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

Output

## Plan for Counting Unholy Matrices **1. Data Structures:** - **Matrix Representation:** Use `unsigned long` (64 bits) to represent an 8x8 binary matrix. Each bit represents a cell in the matrix. - **Dynamic Programming Table:** Use a 2D array `dp[9][1 << 16]`. - `dp[i][j]` stores the count of unholy matrices with `i` rows, where the last two rows are represented by the integer `j`. **2. Functions:** - **`is_unholy(unsigned long matrix)`:** - Takes a matrix represented as an `unsigned long`. - Iterates through each bit (cell) of the matrix. - For each '1' bit, checks if there's another '1' in its four cardinal directions using bitwise operations. - Returns `1` if the matrix is unholy (no isolated '1's), otherwise `0`. - **`count_unholy(int row, int prev_rows)`:** - Base case: If `row == 8`, return `is_unholy(prev_rows)`. - If `dp[row][prev_rows]` is already calculated, return the stored value. - Initialize `count = 0`. - Iterate through all possible combinations of the next two rows (0 to (1 << 16) - 1). - For each combination `next_rows`: - Construct the complete matrix using `prev_rows` and `next_rows`. - Recursively call `count_unholy(row + 2, next_rows)` and add the result to `count`. - Store `count` in `dp[row][prev_rows]`. - Return `count`. **3. Main Function:** - Initialize the `dp` table with -1. - Call `count_unholy(0, 0)` to start the recursion from an empty matrix. - Print the returned value as the total count of unholy matrices. ## C Implementation

#include <stdio.h>

int dp[9][1 << 16];

// Check if a matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int i = 0; i < 8; ++i) {
        for (int j = 0; j < 8; ++j) {
            if ((matrix >> (i * 8 + j)) & 1) { // Check if the cell is '1'
                // Check cardinal neighbors
                int hasNeighbor = (i > 0 && (matrix >> ((i - 1) * 8 + j)) & 1) ||
                                  (i < 7 && (matrix >> ((i + 1) * 8 + j)) & 1) ||
                                  (j > 0 && (matrix >> (i * 8 + j - 1)) & 1) ||
                                  (j < 7 && (matrix >> (i * 8 + j + 1)) & 1);
                if (!hasNeighbor) return 0; // Isolated '1' found
            }
        }
    }
    return 1; // No isolated '1's
}

// Recursively count unholy matrices
long long count_unholy(int row, int prev_rows) {
    if (row == 8) return is_unholy(prev_rows);
    if (dp[row][prev_rows] != -1) return dp[row][prev_rows];

    long long count = 0;
    for (int next_rows = 0; next_rows < (1 << 16); ++next_rows) {
        // Combine previous and next rows to form the matrix
        unsigned long matrix = ((unsigned long)prev_rows << 16) | next_rows;
        count += count_unholy(row + 2, next_rows);
    }

    dp[row][prev_rows] = count;
    return count;
}

int main() {
    for (int i = 0; i < 9; ++i) {
        for (int j = 0; j < (1 << 16); ++j) {
            dp[i][j] = -1;
        }
    }

    long long result = count_unholy(0, 0);
    printf("Total unholy 8x8 matrices: %lld\n", result);
    return 0;
}

This program efficiently calculates the total number of unholy 8x8 matrices using bit manipulation and dynamic programming to avoid redundant calculations.

Extract Code

I extracted the following code from that output:

#include <stdio.h>

int dp[9][1 << 16];

// Check if a matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int i = 0; i < 8; ++i) {
        for (int j = 0; j < 8; ++j) {
            if ((matrix >> (i * 8 + j)) & 1) { // Check if the cell is '1'
                // Check cardinal neighbors
                int hasNeighbor = (i > 0 && (matrix >> ((i - 1) * 8 + j)) & 1) ||
                                  (i < 7 && (matrix >> ((i + 1) * 8 + j)) & 1) ||
                                  (j > 0 && (matrix >> (i * 8 + j - 1)) & 1) ||
                                  (j < 7 && (matrix >> (i * 8 + j + 1)) & 1);
                if (!hasNeighbor) return 0; // Isolated '1' found
            }
        }
    }
    return 1; // No isolated '1's
}

