Try all possible first two rows
    for (int first_row = 0; first_row < MAX_ROW_VAL; first_row++) {
        for (int second_row = 0; second_row < MAX_ROW_VAL; second_row++) {
            if (is_valid_rows(first_row, second_row)) {
                total += solve(first_row, second_row, 2);
            }
        }
    }

    printf("Total number of unholy matrices: %llu\n", total);
    return 0;
}
And got the output: Total number of unholy matrices: 53003772538202116

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

Let's break this down into a clear plan and then implement it. Plan: 1. Represent the 8x8 matrix using an unsigned long (64 bits) 2. Create a function to test if a matrix is unholy: - For each 1 in the matrix, check if it has a neighboring 1 - Use bit operations to check neighbors efficiently 3. Create a dynamic programming solution: - State will be (current_rows, last_two_rows) - For each state, try all possible next two rows - Base case is when we reach 8 rows - Use a 2D array to store intermediate results 4. Main program structure: - Initialize DP array - Call recursive function starting with 0 rows - Print final result Here's the implementation: 

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

// DP array: [number of rows][previous two rows pattern]
unsigned long long dp[9][65536];
int calculated[9][65536];

// Check if a specific row configuration is valid with previous rows
int is_valid_rows(unsigned int prev_two_rows, unsigned int next_two_rows) {
    unsigned int top_row = (prev_two_rows >> 8) & 0xFF;
    unsigned int middle_row = prev_two_rows & 0xFF;
    unsigned int bottom_row = (next_two_rows >> 8) & 0xFF;
    unsigned int last_row = next_two_rows & 0xFF;
    
    // Check middle row's 1s have neighbors
    for(int i = 0; i < 8; i++) {
        if((middle_row & (1 << i)) != 0) {
            unsigned int mask = 1 << i;
            unsigned int has_neighbor = 
                (i > 0 && (middle_row & (mask >> 1))) ||
                (i < 7 && (middle_row & (mask << 1))) ||
                (top_row & mask) ||
                (bottom_row & mask);
            if(!has_neighbor) return 0;
        }
    }
    
    // Check bottom row's 1s have neighbors (except for last row case)
    if(next_two_rows != 0) {
        for(int i = 0; i < 8; i++) {
            if((bottom_row & (1 << i)) != 0) {
                unsigned int mask = 1 << i;
                unsigned int has_neighbor = 
                    (i > 0 && (bottom_row & (mask >> 1))) ||
                    (i < 7 && (bottom_row & (mask << 1))) ||
                    (middle_row & mask) ||
                    (last_row & mask);
                if(!has_neighbor) return 0;
            }
        }
    }
    
    return 1;
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int row, unsigned int prev_two_rows) {
    // Base case: reached end of matrix
    if(row == 8) return 1;
    
    // Return memoized result if available
    if(calculated[row][prev_two_rows]) 
        return dp[row][prev_two_rows];
    
    unsigned long long result = 0;
    
    // Try all possible next two rows
    for(unsigned int next_two_rows = 0; next_two_rows < 65536; next_two_rows++) {
        if(is_valid_rows(prev_two_rows, next_two_rows)) {
            result += count_unholy(row + 2, next_two_rows);
        }
    }
    
    calculated[row][prev_two_rows] = 1;
    dp[row][prev_two_rows] = result;
    return result;
}

int main() {
    // Initialize memoization arrays
    memset(calculated, 0, sizeof(calculated));
    memset(dp, 0, sizeof(dp));
    
    // Start counting from row 0 with no previous rows
    unsigned long long result = count_unholy(0, 0);
    
    printf("Total number of unholy 8x8 matrices: %llu\n", result);
    return 0;
}
This implementation: 1. Uses a 64-bit integer for the matrix representation 2. Implements is_valid_rows() to check if a configuration is valid 3. Uses dynamic programming to avoid recalculating states 4. Processes two rows at a time to ensure proper neighbor checking 5. Memoizes results in a dp array 6. Uses bit operations for efficient checking The program will count all valid unholy matrices by: - Starting with an empty matrix - Trying all possible combinations of two rows at a time - Ensuring no 1s are isolated - Using memoization to avoid recalculating states The final result will be the total number of valid unholy 8x8 matrices.

