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.

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.

Output

# Counting Unholy 8x8 Binary Matrices This problem requires us to count binary matrices where no 1 is isolated. A 1 is considered isolated if it doesn't have another 1 in any of the 4 cardinal directions (up, down, left, right). My approach uses dynamic programming, tracking states as we build the matrix row by row: 

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

// Cache for memoization - indexed by [row][previous row][current row]
uint64_t memo[9][256][256];
bool computed[9][256][256];

// Check if a 1 in middle_row has no neighbors in any of the adjacent rows
// or horizontally in its own row
bool has_isolated_one(int top_row, int middle_row, int bottom_row) {
    for (int i = 0; i < 8; i++) {
        if (middle_row & (1 << i)) {  // If this bit is a 1
            bool has_neighbor = false;
            
            // Check left in current row
            if (i > 0 && (middle_row & (1 << (i-1))))
                has_neighbor = true;
            
            // Check right in current row
            if (i < 7 && (middle_row & (1 << (i+1))))
                has_neighbor = true;
            
            // Check above in top row
            if (top_row & (1 << i))
                has_neighbor = true;
            
            // Check below in bottom row
            if (bottom_row & (1 << i))
                has_neighbor = true;
            
            if (!has_neighbor)
                return true;  // Found an isolated 1
        }
    }
    return false;  // No isolated 1s found
}

// Count unholy matrices starting from the given row
uint64_t count_unholy(int row, int prev_row, int curr_row) {
    // If we've filled all rows
    if (row == 8) {
        // For the last row, check if there are isolated 1s with 
        // no bottom row (bottom_row = 0)
        return has_isolated_one(prev_row, curr_row, 0) ? 0 : 1;
    }
    
    // Return memoized result if available
    if (computed[row][prev_row][curr_row])
        return memo[row][prev_row][curr_row];
    
    uint64_t count = 0;
    // Try all possible configurations for the next row
    for (int next_row = 0; next_row < 256; next_row++) {
        // Check if adding this next row creates an isolated 1 in the current row
        if (!has_isolated_one(prev_row, curr_row, next_row)) {
            count += count_unholy(row + 1, curr_row, next_row);
        }
    }
    
    // Memoize result
    computed[row][prev_row][curr_row] = true;
    memo[row][prev_row][curr_row] = count;
    return count;
}

int main() {
    // Initialize memoization table
    memset(computed, 0, sizeof(computed));
    
    uint64_t total = 0;
    
    // For the first row, try all possible configurations
    for (int first_row = 0; first_row < 256; first_row++) {
        // For the first row, there's no previous row (all 0s)
        for (int second_row = 0; second_row < 256; second_row++) {
            // Check if any 1s in first row would be isolated
            if (!has_isolated_one(0, first_row, second_row)) {
                total += count_unholy(2, first_row, second_row);
            }
        }
    }
    
    printf("Total number of unholy 8x8 matrices: %llu\n", total);
    return 0;
}
## How the Solution Works 1. **State Representation**: I use three key pieces of information to represent a state: - The current row we're filling (row) - The configuration of the previous row (prev_row) - The configuration of the current row (curr_row) 2. **Transition**: For each possible configuration of the next row, we check if it would create any isolated 1s in the current row. If not, we recursively count the matrices from that point. 3. **Memoization**: Since many states repeat during the recursion, I use a 3D array to cache results. 4. **Base Case**: When we reach the 8th row, we check if the final row has any isolated 1s. Time complexity is significantly reduced by using dynamic programming, compared to checking all 2^64 possible matrices.

Extract Code

I extracted the following code from that output: 

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

// Cache for memoization - indexed by [row][previous row][current row]
uint64_t memo[9][256][256];
bool computed[9][256][256];

// Check if a 1 in middle_row has no neighbors in any of the adjacent rows
// or horizontally in its own row
bool has_isolated_one(int top_row, int middle_row, int bottom_row) {
    for (int i = 0; i < 8; i++) {
        if (middle_row & (1 << i)) {  // If this bit is a 1
            bool has_neighbor = false;
            
            // Check left in current row
            if (i > 0 && (middle_row & (1 << (i-1))))
                has_neighbor = true;
            
            // Check right in current row
            if (i < 7 && (middle_row & (1 << (i+1))))
                has_neighbor = true;
            
            // Check above in top row
            if (top_row & (1 << i))
                has_neighbor = true;
            
            // Check below in bottom row
            if (bottom_row & (1 << i))
                has_neighbor = true;
            
            if (!has_neighbor)
                return true;  // Found an isolated 1
        }
    }
    return false;  // No isolated 1s found
}