// Recursively count unholy matrices
long long count_unholy(int row, int prev_rows) {
    if (row == 8) return is_unholy(prev_rows);
    if (dp[row][prev_rows] != -1) return dp[row][prev_rows];

    long long count = 0;
    for (int next_rows = 0; next_rows < (1 << 16); ++next_rows) {
        // Combine previous and next rows to form the matrix
        unsigned long matrix = ((unsigned long)prev_rows << 16) | next_rows;
        count += count_unholy(row + 2, next_rows);
    }

    dp[row][prev_rows] = count;
    return count;
}

int main() {
    for (int i = 0; i < 9; ++i) {
        for (int j = 0; j < (1 << 16); ++j) {
            dp[i][j] = -1;
        }
    }

    long long result = count_unholy(0, 0);
    printf("Total unholy 8x8 matrices: %lld\n", result);
    return 0;
}

Run Code Interpreter

Running the following program:

#include <stdio.h>

int dp[9][1 << 16];

// Check if a matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int i = 0; i < 8; ++i) {
        for (int j = 0; j < 8; ++j) {
            if ((matrix >> (i * 8 + j)) & 1) { // Check if the cell is '1'
                // Check cardinal neighbors
                int hasNeighbor = (i > 0 && (matrix >> ((i - 1) * 8 + j)) & 1) ||
                                  (i < 7 && (matrix >> ((i + 1) * 8 + j)) & 1) ||
                                  (j > 0 && (matrix >> (i * 8 + j - 1)) & 1) ||
                                  (j < 7 && (matrix >> (i * 8 + j + 1)) & 1);
                if (!hasNeighbor) return 0; // Isolated '1' found
            }
        }
    }
    return 1; // No isolated '1's
}

// Recursively count unholy matrices
long long count_unholy(int row, int prev_rows) {
    if (row == 8) return is_unholy(prev_rows);
    if (dp[row][prev_rows] != -1) return dp[row][prev_rows];

    long long count = 0;
    for (int next_rows = 0; next_rows < (1 << 16); ++next_rows) {
        // Combine previous and next rows to form the matrix
        unsigned long matrix = ((unsigned long)prev_rows << 16) | next_rows;
        count += count_unholy(row + 2, next_rows);
    }

    dp[row][prev_rows] = count;
    return count;
}

int main() {
    for (int i = 0; i < 9; ++i) {
        for (int j = 0; j < (1 << 16); ++j) {
            dp[i][j] = -1;
        }
    }

    long long result = count_unholy(0, 0);
    printf("Total unholy 8x8 matrices: %lld\n", result);
    return 0;
}

And got the output: Timeout: function took too long to complete

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False

Initial Query

LLM Generation

Query

Output

## Plan for Counting Unholy Matrices Here's a breakdown of the plan to count unholy 8x8 binary matrices in C: **1. Data Representation:** - **Matrix Representation:** We'll represent an 8x8 binary matrix using an `unsigned long long` (64 bits), where each bit corresponds to a cell in the matrix. - **Row Representation:** We can extract a specific row from the matrix using bit shifting and masking. **2. Function `is_unholy`:** - **Input:** An `unsigned long long` representing the matrix. - **Logic:** - Iterate through each row of the matrix (using bit manipulation). - For each row, iterate through each bit (representing a cell). - If a bit is 1, check its four cardinal neighbors (up, down, left, right) within the matrix boundaries. - If any neighbor is also 1, the matrix might still be unholy, continue checking. - If all neighbors are 0, the matrix is holy (return 0). - **Output:** Return 1 if the matrix is unholy (no isolated 1s), 0 otherwise. **3. Dynamic Programming:** - **State:** `dp[i][prev_row]` represents the count of unholy matrices with `i` rows, where the `(i-1)`-th row is `prev_row`. - **Base Case:** `dp[0][0] = 1` (An empty matrix is considered unholy). - **Recursive Relation:** - Iterate over possible current rows (`curr_row` from 0 to 2^8 - 1). - For each `curr_row`, check if it forms an unholy combination with `prev_row` (using `is_unholy` on a combined temporary matrix). - If unholy, update `dp[i][curr_row]` by adding `dp[i - 1][prev_row]`. - **Final Result:** `dp[8][any_row]` (The count is the same for any final row since the last two rows ensure isolation). **4. Main Function:** - Initialize the `dp` table. - Calculate the unholy matrix count using the dynamic programming approach. - Print the final count. ## C Implementation

