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

# Counting Unholy 8x8 Binary Matrices ## Understanding the Problem An 8x8 binary matrix is considered "unholy" if it has no isolated 1s. A 1 is isolated if there isn't another 1 in any of the four cardinal directions (up, down, left, right). ## Solution Plan 1. **Representation**: Use an unsigned long long to represent each row of the 8x8 matrix (8 bits per row). 2. **Dynamic Programming**: - Define DP[row][prevRow][currRow] where: - row: Number of rows processed (0-8) - prevRow: Configuration of previous row (8 bits) - currRow: Configuration of current row (8 bits) - DP[row][prevRow][currRow] = number of ways to complete the matrix 3. **Transition**: For each possible next row configuration (2^8 = 256 possibilities), check if no 1 in the current row becomes isolated, then update DP[row+1][currRow][nextRow]. 4. **Base Case**: DP[0][0][0] = 1 (one way to have 0 rows filled) 5. **Final Answer**: Sum of DP[8][prevRow][currRow] for all valid prevRow and currRow combinations where no 1 in currRow is isolated. ## Implementation 

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

// Check if a 1 in current row is isolated
int is_isolated(uint8_t prevRow, uint8_t currRow, uint8_t nextRow) {
    for (int i = 0; i < 8; i++) {
        if ((currRow >> i) & 1) {  // If there's a 1 at position i
            // Check left neighbor (if not leftmost column)
            int left = (i > 0) ? ((currRow >> (i-1)) & 1) : 0;
            
            // Check right neighbor (if not rightmost column)
            int right = (i < 7) ? ((currRow >> (i+1)) & 1) : 0;
            
            // Check up neighbor
            int up = (prevRow >> i) & 1;
            
            // Check down neighbor
            int down = (nextRow >> i) & 1;
            
            // If there's no neighbor, the 1 is isolated
            if (!(left || right || up || down)) {
                return 1;  // Isolated 1 found
            }
        }
    }
    return 0;  // No isolated 1 found
}

uint64_t count_unholy_matrices() {
    // DP[row][prevRow][currRow] = count of valid configurations
    uint64_t dp[9][256][256];
    memset(dp, 0, sizeof(dp));
    
    // Base case: one way to have 0 rows
    dp[0][0][0] = 1;
    
    // Fill dp table
    for (int row = 0; row < 8; row++) {
        for (int prevRow = 0; prevRow < 256; prevRow++) {
            for (int currRow = 0; currRow < 256; currRow++) {
                if (dp[row][prevRow][currRow] == 0) continue;
                
                // Try all possible next rows
                for (int nextRow = 0; nextRow < 256; nextRow++) {
                    // Check if no 1 in current row is isolated
                    if (!is_isolated(prevRow, currRow, nextRow)) {
                        dp[row+1][currRow][nextRow] += dp[row][prevRow][currRow];
                    }
                }
            }
        }
    }
    
    // Sum up all valid final states
    uint64_t result = 0;
    for (int prevRow = 0; prevRow < 256; prevRow++) {
        for (int currRow = 0; currRow < 256; currRow++) {
            // For the final row, there's no next row, so we use 0
            // But we still need to check that the last row has no isolated 1s
            if (!is_isolated(prevRow, currRow, 0)) {
                result += dp[8][prevRow][currRow];
            }
        }
    }
    
    return result;
}

int main() {
    uint64_t count = count_unholy_matrices();
    printf("Number of unholy 8x8 matrices: %llu\n", count);
    return 0;
}
Wait, there's a flaw in my reasoning. For the last row, we still need to check that no 1 is isolated, but using 0 as the "next row" isn't correct since that would make any 1 in the last row isolated. Let me fix the implementation: 
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#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <string.h>

// Check if a 1 in current row is isolated
int is_isolated(uint8_t prevRow, uint8_t currRow, uint8_t nextRow) {
    for (int i = 0; i < 8; i++) {
        if ((currRow >> i) & 1) {  // If there's a 1 at position i
            // Check left neighbor (if not leftmost column)
            int left = (i > 0) ? ((currRow >> (i-1)) & 1) : 0;
            
            // Check right neighbor (if not rightmost column)
            int right = (i < 7) ? ((currRow >> (i+1)) & 1) : 0;
            