// Count unholy matrices starting from the given row
uint64_t count_unholy(int row, int prev_row, int curr_row) {
    // If we've filled all rows
    if (row == 8) {
        // For the last row, check if there are isolated 1s with 
        // no bottom row (bottom_row = 0)
        return has_isolated_one(prev_row, curr_row, 0) ? 0 : 1;
    }
    
    // Return memoized result if available
    if (computed[row][prev_row][curr_row])
        return memo[row][prev_row][curr_row];
    
    uint64_t count = 0;
    // Try all possible configurations for the next row
    for (int next_row = 0; next_row < 256; next_row++) {
        // Check if adding this next row creates an isolated 1 in the current row
        if (!has_isolated_one(prev_row, curr_row, next_row)) {
            count += count_unholy(row + 1, curr_row, next_row);
        }
    }
    
    // Memoize result
    computed[row][prev_row][curr_row] = true;
    memo[row][prev_row][curr_row] = count;
    return count;
}

int main() {
    // Initialize memoization table
    memset(computed, 0, sizeof(computed));
    
    uint64_t total = 0;
    
    // For the first row, try all possible configurations
    for (int first_row = 0; first_row < 256; first_row++) {
        // For the first row, there's no previous row (all 0s)
        for (int second_row = 0; second_row < 256; second_row++) {
            // Check if any 1s in first row would be isolated
            if (!has_isolated_one(0, first_row, second_row)) {
                total += count_unholy(2, first_row, second_row);
            }
        }
    }
    
    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 <stdbool.h>
#include <stdint.h>
#include <string.h>

// Cache for memoization - indexed by [row][previous row][current row]
uint64_t memo[9][256][256];
bool computed[9][256][256];

// Check if a 1 in middle_row has no neighbors in any of the adjacent rows
// or horizontally in its own row
bool has_isolated_one(int top_row, int middle_row, int bottom_row) {
    for (int i = 0; i < 8; i++) {
        if (middle_row & (1 << i)) {  // If this bit is a 1
            bool has_neighbor = false;
            
            // Check left in current row
            if (i > 0 && (middle_row & (1 << (i-1))))
                has_neighbor = true;
            
            // Check right in current row
            if (i < 7 && (middle_row & (1 << (i+1))))
                has_neighbor = true;
            
            // Check above in top row
            if (top_row & (1 << i))
                has_neighbor = true;
            
            // Check below in bottom row
            if (bottom_row & (1 << i))
                has_neighbor = true;
            
            if (!has_neighbor)
                return true;  // Found an isolated 1
        }
    }
    return false;  // No isolated 1s found
}

// Count unholy matrices starting from the given row
uint64_t count_unholy(int row, int prev_row, int curr_row) {
    // If we've filled all rows
    if (row == 8) {
        // For the last row, check if there are isolated 1s with 
        // no bottom row (bottom_row = 0)
        return has_isolated_one(prev_row, curr_row, 0) ? 0 : 1;
    }
    
    // Return memoized result if available
    if (computed[row][prev_row][curr_row])
        return memo[row][prev_row][curr_row];
    
    uint64_t count = 0;
    // Try all possible configurations for the next row
    for (int next_row = 0; next_row < 256; next_row++) {
        // Check if adding this next row creates an isolated 1 in the current row
        if (!has_isolated_one(prev_row, curr_row, next_row)) {
            count += count_unholy(row + 1, curr_row, next_row);
        }
    }
    
    // Memoize result
    computed[row][prev_row][curr_row] = true;
    memo[row][prev_row][curr_row] = count;
    return count;
}

int main() {
    // Initialize memoization table
    memset(computed, 0, sizeof(computed));
    
    uint64_t total = 0;
    
    // For the first row, try all possible configurations
    for (int first_row = 0; first_row < 256; first_row++) {
        // For the first row, there's no previous row (all 0s)
        for (int second_row = 0; second_row < 256; second_row++) {
            // Check if any 1s in first row would be isolated
            if (!has_isolated_one(0, first_row, second_row)) {
                total += count_unholy(2, first_row, second_row);
            }
        }
    }
    
    printf("Total number of unholy 8x8 matrices: %llu\n", total);
    return 0;
}
And got the output: Total number of unholy 8x8 matrices: 1121231537486377866 main.c: In function 'main': main.c:86:53: warning: format '%llu' expects argument of type 'long long unsigned int', but argument 2 has type 'uint64_t' {aka 'long unsigned int'} [-Wformat=] 86 | printf("Total number of unholy 8x8 matrices: %llu\n", total); | ~~~^ ~~~~~ | | | | | 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.