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

// Check if a matrix is unholy (no isolated 1s)
int is_unholy(uint64_t matrix) {
    for (int i = 0; i < 8; i++) {
        uint64_t row = (matrix >> (i * 8)) & 0xFF;
        for (int j = 0; j < 8; j++) {
            if ((row & (1ULL << j)) == 0) continue; // Skip if cell is 0

            // Check neighbors (within matrix bounds)
            int has_neighbor = (i > 0 && (matrix >> ((i - 1) * 8) & (1ULL << j))) || // Up
                               (i < 7 && (matrix >> ((i + 1) * 8) & (1ULL << j))) || // Down
                               (j > 0 && (row & (1ULL << (j - 1)))) || // Left
                               (j < 7 && (row & (1ULL << (j + 1)))); // Right

            if (!has_neighbor) return 0; // Isolated 1 found
        }
    }
    return 1; // No isolated 1s found
}

int main() {
    uint64_t dp[9][256] = {0}; // dp table

    dp[0][0] = 1; // Base case

    // Dynamic programming
    for (int i = 1; i <= 8; i++) {
        for (uint64_t prev_row = 0; prev_row < 256; prev_row++) {
            for (uint64_t curr_row = 0; curr_row < 256; curr_row++) {
                // Combine two rows to form a temporary matrix
                uint64_t temp_matrix = (prev_row << 8) | curr_row;

                // Check if unholy
                if (is_unholy(temp_matrix)) {
                    dp[i][curr_row] += dp[i - 1][prev_row];
                }
            }
        }
    }

    uint64_t total_unholy = 0;
    for (int i = 0; i < 256; i++) {
        total_unholy += dp[8][i];
    }

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

    return 0;
}

This C program implements the plan described above to efficiently count the number of unholy 8x8 binary matrices using bit manipulation and dynamic programming.

Extract Code

I extracted the following code from that output:

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

// Check if a matrix is unholy (no isolated 1s)
int is_unholy(uint64_t matrix) {
    for (int i = 0; i < 8; i++) {
        uint64_t row = (matrix >> (i * 8)) & 0xFF;
        for (int j = 0; j < 8; j++) {
            if ((row & (1ULL << j)) == 0) continue; // Skip if cell is 0

            // Check neighbors (within matrix bounds)
            int has_neighbor = (i > 0 && (matrix >> ((i - 1) * 8) & (1ULL << j))) || // Up
                               (i < 7 && (matrix >> ((i + 1) * 8) & (1ULL << j))) || // Down
                               (j > 0 && (row & (1ULL << (j - 1)))) || // Left
                               (j < 7 && (row & (1ULL << (j + 1)))); // Right

            if (!has_neighbor) return 0; // Isolated 1 found
        }
    }
    return 1; // No isolated 1s found
}

int main() {
    uint64_t dp[9][256] = {0}; // dp table

    dp[0][0] = 1; // Base case

    // Dynamic programming
    for (int i = 1; i <= 8; i++) {
        for (uint64_t prev_row = 0; prev_row < 256; prev_row++) {
            for (uint64_t curr_row = 0; curr_row < 256; curr_row++) {
                // Combine two rows to form a temporary matrix
                uint64_t temp_matrix = (prev_row << 8) | curr_row;

                // Check if unholy
                if (is_unholy(temp_matrix)) {
                    dp[i][curr_row] += dp[i - 1][prev_row];
                }
            }
        }
    }

    uint64_t total_unholy = 0;
    for (int i = 0; i < 256; i++) {
        total_unholy += dp[8][i];
    }

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

    return 0;
}

Run Code Interpreter

Running the following program:

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

// Check if a matrix is unholy (no isolated 1s)
int is_unholy(uint64_t matrix) {
    for (int i = 0; i < 8; i++) {
        uint64_t row = (matrix >> (i * 8)) & 0xFF;
        for (int j = 0; j < 8; j++) {
            if ((row & (1ULL << j)) == 0) continue; // Skip if cell is 0

            // Check neighbors (within matrix bounds)
            int has_neighbor = (i > 0 && (matrix >> ((i - 1) * 8) & (1ULL << j))) || // Up
                               (i < 7 && (matrix >> ((i + 1) * 8) & (1ULL << j))) || // Down
                               (j > 0 && (row & (1ULL << (j - 1)))) || // Left
                               (j < 7 && (row & (1ULL << (j + 1)))); // Right

            if (!has_neighbor) return 0; // Isolated 1 found
        }
    }
    return 1; // No isolated 1s found
}

int main() {
    uint64_t dp[9][256] = {0}; // dp table

    dp[0][0] = 1; // Base case

    // Dynamic programming
    for (int i = 1; i <= 8; i++) {
        for (uint64_t prev_row = 0; prev_row < 256; prev_row++) {
            for (uint64_t curr_row = 0; curr_row < 256; curr_row++) {
                // Combine two rows to form a temporary matrix
                uint64_t temp_matrix = (prev_row << 8) | curr_row;