            // Check up neighbor
            int up = (prevRow >> i) & 1;
            
            // Check down neighbor
            int down = (nextRow >> i) & 1;
            
            // If there's no neighbor, the 1 is isolated
            if (!(left || right || up || down)) {
                return 1;  // Isolated 1 found
            }
        }
    }
    return 0;  // No isolated 1 found
}

uint64_t count_unholy_matrices() {
    // DP[row][prevRow][currRow] = count of valid configurations
    uint64_t dp[9][256][256];
    memset(dp, 0, sizeof(dp));
    
    // Base case: one way to have 0 rows
    dp[0][0][0] = 1;
    
    // Fill dp table for rows 0 to 7
    for (int row = 0; row < 8; row++) {
        for (int prevRow = 0; prevRow < 256; prevRow++) {
            for (int currRow = 0; currRow < 256; currRow++) {
                if (dp[row][prevRow][currRow] == 0) continue;
                
                // Try all possible next rows
                for (int nextRow = 0; nextRow < 256; nextRow++) {
                    // Check if no 1 in current row is isolated
                    if (!is_isolated(prevRow, currRow, nextRow)) {
                        dp[row+1][currRow][nextRow] += dp[row][prevRow][currRow];
                    }
                }
            }
        }
    }
    
    // For the 8th row, we need to ensure no 1s are isolated in the last row
    uint64_t result = 0;
    for (int prevRow = 0; prevRow < 256; prevRow++) {
        for (int currRow = 0; currRow < 256; currRow++) {
            if (dp[8][prevRow][currRow] > 0) {
                // For the last row, we check if any 1 in currRow is isolated
                // without considering a next row
                int isolated = 0;
                for (int i = 0; i < 8; i++) {
                    if ((currRow >> i) & 1) {  // If there's a 1 at position i
                        // Check left, right, and up neighbors
                        int left = (i > 0) ? ((currRow >> (i-1)) & 1) : 0;
                        int right = (i < 7) ? ((currRow >> (i+1)) & 1) : 0;
                        int up = (prevRow >> i) & 1;
                        
                        // If there's no neighbor, the 1 is isolated
                        if (!(left || right || up)) {
                            isolated = 1;
                            break;
                        }
                    }
                }
                
                if (!isolated) {
                    result += dp[8][prevRow][currRow];
                }
            }
        }
    }
    
    return result;
}

int main() {
    uint64_t count = count_unholy_matrices();
    printf("Number of unholy 8x8 matrices: %llu\n", count);
    return 0;
}
This implementation uses dynamic programming to count the number of unholy 8x8 matrices by building them row by row. For each state, we try all possible next row configurations and check if that would create any isolated 1s. The final answer is the sum of all valid configurations.

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>

// Check if a 1 in current row is isolated
int is_isolated(uint8_t prevRow, uint8_t currRow, uint8_t nextRow) {
    for (int i = 0; i < 8; i++) {
        if ((currRow >> i) & 1) {  // If there's a 1 at position i
            // Check left neighbor (if not leftmost column)
            int left = (i > 0) ? ((currRow >> (i-1)) & 1) : 0;
            
            // Check right neighbor (if not rightmost column)
            int right = (i < 7) ? ((currRow >> (i+1)) & 1) : 0;
            
            // Check up neighbor
            int up = (prevRow >> i) & 1;
            
            // Check down neighbor
            int down = (nextRow >> i) & 1;
            
            // If there's no neighbor, the 1 is isolated
            if (!(left || right || up || down)) {
                return 1;  // Isolated 1 found
            }
        }
    }
    return 0;  // No isolated 1 found
}

uint64_t count_unholy_matrices() {
    // DP[row][prevRow][currRow] = count of valid configurations
    uint64_t dp[9][256][256];
    memset(dp, 0, sizeof(dp));
    
    // Base case: one way to have 0 rows
    dp[0][0][0] = 1;
    
    // Fill dp table for rows 0 to 7
    for (int row = 0; row < 8; row++) {
        for (int prevRow = 0; prevRow < 256; prevRow++) {
            for (int currRow = 0; currRow < 256; currRow++) {
                if (dp[row][prevRow][currRow] == 0) continue;
                