Extract Code

I extracted the following code from that output: 

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

// DP array: [number of rows][previous two rows pattern]
unsigned long long dp[9][65536];
int calculated[9][65536];

// Check if a specific row configuration is valid with previous rows
int is_valid_rows(unsigned int prev_two_rows, unsigned int next_two_rows) {
    unsigned int top_row = (prev_two_rows >> 8) & 0xFF;
    unsigned int middle_row = prev_two_rows & 0xFF;
    unsigned int bottom_row = (next_two_rows >> 8) & 0xFF;
    unsigned int last_row = next_two_rows & 0xFF;
    
    // Check middle row's 1s have neighbors
    for(int i = 0; i < 8; i++) {
        if((middle_row & (1 << i)) != 0) {
            unsigned int mask = 1 << i;
            unsigned int has_neighbor = 
                (i > 0 && (middle_row & (mask >> 1))) ||
                (i < 7 && (middle_row & (mask << 1))) ||
                (top_row & mask) ||
                (bottom_row & mask);
            if(!has_neighbor) return 0;
        }
    }
    
    // Check bottom row's 1s have neighbors (except for last row case)
    if(next_two_rows != 0) {
        for(int i = 0; i < 8; i++) {
            if((bottom_row & (1 << i)) != 0) {
                unsigned int mask = 1 << i;
                unsigned int has_neighbor = 
                    (i > 0 && (bottom_row & (mask >> 1))) ||
                    (i < 7 && (bottom_row & (mask << 1))) ||
                    (middle_row & mask) ||
                    (last_row & mask);
                if(!has_neighbor) return 0;
            }
        }
    }
    
    return 1;
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int row, unsigned int prev_two_rows) {
    // Base case: reached end of matrix
    if(row == 8) return 1;
    
    // Return memoized result if available
    if(calculated[row][prev_two_rows]) 
        return dp[row][prev_two_rows];
    
    unsigned long long result = 0;
    
    // Try all possible next two rows
    for(unsigned int next_two_rows = 0; next_two_rows < 65536; next_two_rows++) {
        if(is_valid_rows(prev_two_rows, next_two_rows)) {
            result += count_unholy(row + 2, next_two_rows);
        }
    }
    
    calculated[row][prev_two_rows] = 1;
    dp[row][prev_two_rows] = result;
    return result;
}

int main() {
    // Initialize memoization arrays
    memset(calculated, 0, sizeof(calculated));
    memset(dp, 0, sizeof(dp));
    
    // Start counting from row 0 with no previous rows
    unsigned long long result = count_unholy(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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C
#include <stdio.h>
#include <string.h>

// DP array: [number of rows][previous two rows pattern]
unsigned long long dp[9][65536];
int calculated[9][65536];

// Check if a specific row configuration is valid with previous rows
int is_valid_rows(unsigned int prev_two_rows, unsigned int next_two_rows) {
    unsigned int top_row = (prev_two_rows >> 8) & 0xFF;
    unsigned int middle_row = prev_two_rows & 0xFF;
    unsigned int bottom_row = (next_two_rows >> 8) & 0xFF;
    unsigned int last_row = next_two_rows & 0xFF;
    
    // Check middle row's 1s have neighbors
    for(int i = 0; i < 8; i++) {
        if((middle_row & (1 << i)) != 0) {
            unsigned int mask = 1 << i;
            unsigned int has_neighbor = 
                (i > 0 && (middle_row & (mask >> 1))) ||
                (i < 7 && (middle_row & (mask << 1))) ||
                (top_row & mask) ||
                (bottom_row & mask);
            if(!has_neighbor) return 0;
        }
    }
    