                // Check if unholy
                if (is_unholy(temp_matrix)) {
                    dp[i][curr_row] += dp[i - 1][prev_row];
                }
            }
        }
    }

    uint64_t total_unholy = 0;
    for (int i = 0; i < 256; i++) {
        total_unholy += dp[8][i];
    }

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

    return 0;
}

And got the output:

Total unholy 8x8 matrices: 12074480011936412
main.c: In function 'main':
main.c:49:43: warning: format '%llu' expects argument of type 'long long unsigned int', but argument 2 has type 'uint64_t' {aka 'long unsigned int'} [-Wformat=]
   49 |     printf("Total unholy 8x8 matrices: %llu\n", total_unholy);
      |                                        ~~~^     ~~~~~~~~~~~~
      |                                           |     |
      |                                           |     uint64_t {aka long unsigned int}
      |                                           long long unsigned int
      |                                        %lu

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False

Initial Query

LLM Generation

Query

Output

## Plan: **1. Data Representation:** * Use an `unsigned long` to represent an 8x8 binary matrix. Each bit will correspond to a cell in the matrix. * Define constants for bit manipulation: * `ROW_MASK`: Mask to extract a single row from the matrix. * `ISOLATED_1_MASK`: Mask to check for isolated '1's in a 3x3 neighborhood. **2. Function `is_unholy`:** * Input: An `unsigned long` representing the matrix. * Logic: * Iterate through each row of the matrix (except the first and last). * For each row, iterate through each bit (except the first and last). * Use bitwise operations to: * Extract a 3x3 neighborhood around the current bit. * Check if the current bit is '1' and if all its neighbors are '0' using `ISOLATED_1_MASK`. * If an isolated '1' is found, return false. * If no isolated '1's are found, return true. **3. Function `count_unholy_matrices`:** * Input: `rows` (number of rows in the matrix), `matrix` (current matrix). * Logic: * Base Case: If `rows == 8`, check if the matrix is unholy using `is_unholy`. If yes, return 1; otherwise, return 0. * Recursive Step: * Initialize a counter `count` to 0. * Iterate through all possible combinations of the next two rows (0 to 2^16 - 1). * Construct a new matrix by appending the two rows to the current matrix. * Recursively call `count_unholy_matrices` with `rows + 2` and the new matrix. * Add the result to the `count`. * Return `count`. **4. Main Function:** * Call `count_unholy_matrices` with initial values (rows = 0, matrix = 0). * Print the returned count. ## C Implementation:

#include <stdio.h>

#define ROW_MASK 0xFFUL
#define ISOLATED_1_MASK 0x18244281UL

// Function to check if a matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int i = 1; i < 7; i++) {
        unsigned long row = (matrix >> (i * 8)) & ROW_MASK;
        for (int j = 1; j < 7; j++) {
            if ((row & (1UL << j)) && 
                (((matrix >> ((i - 1) * 8)) & ISOLATED_1_MASK) == (1UL << j)) &&
                (((matrix >> (i * 8)) & ISOLATED_1_MASK) == (1UL << j)) &&
                (((matrix >> ((i + 1) * 8)) & ISOLATED_1_MASK) == (1UL << j))) {
                return 0; // Found an isolated '1'
            }
        }
    }
    return 1; // No isolated '1's found
}

// Function to count unholy matrices using dynamic programming
unsigned long count_unholy_matrices(int rows, unsigned long matrix) {
    if (rows == 8) {
        return is_unholy(matrix);
    }

    unsigned long count = 0;
    for (unsigned int next_two_rows = 0; next_two_rows < (1UL << 16); next_two_rows++) {
        unsigned long new_matrix = matrix | ((next_two_rows & 0xFFFFUL) << (rows * 8));
        count += count_unholy_matrices(rows + 2, new_matrix);
    }
    return count;
}

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

**Explanation:** 1. **`is_unholy` Function:** * This function checks if a given matrix is unholy. * It iterates through each cell (except edges) and uses bitwise operations to check for isolated '1's. * `ISOLATED_1_MASK` helps identify if a '1' has any neighboring '1's within its 3x3 neighborhood. 2. **`count_unholy_matrices` Function:** * This function implements the dynamic programming approach. * It recursively explores all possible matrices by adding two rows at a time. * For each complete matrix (8 rows), it calls `is_unholy` to check its validity. 3. **`main` Function:** * Initializes the count and calls `count_unholy_matrices` to get the final result. * Prints the total count of unholy matrices. This program efficiently calculates the number of unholy 8x8 matrices using bit manipulation and dynamic programming.