                // Try all possible next rows
                for (int nextRow = 0; nextRow < 256; nextRow++) {
                    // Check if no 1 in current row is isolated
                    if (!is_isolated(prevRow, currRow, nextRow)) {
                        dp[row+1][currRow][nextRow] += dp[row][prevRow][currRow];
                    }
                }
            }
        }
    }
    
    // For the 8th row, we need to ensure no 1s are isolated in the last row
    uint64_t result = 0;
    for (int prevRow = 0; prevRow < 256; prevRow++) {
        for (int currRow = 0; currRow < 256; currRow++) {
            if (dp[8][prevRow][currRow] > 0) {
                // For the last row, we check if any 1 in currRow is isolated
                // without considering a next row
                int isolated = 0;
                for (int i = 0; i < 8; i++) {
                    if ((currRow >> i) & 1) {  // If there's a 1 at position i
                        // Check left, right, and up neighbors
                        int left = (i > 0) ? ((currRow >> (i-1)) & 1) : 0;
                        int right = (i < 7) ? ((currRow >> (i+1)) & 1) : 0;
                        int up = (prevRow >> i) & 1;
                        
                        // If there's no neighbor, the 1 is isolated
                        if (!(left || right || up)) {
                            isolated = 1;
                            break;
                        }
                    }
                }
                
                if (!isolated) {
                    result += dp[8][prevRow][currRow];
                }
            }
        }
    }
    
    return result;
}

int main() {
    uint64_t count = count_unholy_matrices();
    printf("Number of unholy 8x8 matrices: %llu\n", count);
    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>

// Check if a 1 in current row is isolated
int is_isolated(uint8_t prevRow, uint8_t currRow, uint8_t nextRow) {
    for (int i = 0; i < 8; i++) {
        if ((currRow >> i) & 1) {  // If there's a 1 at position i
            // Check left neighbor (if not leftmost column)
            int left = (i > 0) ? ((currRow >> (i-1)) & 1) : 0;
            
            // Check right neighbor (if not rightmost column)
            int right = (i < 7) ? ((currRow >> (i+1)) & 1) : 0;
            
            // Check up neighbor
            int up = (prevRow >> i) & 1;
            
            // Check down neighbor
            int down = (nextRow >> i) & 1;
            
            // If there's no neighbor, the 1 is isolated
            if (!(left || right || up || down)) {
                return 1;  // Isolated 1 found
            }
        }
    }
    return 0;  // No isolated 1 found
}

uint64_t count_unholy_matrices() {
    // DP[row][prevRow][currRow] = count of valid configurations
    uint64_t dp[9][256][256];
    memset(dp, 0, sizeof(dp));
    
    // Base case: one way to have 0 rows
    dp[0][0][0] = 1;
    
    // Fill dp table for rows 0 to 7
    for (int row = 0; row < 8; row++) {
        for (int prevRow = 0; prevRow < 256; prevRow++) {
            for (int currRow = 0; currRow < 256; currRow++) {
                if (dp[row][prevRow][currRow] == 0) continue;
                
                // Try all possible next rows
                for (int nextRow = 0; nextRow < 256; nextRow++) {
                    // Check if no 1 in current row is isolated
                    if (!is_isolated(prevRow, currRow, nextRow)) {
                        dp[row+1][currRow][nextRow] += dp[row][prevRow][currRow];
                    }
                }
            }
        }
    }
    
    // For the 8th row, we need to ensure no 1s are isolated in the last row
    uint64_t result = 0;
    for (int prevRow = 0; prevRow < 256; prevRow++) {
        for (int currRow = 0; currRow < 256; currRow++) {
            if (dp[8][prevRow][currRow] > 0) {
                // For the last row, we check if any 1 in currRow is isolated
                // without considering a next row
                int isolated = 0;
                for (int i = 0; i < 8; i++) {
                    if ((currRow >> i) & 1) {  // If there's a 1 at position i
                        // Check left, right, and up neighbors
                        int left = (i > 0) ? ((currRow >> (i-1)) & 1) : 0;
                        int right = (i < 7) ? ((currRow >> (i+1)) & 1) : 0;
                        int up = (prevRow >> i) & 1;
                        