    // Check bottom row's 1s have neighbors (except for last row case)
    if(next_two_rows != 0) {
        for(int i = 0; i < 8; i++) {
            if((bottom_row & (1 << i)) != 0) {
                unsigned int mask = 1 << i;
                unsigned int has_neighbor = 
                    (i > 0 && (bottom_row & (mask >> 1))) ||
                    (i < 7 && (bottom_row & (mask << 1))) ||
                    (middle_row & mask) ||
                    (last_row & mask);
                if(!has_neighbor) return 0;
            }
        }
    }
    
    return 1;
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int row, unsigned int prev_two_rows) {
    // Base case: reached end of matrix
    if(row == 8) return 1;
    
    // Return memoized result if available
    if(calculated[row][prev_two_rows]) 
        return dp[row][prev_two_rows];
    
    unsigned long long result = 0;
    
    // Try all possible next two rows
    for(unsigned int next_two_rows = 0; next_two_rows < 65536; next_two_rows++) {
        if(is_valid_rows(prev_two_rows, next_two_rows)) {
            result += count_unholy(row + 2, next_two_rows);
        }
    }
    
    calculated[row][prev_two_rows] = 1;
    dp[row][prev_two_rows] = result;
    return result;
}

int main() {
    // Initialize memoization arrays
    memset(calculated, 0, sizeof(calculated));
    memset(dp, 0, sizeof(dp));
    
    // Start counting from row 0 with no previous rows
    unsigned long long result = count_unholy(0, 0);
    
    printf("Total number of unholy 8x8 matrices: %llu\n", result);
    return 0;
}
And got the output: main.c:1:2: error: expected ';' before 'typedef' 1 | C | ^ | ; main.sh: line 2: ./a.out: No such file or directory

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

Let's break this down into a clear plan and then implement it. Plan: 1. Represent the 8x8 matrix using an unsigned long (64 bits) 2. Create a function to test if a matrix is unholy: - For each 1 in the matrix, check if it has another 1 adjacent to it - Use bit operations to check adjacency 3. Create a dynamic programming solution: - State: current row pair and previous row pair - Base case: when we've filled all 8 rows - For each state, try all possible combinations for the next row pair - Memoize results using a hash table or array 4. Main program: - Initialize memoization structure - Call recursive function starting with first two rows - Return final count Here's the implementation: 

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

// Memoization table
unsigned long long memo[256][256];  // [prev_rows][curr_rows]

// Check if a three-row sequence contains any isolated 1s in the middle row
int has_isolated_ones(unsigned int top, unsigned int mid, unsigned int bot) {
    unsigned int isolated;
    
    // A 1 is not isolated if it has a neighbor above, below, left, or right
    // Get all 1s in middle row
    unsigned int ones = mid;
    
    // Get ones that have neighbors above or below
    unsigned int vertical_neighbors = top | bot;
    
    // Get ones that have neighbors to the left or right
    unsigned int horizontal_neighbors = (mid << 1) | (mid >> 1);
    
    // A 1 is isolated if it has no neighbors
    isolated = ones & ~vertical_neighbors & ~horizontal_neighbors;
    
    return isolated != 0;
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int row, unsigned int prev_pair, unsigned int curr_pair) {
    // Base case: we've filled all rows
    if (row >= 7)
        return 1;
    
    // Check memoization table
    if (memo[prev_pair][curr_pair] != -1)
        return memo[prev_pair][curr_pair];
    
    unsigned long long count = 0;
    unsigned int prev_top = (prev_pair >> 8) & 0xFF;
    unsigned int prev_bot = prev_pair & 0xFF;
    unsigned int curr_top = (curr_pair >> 8) & 0xFF;
    unsigned int curr_bot = curr_pair & 0xFF;
    
    // Try all possibilities for next row pair
    for (unsigned int next_pair = 0; next_pair < (1 << 16); next_pair++) {
        unsigned int next_top = (next_pair >> 8) & 0xFF;
        unsigned int next_bot = next_pair & 0xFF;
        
        // Check if current configuration creates any isolated ones
        if (!has_isolated_ones(prev_bot, curr_top, next_top) &&
            !has_isolated_ones(curr_top, curr_bot, next_bot)) {
            count += count_unholy(row + 2, curr_pair, next_pair);
        }
    }
    
    memo[prev_pair][curr_pair] = count;
    return count;
}

int main() {
    unsigned long long total = 0;
    