Extract Code

I extracted the following code from that output:

#include <stdio.h>

#define ROW_MASK 0xFFUL
#define ISOLATED_1_MASK 0x18244281UL

// Function to check if a matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int i = 1; i < 7; i++) {
        unsigned long row = (matrix >> (i * 8)) & ROW_MASK;
        for (int j = 1; j < 7; j++) {
            if ((row & (1UL << j)) && 
                (((matrix >> ((i - 1) * 8)) & ISOLATED_1_MASK) == (1UL << j)) &&
                (((matrix >> (i * 8)) & ISOLATED_1_MASK) == (1UL << j)) &&
                (((matrix >> ((i + 1) * 8)) & ISOLATED_1_MASK) == (1UL << j))) {
                return 0; // Found an isolated '1'
            }
        }
    }
    return 1; // No isolated '1's found
}

// Function to count unholy matrices using dynamic programming
unsigned long count_unholy_matrices(int rows, unsigned long matrix) {
    if (rows == 8) {
        return is_unholy(matrix);
    }

    unsigned long count = 0;
    for (unsigned int next_two_rows = 0; next_two_rows < (1UL << 16); next_two_rows++) {
        unsigned long new_matrix = matrix | ((next_two_rows & 0xFFFFUL) << (rows * 8));
        count += count_unholy_matrices(rows + 2, new_matrix);
    }
    return count;
}

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

Run Code Interpreter

Running the following program:

#include <stdio.h>

#define ROW_MASK 0xFFUL
#define ISOLATED_1_MASK 0x18244281UL

// Function to check if a matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int i = 1; i < 7; i++) {
        unsigned long row = (matrix >> (i * 8)) & ROW_MASK;
        for (int j = 1; j < 7; j++) {
            if ((row & (1UL << j)) && 
                (((matrix >> ((i - 1) * 8)) & ISOLATED_1_MASK) == (1UL << j)) &&
                (((matrix >> (i * 8)) & ISOLATED_1_MASK) == (1UL << j)) &&
                (((matrix >> ((i + 1) * 8)) & ISOLATED_1_MASK) == (1UL << j))) {
                return 0; // Found an isolated '1'
            }
        }
    }
    return 1; // No isolated '1's found
}

// Function to count unholy matrices using dynamic programming
unsigned long count_unholy_matrices(int rows, unsigned long matrix) {
    if (rows == 8) {
        return is_unholy(matrix);
    }

    unsigned long count = 0;
    for (unsigned int next_two_rows = 0; next_two_rows < (1UL << 16); next_two_rows++) {
        unsigned long new_matrix = matrix | ((next_two_rows & 0xFFFFUL) << (rows * 8));
        count += count_unholy_matrices(rows + 2, new_matrix);
    }
    return count;
}

int main() {
    unsigned long count = count_unholy_matrices(0, 0);
    printf("Total number of unholy 8x8 matrices: %lu\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

Initial Query

LLM Generation

Query

Output

## Plan for C program to count unholy 8x8 binary matrices: **1. Data Representation:** - Use `unsigned long long` (64 bits) to represent an 8x8 matrix. Each bit represents a cell in the matrix. - Define macros for easier bit manipulation: - `SET_BIT(matrix, row, col)`: Sets the bit at (row, col) to 1. - `GET_BIT(matrix, row, col)`: Returns the bit value at (row, col). **2. Function: `is_unholy(unsigned long long matrix)`** - Iterate through each cell of the matrix (excluding the border). - For each cell: - If the cell is 1: - Check if any of the 4 neighbors (up, down, left, right) are also 1. - If no neighbors are 1, return 0 (not unholy). - If all 1s have at least one neighboring 1, return 1 (unholy). **3. Function: `count_unholy(int rows, unsigned long long prev_row, unsigned long long prev_prev_row)`** - **Base case:** If `rows == 8`, return `is_unholy(prev_row)`. - **Recursive step:** - Initialize a counter `count` to 0. - Iterate through all possible combinations for the next two rows (2^16 total): - Construct the current two rows using bit manipulation. - Check for conflicts with previous rows to ensure no isolated 1s are created at the boundary. - If no conflicts, recursively call `count_unholy` with `rows + 2`, the newly constructed second row as `prev_row`, and the first new row as `prev_prev_row`. - Add the result of the recursive call to `count`. - Return `count`. **4. Main Function:** - Initialize a variable `total_unholy` to store the result. - Call `count_unholy(0, 0, 0)` to start the dynamic programming recursion. - Print the value of `total_unholy`. ## C Implementation:

#include <stdio.h>

#define SET_BIT(matrix, row, col) (matrix |= (1ULL << ((row) * 8 + (col))))
#define GET_BIT(matrix, row, col) ((matrix >> ((row) * 8 + (col))) & 1ULL)

// Check if a matrix is unholy
int is_unholy(unsigned long long matrix) {
  for (int i = 1; i < 7; i++) {
    for (int j = 1; j < 7; j++) {
      if (GET_BIT(matrix, i, j)) {
        if (!(GET_BIT(matrix, i - 1, j) || GET_BIT(matrix, i + 1, j) ||
              GET_BIT(matrix, i, j - 1) || GET_BIT(matrix, i, j + 1))) {
          return 0; // Found an isolated 1
        }
      }
    }
  }
  return 1; // No isolated 1s found
}

// Dynamic programming function to count unholy matrices
unsigned long long count_unholy(int rows, unsigned long long prev_row,
                               unsigned long long prev_prev_row) {
  if (rows == 8) {
    return is_unholy(prev_row);
  }
  unsigned long long count = 0;
  for (int i = 0; i < 65536; i++) { // Iterate through all 2^16 combinations
    unsigned long long row1 = i;
    unsigned long long row2 = (i >> 8) | ((i & 0xFF) << 8); // Rotate for next row

    // Check for conflicts at the boundary with previous rows
    if (rows > 0 && ((row1 & prev_row) != row1) ||
        (rows > 1 && ((row1 & prev_prev_row) != row1))) {
      continue; // Skip invalid combinations
    }

    count += count_unholy(rows + 2, row2, row1);
  }
  return count;
}

int main() {
  unsigned long long total_unholy = count_unholy(0, 0, 0);
  printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy);
  return 0;
}

This program utilizes bit manipulation and dynamic programming to efficiently count the total number of unholy 8x8 binary matrices. The `is_unholy` function checks for isolated 1s, while the `count_unholy` function recursively explores all valid matrix configurations to count the unholy ones.

Extract Code

I extracted the following code from that output:

#include <stdio.h>

#define SET_BIT(matrix, row, col) (matrix |= (1ULL << ((row) * 8 + (col))))
#define GET_BIT(matrix, row, col) ((matrix >> ((row) * 8 + (col))) & 1ULL)

// Check if a matrix is unholy
int is_unholy(unsigned long long matrix) {
  for (int i = 1; i < 7; i++) {
    for (int j = 1; j < 7; j++) {
      if (GET_BIT(matrix, i, j)) {
        if (!(GET_BIT(matrix, i - 1, j) || GET_BIT(matrix, i + 1, j) ||
              GET_BIT(matrix, i, j - 1) || GET_BIT(matrix, i, j + 1))) {
          return 0; // Found an isolated 1
        }
      }
    }
  }
  return 1; // No isolated 1s found
}

// Dynamic programming function to count unholy matrices
unsigned long long count_unholy(int rows, unsigned long long prev_row,
                               unsigned long long prev_prev_row) {
  if (rows == 8) {
    return is_unholy(prev_row);
  }
  unsigned long long count = 0;
  for (int i = 0; i < 65536; i++) { // Iterate through all 2^16 combinations
    unsigned long long row1 = i;
    unsigned long long row2 = (i >> 8) | ((i & 0xFF) << 8); // Rotate for next row

    // Check for conflicts at the boundary with previous rows
    if (rows > 0 && ((row1 & prev_row) != row1) ||
        (rows > 1 && ((row1 & prev_prev_row) != row1))) {
      continue; // Skip invalid combinations
    }

    count += count_unholy(rows + 2, row2, row1);
  }
  return count;
}

int main() {
  unsigned long long total_unholy = count_unholy(0, 0, 0);
  printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy);
  return 0;
}