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.

Output

# Solution: Counting Unholy 8x8 Binary Matrices I'll write a C program to count all unholy 8x8 matrices - matrices where no 1 is isolated (every 1 must have at least one adjacent 1 in a cardinal direction). This is a combinatorial problem that would be infeasible to solve with brute force (2^64 possible matrices). Instead, I'll use dynamic programming with bit manipulation for efficiency. ## Approach 1. Build the matrix row by row 2. Use DP where each state represents the previous row's pattern and current row's pattern 3. For each valid state, count all possible ways to build the rest of the matrix 4. Ensure that no 1 is isolated in any row ## Implementation 

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

// dp[row][prev_row_pattern][curr_row_pattern]
long long dp[9][256][256];
bool visited[9][256][256];

// Check if a 1 at position i in mask has an adjacent 1 in the same row
bool has_adjacent_in_row(int mask, int i) {
    // Check left
    if (i > 0 && (mask & (1 << (i-1))))
        return true;
    // Check right
    if (i < 7 && (mask & (1 << (i+1))))
        return true;
    return false;
}

// Check if the given configuration is valid (no isolated 1s)
bool is_valid(int prev_row, int curr_row, int next_row) {
    for (int i = 0; i < 8; i++) {
        // Check if there's a 1 at this position in current row
        if (curr_row & (1 << i)) {
            bool has_adjacent = has_adjacent_in_row(curr_row, i);
            // Check above
            if (!has_adjacent && (prev_row & (1 << i)))
                has_adjacent = true;
            // Check below
            if (!has_adjacent && (next_row & (1 << i)))
                has_adjacent = true;
            
            // If still no adjacent 1s, this 1 is isolated
            if (!has_adjacent)
                return false;
        }
    }
    return true;
}

// Count unholy matrices recursively
long long count_unholy(int row, int prev_row, int curr_row) {
    // If we've filled all rows, we have a valid matrix
    if (row == 8)
        return 1;
    
    // If we've already calculated this state
    if (visited[row][prev_row][curr_row])
        return dp[row][prev_row][curr_row];
    
    visited[row][prev_row][curr_row] = true;
    long long result = 0;
    
    // Try all possible configurations for the next row
    for (int next_row = 0; next_row < 256; next_row++) {
        if (is_valid(prev_row, curr_row, next_row) && is_valid(curr_row, next_row, 0)) {
            result += count_unholy(row + 1, curr_row, next_row);
        }
    }
    
    dp[row][prev_row][curr_row] = result;
    return result;
}

int main() {
    // Initialize DP table
    memset(visited, 0, sizeof(visited));
    
    long long total = 0;
    
    // For the first row, previous row is all 0s
    for (int first_row = 0; first_row < 256; first_row++) {
        // For the second row
        for (int second_row = 0; second_row < 256; second_row++) {
            // Check if first and second rows form a valid configuration
            if (is_valid(0, first_row, second_row) && is_valid(first_row, second_row, 0)) {
                total += count_unholy(2, first_row, second_row);
            }
        }
    }
    
    printf("Total number of unholy 8x8 matrices: %lld\n", total);
    return 0;
}
## Explanation 1. I represent each row as an 8-bit integer (values 0-255) 2. The dynamic programming state includes: - Current row number - Pattern of the previous row - Pattern of the current row 3. The `is_valid` function checks if a configuration has no isolated 1s: - For each 1 in the current row, it checks if there's an adjacent 1 in the same row, the row above, or the row below - If any 1 is isolated, the configuration is invalid 4. The recursive function `count_unholy` builds the matrix row by row, trying all valid next-row configurations and summing the results 5. The main function initializes the process by trying all valid combinations for the first two rows This solution efficiently handles the problem by: - Using bit manipulation to represent rows compactly - Employing memoization to avoid redundant calculations - Building the matrix systematically row by row The output will give us the total count of unholy 8x8 binary matrices.