    // Initialize memoization table
    memset(memo, -1, sizeof(memo));
    
    // Try all possibilities for first two rows
    for (unsigned int first_pair = 0; first_pair < (1 << 16); first_pair++) {
        unsigned int top = (first_pair >> 8) & 0xFF;
        unsigned int bot = first_pair & 0xFF;
        
        // Check if first two rows are valid (no isolated ones in first row)
        if (!has_isolated_ones(0, top, bot)) {
            total += count_unholy(2, 0, first_pair);
        }
    }
    
    printf("Total number of unholy 8x8 matrices: %llu\n", total);
    return 0;
}
This implementation: 1. Uses bit manipulation to efficiently check for isolated ones 2. Uses dynamic programming with memoization to avoid recalculating states 3. Processes the matrix two rows at a time to reduce the state space 4. Uses unsigned integers to represent row pairs 5. Checks for isolated ones in each configuration The program will count all valid unholy matrices by: 1. Trying all possibilities for the first two rows 2. For each valid first two rows, recursively trying all possibilities for subsequent row pairs 3. Using memoization to store results for previously seen states 4. Combining the counts to get the final total The time complexity is O(2^16 * 2^16 * 4), as we need to try all possibilities for each pair of rows, and we have 4 pairs of rows total. The space complexity is O(2^16 * 2^16) for the memoization table.

Extract Code

I extracted the following code from that output: 

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

// Memoization table
unsigned long long memo[256][256];  // [prev_rows][curr_rows]

// Check if a three-row sequence contains any isolated 1s in the middle row
int has_isolated_ones(unsigned int top, unsigned int mid, unsigned int bot) {
    unsigned int isolated;
    
    // A 1 is not isolated if it has a neighbor above, below, left, or right
    // Get all 1s in middle row
    unsigned int ones = mid;
    
    // Get ones that have neighbors above or below
    unsigned int vertical_neighbors = top | bot;
    
    // Get ones that have neighbors to the left or right
    unsigned int horizontal_neighbors = (mid << 1) | (mid >> 1);
    
    // A 1 is isolated if it has no neighbors
    isolated = ones & ~vertical_neighbors & ~horizontal_neighbors;
    
    return isolated != 0;
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int row, unsigned int prev_pair, unsigned int curr_pair) {
    // Base case: we've filled all rows
    if (row >= 7)
        return 1;
    
    // Check memoization table
    if (memo[prev_pair][curr_pair] != -1)
        return memo[prev_pair][curr_pair];
    
    unsigned long long count = 0;
    unsigned int prev_top = (prev_pair >> 8) & 0xFF;
    unsigned int prev_bot = prev_pair & 0xFF;
    unsigned int curr_top = (curr_pair >> 8) & 0xFF;
    unsigned int curr_bot = curr_pair & 0xFF;
    
    // Try all possibilities for next row pair
    for (unsigned int next_pair = 0; next_pair < (1 << 16); next_pair++) {
        unsigned int next_top = (next_pair >> 8) & 0xFF;
        unsigned int next_bot = next_pair & 0xFF;
        
        // Check if current configuration creates any isolated ones
        if (!has_isolated_ones(prev_bot, curr_top, next_top) &&
            !has_isolated_ones(curr_top, curr_bot, next_bot)) {
            count += count_unholy(row + 2, curr_pair, next_pair);
        }
    }
    
    memo[prev_pair][curr_pair] = count;
    return count;
}

int main() {
    unsigned long long total = 0;
    
    // Initialize memoization table
    memset(memo, -1, sizeof(memo));
    
    // Try all possibilities for first two rows
    for (unsigned int first_pair = 0; first_pair < (1 << 16); first_pair++) {
        unsigned int top = (first_pair >> 8) & 0xFF;
        unsigned int bot = first_pair & 0xFF;
        