Run Code Interpreter

Running the following program:

#include <stdio.h>

#define SET_BIT(matrix, row, col) (matrix |= (1ULL << ((row) * 8 + (col))))
#define GET_BIT(matrix, row, col) ((matrix >> ((row) * 8 + (col))) & 1ULL)

// Check if a matrix is unholy
int is_unholy(unsigned long long matrix) {
  for (int i = 1; i < 7; i++) {
    for (int j = 1; j < 7; j++) {
      if (GET_BIT(matrix, i, j)) {
        if (!(GET_BIT(matrix, i - 1, j) || GET_BIT(matrix, i + 1, j) ||
              GET_BIT(matrix, i, j - 1) || GET_BIT(matrix, i, j + 1))) {
          return 0; // Found an isolated 1
        }
      }
    }
  }
  return 1; // No isolated 1s found
}

// Dynamic programming function to count unholy matrices
unsigned long long count_unholy(int rows, unsigned long long prev_row,
                               unsigned long long prev_prev_row) {
  if (rows == 8) {
    return is_unholy(prev_row);
  }
  unsigned long long count = 0;
  for (int i = 0; i < 65536; i++) { // Iterate through all 2^16 combinations
    unsigned long long row1 = i;
    unsigned long long row2 = (i >> 8) | ((i & 0xFF) << 8); // Rotate for next row

    // Check for conflicts at the boundary with previous rows
    if (rows > 0 && ((row1 & prev_row) != row1) ||
        (rows > 1 && ((row1 & prev_prev_row) != row1))) {
      continue; // Skip invalid combinations
    }

    count += count_unholy(rows + 2, row2, row1);
  }
  return count;
}

int main() {
  unsigned long long total_unholy = count_unholy(0, 0, 0);
  printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy);
  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

LLM Generation

Query

Output

## Plan for the C program: **1. Data Structures and Functions:** - **`typedef unsigned long long uint64;`**: Define `uint64` for better readability, as we'll use it to represent our 8x8 matrix. - **`int is_unholy(uint64 matrix)`**: This function takes a 64-bit integer representing the matrix and returns 1 if it's unholy, 0 otherwise. It uses bit manipulation to check for isolated 1s. - **`uint64 count_unholy(int rows, uint64 prev_row, uint64 prev_prev_row)`**: This recursive function takes the number of rows processed so far (`rows`), the previous row (`prev_row`), and the row before the previous row (`prev_prev_row`) as input. It returns the count of unholy matrices possible with the given configuration. **2. `is_unholy(uint64 matrix)` Function:** - Iterate through each bit of the `matrix` (representing a cell in the 8x8 grid). - For each '1' bit, check if it has at least one adjacent '1' in the north, south, east, or west directions using bitwise operations: - **North:** `(matrix >> 8) & (1ULL << i)` - **South:** `(matrix << 8) & (1ULL << i)` - **East:** `(matrix >> 1) & (1ULL << i)` - **West:** `(matrix << 1) & (1ULL << i)` - If any '1' bit has no adjacent '1's, the matrix is not unholy. Return 0. - If all '1' bits have at least one adjacent '1', the matrix is unholy. Return 1. **3. `count_unholy(int rows, uint64 prev_row, uint64 prev_prev_row)` Function:** - **Base Case:** If `rows == 8`, we have processed all 8 rows. - Check if the last two rows (`prev_row`, `prev_prev_row`) form an unholy combination using `is_unholy()`. - Return 1 if unholy, 0 otherwise. - **Recursive Step:** - Iterate through all possible combinations of the next two rows (0 to 2^16 - 1). - For each combination: - Construct the current two rows (`curr_row`, `prev_row`). - Check if this combination is valid (i.e., no isolated 1s are created with the previous rows). - If valid, recursively call `count_unholy(rows + 2, curr_row, prev_row)` and add the result to the count. - Return the total count. **4. Main Function:** - Initialize a variable `total_count` to store the count of unholy matrices. - Call the recursive function `count_unholy(0, 0, 0)` to start the counting process. - Print the `total_count`. ## C Implementation:

#include <stdio.h>

typedef unsigned long long uint64;