Extract Code

I extracted the following code from that output: 

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

// dp[row][prev_row_pattern][curr_row_pattern]
long long dp[9][256][256];
bool visited[9][256][256];

// Check if a 1 at position i in mask has an adjacent 1 in the same row
bool has_adjacent_in_row(int mask, int i) {
    // Check left
    if (i > 0 && (mask & (1 << (i-1))))
        return true;
    // Check right
    if (i < 7 && (mask & (1 << (i+1))))
        return true;
    return false;
}

// Check if the given configuration is valid (no isolated 1s)
bool is_valid(int prev_row, int curr_row, int next_row) {
    for (int i = 0; i < 8; i++) {
        // Check if there's a 1 at this position in current row
        if (curr_row & (1 << i)) {
            bool has_adjacent = has_adjacent_in_row(curr_row, i);
            // Check above
            if (!has_adjacent && (prev_row & (1 << i)))
                has_adjacent = true;
            // Check below
            if (!has_adjacent && (next_row & (1 << i)))
                has_adjacent = true;
            
            // If still no adjacent 1s, this 1 is isolated
            if (!has_adjacent)
                return false;
        }
    }
    return true;
}

// Count unholy matrices recursively
long long count_unholy(int row, int prev_row, int curr_row) {
    // If we've filled all rows, we have a valid matrix
    if (row == 8)
        return 1;
    
    // If we've already calculated this state
    if (visited[row][prev_row][curr_row])
        return dp[row][prev_row][curr_row];
    
    visited[row][prev_row][curr_row] = true;
    long long result = 0;
    
    // Try all possible configurations for the next row
    for (int next_row = 0; next_row < 256; next_row++) {
        if (is_valid(prev_row, curr_row, next_row) && is_valid(curr_row, next_row, 0)) {
            result += count_unholy(row + 1, curr_row, next_row);
        }
    }
    
    dp[row][prev_row][curr_row] = result;
    return result;
}

int main() {
    // Initialize DP table
    memset(visited, 0, sizeof(visited));
    
    long long total = 0;
    
    // For the first row, previous row is all 0s
    for (int first_row = 0; first_row < 256; first_row++) {
        // For the second row
        for (int second_row = 0; second_row < 256; second_row++) {
            // Check if first and second rows form a valid configuration
            if (is_valid(0, first_row, second_row) && is_valid(first_row, second_row, 0)) {
                total += count_unholy(2, first_row, second_row);
            }
        }
    }
    
    printf("Total number of unholy 8x8 matrices: %lld\n", total);
    return 0;
}

Run Code Interpreter

Running the following program: 

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

// dp[row][prev_row_pattern][curr_row_pattern]
long long dp[9][256][256];
bool visited[9][256][256];

// Check if a 1 at position i in mask has an adjacent 1 in the same row
bool has_adjacent_in_row(int mask, int i) {
    // Check left
    if (i > 0 && (mask & (1 << (i-1))))
        return true;
    // Check right
    if (i < 7 && (mask & (1 << (i+1))))
        return true;
    return false;
}

// Check if the given configuration is valid (no isolated 1s)
bool is_valid(int prev_row, int curr_row, int next_row) {
    for (int i = 0; i < 8; i++) {
        // Check if there's a 1 at this position in current row
        if (curr_row & (1 << i)) {
            bool has_adjacent = has_adjacent_in_row(curr_row, i);
            // Check above
            if (!has_adjacent && (prev_row & (1 << i)))
                has_adjacent = true;
            // Check below
            if (!has_adjacent && (next_row & (1 << i)))
                has_adjacent = true;
            
            // If still no adjacent 1s, this 1 is isolated
            if (!has_adjacent)
                return false;
        }
    }
    return true;
}

// Count unholy matrices recursively
long long count_unholy(int row, int prev_row, int curr_row) {
    // If we've filled all rows, we have a valid matrix
    if (row == 8)
        return 1;
    
    // If we've already calculated this state
    if (visited[row][prev_row][curr_row])
        return dp[row][prev_row][curr_row];
    
    visited[row][prev_row][curr_row] = true;
    long long result = 0;
    
    // Try all possible configurations for the next row
    for (int next_row = 0; next_row < 256; next_row++) {
        if (is_valid(prev_row, curr_row, next_row) && is_valid(curr_row, next_row, 0)) {
            result += count_unholy(row + 1, curr_row, next_row);
        }
    }
    
    dp[row][prev_row][curr_row] = result;
    return result;
}

int main() {
    // Initialize DP table
    memset(visited, 0, sizeof(visited));
    
    long long total = 0;
    
    // For the first row, previous row is all 0s
    for (int first_row = 0; first_row < 256; first_row++) {
        // For the second row
        for (int second_row = 0; second_row < 256; second_row++) {
            // Check if first and second rows form a valid configuration
            if (is_valid(0, first_row, second_row) && is_valid(first_row, second_row, 0)) {
                total += count_unholy(2, first_row, second_row);
            }
        }
    }
    
    printf("Total number of unholy 8x8 matrices: %lld\n", total);
    return 0;
}
And got the output: Total number of unholy 8x8 matrices: 132797167652221230

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False