        // Check if first two rows are valid (no isolated ones in first row)
        if (!has_isolated_ones(0, top, bot)) {
            total += count_unholy(2, 0, first_pair);
        }
    }
    
    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 <string.h>

// Memoization table
unsigned long long memo[256][256];  // [prev_rows][curr_rows]

// Check if a three-row sequence contains any isolated 1s in the middle row
int has_isolated_ones(unsigned int top, unsigned int mid, unsigned int bot) {
    unsigned int isolated;
    
    // A 1 is not isolated if it has a neighbor above, below, left, or right
    // Get all 1s in middle row
    unsigned int ones = mid;
    
    // Get ones that have neighbors above or below
    unsigned int vertical_neighbors = top | bot;
    
    // Get ones that have neighbors to the left or right
    unsigned int horizontal_neighbors = (mid << 1) | (mid >> 1);
    
    // A 1 is isolated if it has no neighbors
    isolated = ones & ~vertical_neighbors & ~horizontal_neighbors;
    
    return isolated != 0;
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int row, unsigned int prev_pair, unsigned int curr_pair) {
    // Base case: we've filled all rows
    if (row >= 7)
        return 1;
    
    // Check memoization table
    if (memo[prev_pair][curr_pair] != -1)
        return memo[prev_pair][curr_pair];
    
    unsigned long long count = 0;
    unsigned int prev_top = (prev_pair >> 8) & 0xFF;
    unsigned int prev_bot = prev_pair & 0xFF;
    unsigned int curr_top = (curr_pair >> 8) & 0xFF;
    unsigned int curr_bot = curr_pair & 0xFF;
    
    // Try all possibilities for next row pair
    for (unsigned int next_pair = 0; next_pair < (1 << 16); next_pair++) {
        unsigned int next_top = (next_pair >> 8) & 0xFF;
        unsigned int next_bot = next_pair & 0xFF;
        
        // Check if current configuration creates any isolated ones
        if (!has_isolated_ones(prev_bot, curr_top, next_top) &&
            !has_isolated_ones(curr_top, curr_bot, next_bot)) {
            count += count_unholy(row + 2, curr_pair, next_pair);
        }
    }
    
    memo[prev_pair][curr_pair] = count;
    return count;
}

int main() {
    unsigned long long total = 0;
    
    // Initialize memoization table
    memset(memo, -1, sizeof(memo));
    
    // Try all possibilities for first two rows
    for (unsigned int first_pair = 0; first_pair < (1 << 16); first_pair++) {
        unsigned int top = (first_pair >> 8) & 0xFF;
        unsigned int bot = first_pair & 0xFF;
        
        // Check if first two rows are valid (no isolated ones in first row)
        if (!has_isolated_ones(0, top, bot)) {
            total += count_unholy(2, 0, first_pair);
        }
    }
    
    printf("Total number of unholy 8x8 matrices: %llu\n", total);
    return 0;
}
And got the output: Total number of unholy 8x8 matrices: 4475378688

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

Let's break this down into a clear plan and then implement it. Plan: 1. Represent the 8x8 matrix using 64 bits (unsigned long long) 2. Create helper functions: - is_unholy(): Check if a matrix has no isolated 1s - get_bit(): Get a specific bit from the matrix - count_unholy(): Main dynamic programming function 3. Dynamic Programming Approach: - State: (current_rows, previous_two_rows) - For each state, try all possible next two rows - Check if current configuration is valid - Base case: When all 8 rows are filled - Use memoization to store results 4. Main steps: - Initialize memoization table - Start recursion from row 0 - Sum up all valid configurations Here's the implementation: 

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

#define ROWS 8
#define COLS 8
#define MAX_TWO_ROWS (1 << 16)

unsigned long long memo[ROWS/2][MAX_TWO_ROWS];
int computed[ROWS/2][MAX_TWO_ROWS];

// Get bit at position (row, col)
int get_bit(unsigned long long matrix, int row, int col) {
    return (matrix >> (row * COLS + col)) & 1;
}

// Check if a 1 at (row, col) is isolated
int is_isolated(unsigned long long matrix, int row, int col) {
    if (!get_bit(matrix, row, col)) return 0;
    