// Checks if a matrix is unholy (no isolated 1s)
int is_unholy(uint64 matrix) {
    for (int i = 0; i < 64; i++) {
        if ((matrix >> i) & 1ULL) {  // If the bit is 1
            if (!(((matrix >> 8) & (1ULL << i)) |  // North
                  ((matrix << 8) & (1ULL << i)) |  // South
                  ((matrix >> 1) & (1ULL << i)) |  // East
                  ((matrix << 1) & (1ULL << i))))   // West
                return 0;  // Isolated 1 found
        }
    }
    return 1;  // No isolated 1s found
}

// Recursively counts unholy matrices
uint64 count_unholy(int rows, uint64 prev_row, uint64 prev_prev_row) {
    if (rows == 8) {
        return is_unholy((prev_prev_row << 8) | prev_row);
    }
    
    uint64 count = 0;
    for (uint64 next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        uint64 curr_row = next_rows >> 8;
        uint64 next_next_row = next_rows & 0xFF;

        // Check if the current combination is valid
        if (is_unholy((prev_prev_row << 16) | (prev_row << 8) | curr_row)) {
            count += count_unholy(rows + 2, curr_row, next_next_row);
        }
    }
    return count;
}

int main() {
    uint64 total_count = count_unholy(0, 0, 0);
    printf("Total number of unholy 8x8 matrices: %llu\n", total_count);
    return 0;
}

This program defines the required functions and uses dynamic programming with bit manipulation to efficiently calculate the total number of unholy 8x8 matrices.

Extract Code

I extracted the following code from that output:

#include <stdio.h>

typedef unsigned long long uint64;

// Checks if a matrix is unholy (no isolated 1s)
int is_unholy(uint64 matrix) {
    for (int i = 0; i < 64; i++) {
        if ((matrix >> i) & 1ULL) {  // If the bit is 1
            if (!(((matrix >> 8) & (1ULL << i)) |  // North
                  ((matrix << 8) & (1ULL << i)) |  // South
                  ((matrix >> 1) & (1ULL << i)) |  // East
                  ((matrix << 1) & (1ULL << i))))   // West
                return 0;  // Isolated 1 found
        }
    }
    return 1;  // No isolated 1s found
}

// Recursively counts unholy matrices
uint64 count_unholy(int rows, uint64 prev_row, uint64 prev_prev_row) {
    if (rows == 8) {
        return is_unholy((prev_prev_row << 8) | prev_row);
    }
    
    uint64 count = 0;
    for (uint64 next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        uint64 curr_row = next_rows >> 8;
        uint64 next_next_row = next_rows & 0xFF;

        // Check if the current combination is valid
        if (is_unholy((prev_prev_row << 16) | (prev_row << 8) | curr_row)) {
            count += count_unholy(rows + 2, curr_row, next_next_row);
        }
    }
    return count;
}

int main() {
    uint64 total_count = count_unholy(0, 0, 0);
    printf("Total number of unholy 8x8 matrices: %llu\n", total_count);
    return 0;
}

Run Code Interpreter

Running the following program:

#include <stdio.h>

typedef unsigned long long uint64;

// Checks if a matrix is unholy (no isolated 1s)
int is_unholy(uint64 matrix) {
    for (int i = 0; i < 64; i++) {
        if ((matrix >> i) & 1ULL) {  // If the bit is 1
            if (!(((matrix >> 8) & (1ULL << i)) |  // North
                  ((matrix << 8) & (1ULL << i)) |  // South
                  ((matrix >> 1) & (1ULL << i)) |  // East
                  ((matrix << 1) & (1ULL << i))))   // West
                return 0;  // Isolated 1 found
        }
    }
    return 1;  // No isolated 1s found
}

// Recursively counts unholy matrices
uint64 count_unholy(int rows, uint64 prev_row, uint64 prev_prev_row) {
    if (rows == 8) {
        return is_unholy((prev_prev_row << 8) | prev_row);
    }
    
    uint64 count = 0;
    for (uint64 next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        uint64 curr_row = next_rows >> 8;
        uint64 next_next_row = next_rows & 0xFF;

        // Check if the current combination is valid
        if (is_unholy((prev_prev_row << 16) | (prev_row << 8) | curr_row)) {
            count += count_unholy(rows + 2, curr_row, next_next_row);
        }
    }
    return count;
}

int main() {
    uint64 total_count = count_unholy(0, 0, 0);
    printf("Total number of unholy 8x8 matrices: %llu\n", total_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