                        // If there's no neighbor, the 1 is isolated
                        if (!(left || right || up)) {
                            isolated = 1;
                            break;
                        }
                    }
                }
                
                if (!isolated) {
                    result += dp[8][prevRow][currRow];
                }
            }
        }
    }
    
    return result;
}

int main() {
    uint64_t count = count_unholy_matrices();
    printf("Number of unholy 8x8 matrices: %llu\n", count);
    return 0;
}
And got the output: Number of unholy 8x8 matrices: 1121231537486377866 main.c: In function 'main': main.c:92:47: warning: format '%llu' expects argument of type 'long long unsigned int', but argument 2 has type 'uint64_t' {aka 'long unsigned int'} [-Wformat=] 92 | printf("Number of unholy 8x8 matrices: %llu\n", count); | ~~~^ ~~~~~ | | | | | uint64_t {aka long unsigned int} | long long unsigned int | %lu

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: True

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

# Counting Unholy 8x8 Matrices in C I'll create a program to count 8x8 binary matrices that have no isolated 1s, using bit manipulation and dynamic programming. ## Understanding the Problem - An 8x8 binary matrix is represented as 64 bits (can fit in an unsigned long long) - A 1 is "isolated" if it has no 1s in the four adjacent positions (up, down, left, right) - A matrix is "unholy" if it contains no isolated 1s - We'll count all possible unholy 8x8 matrices using dynamic programming ## Approach 1. We'll recurse on rows, building the matrix two rows at a time 2. Each state in our DP table represents the number of unholy matrices with the last two rows in a specific configuration 3. For each valid configuration of the last two rows, we'll try all possible 2^16 configurations for the next two rows 4. A configuration is valid if it doesn't create any isolated 1s ## Implementation Plan 1. Create a function to check if a matrix is unholy 2. Implement a DP solution that builds the matrix row by row 3. Use bit operations for efficiency 4. Calculate the total count of unholy matrices Here's the C implementation: 

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

#define ROW_SIZE 8
#define MATRIX_ROWS 8
#define MAX_PATTERN (1 << 16)  // 2^16 possible patterns for two rows

// Function to check if a 1 at position (row, col) is isolated
bool is_isolated(uint64_t matrix, int row, int col) {
    // Check if there's a 1 bit to the left
    if (col > 0 && (matrix & (1ULL << ((row * ROW_SIZE) + col - 1))))
        return false;
    
    // Check if there's a 1 bit to the right
    if (col < ROW_SIZE - 1 && (matrix & (1ULL << ((row * ROW_SIZE) + col + 1))))
        return false;
    
    // Check if there's a 1 bit above
    if (row > 0 && (matrix & (1ULL << (((row - 1) * ROW_SIZE) + col))))
        return false;
    
    // Check if there's a 1 bit below
    if (row < MATRIX_ROWS - 1 && (matrix & (1ULL << (((row + 1) * ROW_SIZE) + col))))
        return false;
    
    // If we didn't find any adjacent 1s, the bit is isolated
    return true;
}

// Function to check if two rows (current and next) have any isolated 1s
bool has_isolated_ones(uint16_t prev_rows, uint16_t curr_rows) {
    // Extract individual rows
    uint8_t row1 = (prev_rows >> 8) & 0xFF;  // Second row of prev_rows
    uint8_t row2 = prev_rows & 0xFF;         // First row of curr_rows
    uint8_t row3 = (curr_rows >> 8) & 0xFF;  // Second row of curr_rows
    uint8_t row4 = curr_rows & 0xFF;         // First row of next set
    
    // Check for isolated 1s in row2 (which is between row1 and row3)
    for (int col = 0; col < ROW_SIZE; col++) {
        if ((row2 & (1 << col))) {  // If there's a 1 at this position
            bool has_adjacent = false;
            
            // Check left
            if (col > 0 && (row2 & (1 << (col - 1))))
                has_adjacent = true;
            // Check right
            else if (col < ROW_SIZE - 1 && (row2 & (1 << (col + 1))))
                has_adjacent = true;
            // Check above (row1)
            else if (row1 & (1 << col))
                has_adjacent = true;
            // Check below (row3)
            else if (row3 & (1 << col))
                has_adjacent = true;
            
            if (!has_adjacent)
                return true;  // Found an isolated 1
        }
    }
    