    // Check four directions
    if (row > 0 && get_bit(matrix, row-1, col)) return 0;
    if (row < ROWS-1 && get_bit(matrix, row+1, col)) return 0;
    if (col > 0 && get_bit(matrix, row, col-1)) return 0;
    if (col < COLS-1 && get_bit(matrix, row, col+1)) return 0;
    
    return 1;
}

// Check if matrix has any isolated 1s
int has_isolated_ones(unsigned long long matrix) {
    for(int i = 0; i < ROWS; i++) {
        for(int j = 0; j < COLS; j++) {
            if (is_isolated(matrix, i, j)) return 1;
        }
    }
    return 0;
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int current_row, unsigned int prev_two_rows) {
    // Base case: all rows filled
    if (current_row == ROWS/2) return 1;
    
    // Check memoization
    if (computed[current_row][prev_two_rows]) 
        return memo[current_row][prev_two_rows];
        
    unsigned long long count = 0;
    
    // Try all possible next two rows
    for(unsigned int next_two_rows = 0; next_two_rows < MAX_TWO_ROWS; next_two_rows++) {
        unsigned long long partial_matrix = 0;
        
        // Construct partial matrix for validation
        if (current_row > 0) {
            partial_matrix |= ((unsigned long long)prev_two_rows) << (2*COLS*(current_row-1));
        }
        partial_matrix |= ((unsigned long long)next_two_rows) << (2*COLS*current_row);
        
        // Check if this partial configuration is valid
        if (!has_isolated_ones(partial_matrix)) {
            count += count_unholy(current_row + 1, next_two_rows);
        }
    }
    
    computed[current_row][prev_two_rows] = 1;
    memo[current_row][prev_two_rows] = count;
    return count;
}

int main() {
    // Initialize memoization arrays
    memset(computed, 0, sizeof(computed));
    memset(memo, 0, sizeof(memo));
    
    // Start counting from row 0 with empty previous rows
    unsigned long long result = count_unholy(0, 0);
    
    printf("Number of unholy 8x8 matrices: %llu\n", result);
    return 0;
}
This implementation: 1. Uses bit manipulation to efficiently represent and check matrices 2. Uses dynamic programming with memoization to avoid recalculating states 3. Checks for isolated 1s by examining neighbors in all four directions 4. Processes two rows at a time to optimize the solution 5. Uses a memoization table to store intermediate results The program will count all possible unholy 8x8 binary matrices. Note that this is a computationally intensive problem, so it might take some time to run. Some potential optimizations could include: 1. Precomputing valid two-row configurations 2. Using more bit manipulation tricks to check for isolated 1s 3. Parallelizing the computation for different initial states Would you like me to explain any part in more detail or suggest additional optimizations?

Extract Code

I extracted the following code from that output: 

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

#define ROWS 8
#define COLS 8
#define MAX_TWO_ROWS (1 << 16)

unsigned long long memo[ROWS/2][MAX_TWO_ROWS];
int computed[ROWS/2][MAX_TWO_ROWS];

// Get bit at position (row, col)
int get_bit(unsigned long long matrix, int row, int col) {
    return (matrix >> (row * COLS + col)) & 1;
}

// Check if a 1 at (row, col) is isolated
int is_isolated(unsigned long long matrix, int row, int col) {
    if (!get_bit(matrix, row, col)) return 0;
    
    // Check four directions
    if (row > 0 && get_bit(matrix, row-1, col)) return 0;
    if (row < ROWS-1 && get_bit(matrix, row+1, col)) return 0;
    if (col > 0 && get_bit(matrix, row, col-1)) return 0;
    if (col < COLS-1 && get_bit(matrix, row, col+1)) return 0;
    
    return 1;
}

// Check if matrix has any isolated 1s
int has_isolated_ones(unsigned long long matrix) {
    for(int i = 0; i < ROWS; i++) {
        for(int j = 0; j < COLS; j++) {
            if (is_isolated(matrix, i, j)) return 1;
        }
    }
    return 0;
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int current_row, unsigned int prev_two_rows) {
    // Base case: all rows filled
    if (current_row == ROWS/2) return 1;
    