    // Check for isolated 1s in row3 (which is between row2 and row4)
    for (int col = 0; col < ROW_SIZE; col++) {
        if ((row3 & (1 << col))) {  // If there's a 1 at this position
            bool has_adjacent = false;
            
            // Check left
            if (col > 0 && (row3 & (1 << (col - 1))))
                has_adjacent = true;
            // Check right
            else if (col < ROW_SIZE - 1 && (row3 & (1 << (col + 1))))
                has_adjacent = true;
            // Check above (row2)
            else if (row2 & (1 << col))
                has_adjacent = true;
            // Check below (row4)
            else if (row4 & (1 << col))
                has_adjacent = true;
            
            if (!has_adjacent)
                return true;  // Found an isolated 1
        }
    }
    
    return false;  // No isolated 1s found
}

// Function to count unholy matrices
uint64_t count_unholy_matrices() {
    // DP[r][pattern] = number of unholy matrices with r rows where the last two rows form the pattern
    uint64_t dp[MATRIX_ROWS/2 + 1][MAX_PATTERN] = {0};
    
    // Base case: Initialize the DP table for the first two rows
    for (uint32_t pattern = 0; pattern < MAX_PATTERN; pattern++) {
        // Check if the 2-row pattern has any isolated 1s
        bool isolated = false;
        uint8_t row1 = (pattern >> 8) & 0xFF;
        uint8_t row2 = pattern & 0xFF;
        
        for (int col = 0; col < ROW_SIZE; col++) {
            // Check row1
            if ((row1 & (1 << col))) {
                bool has_adjacent = false;
                // Check left
                if (col > 0 && (row1 & (1 << (col - 1))))
                    has_adjacent = true;
                // Check right
                else if (col < ROW_SIZE - 1 && (row1 & (1 << (col + 1))))
                    has_adjacent = true;
                // Check below (row2)
                else if (row2 & (1 << col))
                    has_adjacent = true;
                
                if (!has_adjacent) {
                    isolated = true;
                    break;
                }
            }
            
            // Check row2
            if ((row2 & (1 << col))) {
                bool has_adjacent = false;
                // Check left
                if (col > 0 && (row2 & (1 << (col - 1))))
                    has_adjacent = true;
                // Check right
                else if (col < ROW_SIZE - 1 && (row2 & (1 << (col + 1))))
                    has_adjacent = true;
                // Check above (row1)
                else if (row1 & (1 << col))
                    has_adjacent = true;
                
                if (!has_adjacent) {
                    isolated = true;
                    break;
                }
            }
        }
        
        if (!isolated)
            dp[1][pattern] = 1;  // This 2-row pattern is unholy
    }
    
    // Fill the DP table
    for (int r = 2; r <= MATRIX_ROWS/2; r++) {
        for (uint32_t prev_pattern = 0; prev_pattern < MAX_PATTERN; prev_pattern++) {
            if (dp[r-1][prev_pattern] == 0)
                continue;  // Skip if there are no unholy matrices with this pattern
            
            for (uint32_t curr_pattern = 0; curr_pattern < MAX_PATTERN; curr_pattern++) {
                // Check if adding the next two rows creates any isolated 1s
                if (!has_isolated_ones(prev_pattern, curr_pattern)) {
                    dp[r][curr_pattern] += dp[r-1][prev_pattern];
                }
            }
        }
    }
    
    // Sum up all the possible unholy matrices with 8 rows
    uint64_t total = 0;
    for (uint32_t pattern = 0; pattern < MAX_PATTERN; pattern++) {
        total += dp[MATRIX_ROWS/2][pattern];
    }
    
    return total;
}

int main() {
    uint64_t count = count_unholy_matrices();
    printf("Total number of unholy 8x8 matrices: %llu\n", count);
    return 0;
}
## Explanation 1. **Representation**: Each row is 8 bits, and two consecutive rows form a 16-bit pattern. The entire 8x8 matrix can be represented as 4 pairs of rows. 2. **Dynamic Programming**: - The DP table `dp[r][pattern]` stores the number of unholy matrices with `r` pairs of rows (2*r total rows), where the last two rows form the given pattern. - Base case: For r=1 (first two rows), we enumerate all 2^16 patterns and check if any have isolated 1s. - For r=2,3,4: We try all possible next-two-row patterns and verify they don't create isolated 1s when combined with the previous pattern. 3. **Isolation Check**: The `has_isolated_ones` function checks if there are any isolated 1s when two row-pairs are combined. 4. **Final Count**: The total number of unholy matrices is the sum of all entries in `dp[4][pattern]` for all valid patterns. This program efficiently counts all unholy 8x8 matrices using bit manipulation and dynamic programming.

Extract Code

I extracted the following code from that output: 

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

#define ROW_SIZE 8
#define MATRIX_ROWS 8
#define MAX_PATTERN (1 << 16)  // 2^16 possible patterns for two rows

// Function to check if a 1 at position (row, col) is isolated
bool is_isolated(uint64_t matrix, int row, int col) {
    // Check if there's a 1 bit to the left
    if (col > 0 && (matrix & (1ULL << ((row * ROW_SIZE) + col - 1))))
        return false;
    
    // Check if there's a 1 bit to the right
    if (col < ROW_SIZE - 1 && (matrix & (1ULL << ((row * ROW_SIZE) + col + 1))))
        return false;
    
    // Check if there's a 1 bit above
    if (row > 0 && (matrix & (1ULL << (((row - 1) * ROW_SIZE) + col))))
        return false;
    
    // Check if there's a 1 bit below
    if (row < MATRIX_ROWS - 1 && (matrix & (1ULL << (((row + 1) * ROW_SIZE) + col))))
        return false;
    
    // If we didn't find any adjacent 1s, the bit is isolated
    return true;
}

// Function to check if two rows (current and next) have any isolated 1s
bool has_isolated_ones(uint16_t prev_rows, uint16_t curr_rows) {
    // Extract individual rows
    uint8_t row1 = (prev_rows >> 8) & 0xFF;  // Second row of prev_rows
    uint8_t row2 = prev_rows & 0xFF;         // First row of curr_rows
    uint8_t row3 = (curr_rows >> 8) & 0xFF;  // Second row of curr_rows
    uint8_t row4 = curr_rows & 0xFF;         // First row of next set
    
    // Check for isolated 1s in row2 (which is between row1 and row3)
    for (int col = 0; col < ROW_SIZE; col++) {
        if ((row2 & (1 << col))) {  // If there's a 1 at this position
            bool has_adjacent = false;
            
            // Check left
            if (col > 0 && (row2 & (1 << (col - 1))))
                has_adjacent = true;
            // Check right
            else if (col < ROW_SIZE - 1 && (row2 & (1 << (col + 1))))
                has_adjacent = true;
            // Check above (row1)
            else if (row1 & (1 << col))
                has_adjacent = true;
            // Check below (row3)
            else if (row3 & (1 << col))
                has_adjacent = true;
            
            if (!has_adjacent)
                return true;  // Found an isolated 1
        }
    }
    
    // Check for isolated 1s in row3 (which is between row2 and row4)
    for (int col = 0; col < ROW_SIZE; col++) {
        if ((row3 & (1 << col))) {  // If there's a 1 at this position
            bool has_adjacent = false;
            
            // Check left
            if (col > 0 && (row3 & (1 << (col - 1))))
                has_adjacent = true;
            // Check right
            else if (col < ROW_SIZE - 1 && (row3 & (1 << (col + 1))))
                has_adjacent = true;
            // Check above (row2)
            else if (row2 & (1 << col))
                has_adjacent = true;
            // Check below (row4)
            else if (row4 & (1 << col))
                has_adjacent = true;
            
            if (!has_adjacent)
                return true;  // Found an isolated 1
        }
    }
    
    return false;  // No isolated 1s found
}

// Function to count unholy matrices
uint64_t count_unholy_matrices() {
    // DP[r][pattern] = number of unholy matrices with r rows where the last two rows form the pattern
    uint64_t dp[MATRIX_ROWS/2 + 1][MAX_PATTERN] = {0};
    