    // Check memoization
    if (computed[current_row][prev_two_rows]) 
        return memo[current_row][prev_two_rows];
        
    unsigned long long count = 0;
    
    // Try all possible next two rows
    for(unsigned int next_two_rows = 0; next_two_rows < MAX_TWO_ROWS; next_two_rows++) {
        unsigned long long partial_matrix = 0;
        
        // Construct partial matrix for validation
        if (current_row > 0) {
            partial_matrix |= ((unsigned long long)prev_two_rows) << (2*COLS*(current_row-1));
        }
        partial_matrix |= ((unsigned long long)next_two_rows) << (2*COLS*current_row);
        
        // Check if this partial configuration is valid
        if (!has_isolated_ones(partial_matrix)) {
            count += count_unholy(current_row + 1, next_two_rows);
        }
    }
    
    computed[current_row][prev_two_rows] = 1;
    memo[current_row][prev_two_rows] = count;
    return count;
}

int main() {
    // Initialize memoization arrays
    memset(computed, 0, sizeof(computed));
    memset(memo, 0, sizeof(memo));
    
    // Start counting from row 0 with empty previous rows
    unsigned long long result = count_unholy(0, 0);
    
    printf("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 <string.h>

#define ROWS 8
#define COLS 8
#define MAX_TWO_ROWS (1 << 16)

unsigned long long memo[ROWS/2][MAX_TWO_ROWS];
int computed[ROWS/2][MAX_TWO_ROWS];

// Get bit at position (row, col)
int get_bit(unsigned long long matrix, int row, int col) {
    return (matrix >> (row * COLS + col)) & 1;
}

// Check if a 1 at (row, col) is isolated
int is_isolated(unsigned long long matrix, int row, int col) {
    if (!get_bit(matrix, row, col)) return 0;
    
    // Check four directions
    if (row > 0 && get_bit(matrix, row-1, col)) return 0;
    if (row < ROWS-1 && get_bit(matrix, row+1, col)) return 0;
    if (col > 0 && get_bit(matrix, row, col-1)) return 0;
    if (col < COLS-1 && get_bit(matrix, row, col+1)) return 0;
    
    return 1;
}

// Check if matrix has any isolated 1s
int has_isolated_ones(unsigned long long matrix) {
    for(int i = 0; i < ROWS; i++) {
        for(int j = 0; j < COLS; j++) {
            if (is_isolated(matrix, i, j)) return 1;
        }
    }
    return 0;
}

// Recursive function to count unholy matrices
unsigned long long count_unholy(int current_row, unsigned int prev_two_rows) {
    // Base case: all rows filled
    if (current_row == ROWS/2) return 1;
    
    // Check memoization
    if (computed[current_row][prev_two_rows]) 
        return memo[current_row][prev_two_rows];
        
    unsigned long long count = 0;
    
    // Try all possible next two rows
    for(unsigned int next_two_rows = 0; next_two_rows < MAX_TWO_ROWS; next_two_rows++) {
        unsigned long long partial_matrix = 0;
        
        // Construct partial matrix for validation
        if (current_row > 0) {
            partial_matrix |= ((unsigned long long)prev_two_rows) << (2*COLS*(current_row-1));
        }
        partial_matrix |= ((unsigned long long)next_two_rows) << (2*COLS*current_row);
        
        // Check if this partial configuration is valid
        if (!has_isolated_ones(partial_matrix)) {
            count += count_unholy(current_row + 1, next_two_rows);
        }
    }
    
    computed[current_row][prev_two_rows] = 1;
    memo[current_row][prev_two_rows] = count;
    return count;
}

int main() {
    // Initialize memoization arrays
    memset(computed, 0, sizeof(computed));
    memset(memo, 0, sizeof(memo));
    
    // Start counting from row 0 with empty previous rows
    unsigned long long result = count_unholy(0, 0);
    
    printf("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