    // Base case: Initialize the DP table for the first two rows
    for (uint32_t pattern = 0; pattern < MAX_PATTERN; pattern++) {
        // Check if the 2-row pattern has any isolated 1s
        bool isolated = false;
        uint8_t row1 = (pattern >> 8) & 0xFF;
        uint8_t row2 = pattern & 0xFF;
        
        for (int col = 0; col < ROW_SIZE; col++) {
            // Check row1
            if ((row1 & (1 << col))) {
                bool has_adjacent = false;
                // Check left
                if (col > 0 && (row1 & (1 << (col - 1))))
                    has_adjacent = true;
                // Check right
                else if (col < ROW_SIZE - 1 && (row1 & (1 << (col + 1))))
                    has_adjacent = true;
                // Check below (row2)
                else if (row2 & (1 << col))
                    has_adjacent = true;
                
                if (!has_adjacent) {
                    isolated = true;
                    break;
                }
            }
            
            // Check row2
            if ((row2 & (1 << col))) {
                bool has_adjacent = false;
                // Check left
                if (col > 0 && (row2 & (1 << (col - 1))))
                    has_adjacent = true;
                // Check right
                else if (col < ROW_SIZE - 1 && (row2 & (1 << (col + 1))))
                    has_adjacent = true;
                // Check above (row1)
                else if (row1 & (1 << col))
                    has_adjacent = true;
                
                if (!has_adjacent) {
                    isolated = true;
                    break;
                }
            }
        }
        
        if (!isolated)
            dp[1][pattern] = 1;  // This 2-row pattern is unholy
    }
    
    // Fill the DP table
    for (int r = 2; r <= MATRIX_ROWS/2; r++) {
        for (uint32_t prev_pattern = 0; prev_pattern < MAX_PATTERN; prev_pattern++) {
            if (dp[r-1][prev_pattern] == 0)
                continue;  // Skip if there are no unholy matrices with this pattern
            
            for (uint32_t curr_pattern = 0; curr_pattern < MAX_PATTERN; curr_pattern++) {
                // Check if adding the next two rows creates any isolated 1s
                if (!has_isolated_ones(prev_pattern, curr_pattern)) {
                    dp[r][curr_pattern] += dp[r-1][prev_pattern];
                }
            }
        }
    }
    
    // Sum up all the possible unholy matrices with 8 rows
    uint64_t total = 0;
    for (uint32_t pattern = 0; pattern < MAX_PATTERN; pattern++) {
        total += dp[MATRIX_ROWS/2][pattern];
    }
    
    return total;
}

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

Run Code Interpreter

Running the following program: 

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

#define ROW_SIZE 8
#define MATRIX_ROWS 8
#define MAX_PATTERN (1 << 16)  // 2^16 possible patterns for two rows

// Function to check if a 1 at position (row, col) is isolated
bool is_isolated(uint64_t matrix, int row, int col) {
    // Check if there's a 1 bit to the left
    if (col > 0 && (matrix & (1ULL << ((row * ROW_SIZE) + col - 1))))
        return false;
    
    // Check if there's a 1 bit to the right
    if (col < ROW_SIZE - 1 && (matrix & (1ULL << ((row * ROW_SIZE) + col + 1))))
        return false;
    
    // Check if there's a 1 bit above
    if (row > 0 && (matrix & (1ULL << (((row - 1) * ROW_SIZE) + col))))
        return false;
    
    // Check if there's a 1 bit below
    if (row < MATRIX_ROWS - 1 && (matrix & (1ULL << (((row + 1) * ROW_SIZE) + col))))
        return false;
    
    // If we didn't find any adjacent 1s, the bit is isolated
    return true;
}

// Function to check if two rows (current and next) have any isolated 1s
bool has_isolated_ones(uint16_t prev_rows, uint16_t curr_rows) {
    // Extract individual rows
    uint8_t row1 = (prev_rows >> 8) & 0xFF;  // Second row of prev_rows
    uint8_t row2 = prev_rows & 0xFF;         // First row of curr_rows
    uint8_t row3 = (curr_rows >> 8) & 0xFF;  // Second row of curr_rows
    uint8_t row4 = curr_rows & 0xFF;         // First row of next set
    
    // Check for isolated 1s in row2 (which is between row1 and row3)
    for (int col = 0; col < ROW_SIZE; col++) {
        if ((row2 & (1 << col))) {  // If there's a 1 at this position
            bool has_adjacent = false;
            
            // Check left
            if (col > 0 && (row2 & (1 << (col - 1))))
                has_adjacent = true;
            // Check right
            else if (col < ROW_SIZE - 1 && (row2 & (1 << (col + 1))))
                has_adjacent = true;
            // Check above (row1)
            else if (row1 & (1 << col))
                has_adjacent = true;
            // Check below (row3)
            else if (row3 & (1 << col))
                has_adjacent = true;
            
            if (!has_adjacent)
                return true;  // Found an isolated 1
        }
    }
    
    // Check for isolated 1s in row3 (which is between row2 and row4)
    for (int col = 0; col < ROW_SIZE; col++) {
        if ((row3 & (1 << col))) {  // If there's a 1 at this position
            bool has_adjacent = false;
            
            // Check left
            if (col > 0 && (row3 & (1 << (col - 1))))
                has_adjacent = true;
            // Check right
            else if (col < ROW_SIZE - 1 && (row3 & (1 << (col + 1))))
                has_adjacent = true;
            // Check above (row2)
            else if (row2 & (1 << col))
                has_adjacent = true;
            // Check below (row4)
            else if (row4 & (1 << col))
                has_adjacent = true;
            
            if (!has_adjacent)
                return true;  // Found an isolated 1
        }
    }
    
    return false;  // No isolated 1s found
}

// Function to count unholy matrices
uint64_t count_unholy_matrices() {
    // DP[r][pattern] = number of unholy matrices with r rows where the last two rows form the pattern
    uint64_t dp[MATRIX_ROWS/2 + 1][MAX_PATTERN] = {0};
    
    // Base case: Initialize the DP table for the first two rows
    for (uint32_t pattern = 0; pattern < MAX_PATTERN; pattern++) {
        // Check if the 2-row pattern has any isolated 1s
        bool isolated = false;
        uint8_t row1 = (pattern >> 8) & 0xFF;
        uint8_t row2 = pattern & 0xFF;
        
        for (int col = 0; col < ROW_SIZE; col++) {
            // Check row1
            if ((row1 & (1 << col))) {
                bool has_adjacent = false;
                // Check left
                if (col > 0 && (row1 & (1 << (col - 1))))
                    has_adjacent = true;
                // Check right
                else if (col < ROW_SIZE - 1 && (row1 & (1 << (col + 1))))
                    has_adjacent = true;
                // Check below (row2)
                else if (row2 & (1 << col))
                    has_adjacent = true;
                
                if (!has_adjacent) {
                    isolated = true;
                    break;
                }
            }
            
            // Check row2
            if ((row2 & (1 << col))) {
                bool has_adjacent = false;
                // Check left
                if (col > 0 && (row2 & (1 << (col - 1))))
                    has_adjacent = true;
                // Check right
                else if (col < ROW_SIZE - 1 && (row2 & (1 << (col + 1))))
                    has_adjacent = true;
                // Check above (row1)
                else if (row1 & (1 << col))
                    has_adjacent = true;
                
                if (!has_adjacent) {
                    isolated = true;
                    break;
                }
            }
        }
        
        if (!isolated)
            dp[1][pattern] = 1;  // This 2-row pattern is unholy
    }
    
    // Fill the DP table
    for (int r = 2; r <= MATRIX_ROWS/2; r++) {
        for (uint32_t prev_pattern = 0; prev_pattern < MAX_PATTERN; prev_pattern++) {
            if (dp[r-1][prev_pattern] == 0)
                continue;  // Skip if there are no unholy matrices with this pattern
            
            for (uint32_t curr_pattern = 0; curr_pattern < MAX_PATTERN; curr_pattern++) {
                // Check if adding the next two rows creates any isolated 1s
                if (!has_isolated_ones(prev_pattern, curr_pattern)) {
                    dp[r][curr_pattern] += dp[r-1][prev_pattern];
                }
            }
        }
    }
    
    // Sum up all the possible unholy matrices with 8 rows
    uint64_t total = 0;
    for (uint32_t pattern = 0; pattern < MAX_PATTERN; pattern++) {
        total += dp[MATRIX_ROWS/2][pattern];
    }
    
    return total;
}

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