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

Certainly! Let's break down the plan for the program step by step and then implement it in C. ### Plan 1. **Matrix Representation**: - Use an `unsigned long` to represent the matrix. Each bit of the `unsigned long` will represent a cell in the 8x8 matrix. - The 8x8 matrix has 64 cells, so we need 64 bits. Since an `unsigned long` typically has at least 64 bits, it’s suitable for this purpose. 2. **Function to Check if Matrix is Unholy**: - Implement a function `is_unholy` that takes an `unsigned long` matrix and returns whether it is unholy. - Use bit-shifting and masking to check all the neighbors of each 1 in the matrix. 3. **Dynamic Programming Approach**: - Use dynamic programming to count the number of unholy matrices. - Recurse on the number of rows the matrix has. - Enumerate all possible configurations for the next two rows (2^16 possibilities) and check if they form an unholy matrix when combined with the already verified rows. ### Implementation in C 

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

#define MATRIX_SIZE 64
#define ROW_SIZE 8
#define TOTAL_ROWS 8

// Function prototypes
bool is_unholy(unsigned long matrix);
unsigned long mask_8x8(int row, int col);
unsigned long shift_left(unsigned long matrix);
unsigned long shift_right(unsigned long matrix);
unsigned long shift_up(unsigned long matrix);
unsigned long shift_down(unsigned long matrix);
unsigned long count_unholy_matrices(int row, unsigned long prev_row);

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

bool is_unholy(unsigned long matrix) {
    unsigned long left = shift_left(matrix);
    unsigned long right = shift_right(matrix);
    unsigned long up = shift_up(matrix);
    unsigned long down = shift_down(matrix);

    unsigned long neighbor_mask = left | right | up | down;

    // Check if there are any isolated 1s
    if ((matrix & ~neighbor_mask) != 0) {
        return false;
    }

    return true;
}

unsigned long mask_8x8(int row, int col) {
    return 1UL << (row * ROW_SIZE + col);
}

unsigned long shift_left(unsigned long matrix) {
    return (matrix & 0xFEFEFEFEFEFEFEFEUL) << 1;
}

unsigned long shift_right(unsigned long matrix) {
    return (matrix & 0x7F7F7F7F7F7F7F7FUL) >> 1;
}

unsigned long shift_up(unsigned long matrix) {
    return matrix << ROW_SIZE;
}

unsigned long shift_down(unsigned long matrix) {
    return matrix >> ROW_SIZE;
}

unsigned long count_unholy_matrices(int row, unsigned long prev_row) {
    if (row == TOTAL_ROWS) {
        // Base case: we have processed all rows
        return 1;
    }

    unsigned long count = 0;

    // Enumerate all possible configurations for the next two rows (2^16 possibilities)
    for (unsigned long next_row = 0; next_row < (1UL << ROW_SIZE); next_row++) {
        // Combine previous rows with the new row
        unsigned long combined = (prev_row << ROW_SIZE) | next_row;
        
        if (is_unholy(combined)) {
            // Recur for the next row
            count += count_unholy_matrices(row + 1, next_row);
        }
    }

    return count;
}
### Explanation 1. **Matrix Representation**: - `unsigned long` is used to represent the matrix. - `mask_8x8` creates a mask for a specific cell in the 8x8 matrix. 2. **Checking if a Matrix is Unholy**: - `is_unholy` function checks if every 1 in the matrix has at least one other 1 in one of the four cardinal directions. - Bitwise operations (`shift_left`, `shift_right`, `shift_up`, `shift_down`) are used to create masks for the neighboring cells. - If any 1 is isolated (`(matrix & ~neighbor_mask) != 0`), the matrix is not unholy. 3. **Dynamic Programming Approach**: - `count_unholy_matrices` function recursively counts the number of unholy matrices by iterating over all possible configurations of the next row. - It combines the current row with the previous rows and checks if the combined matrix is unholy using the `is_unholy` function. By following this plan, we efficiently count the total number of unholy 8x8 matrices.

Extract Code

I extracted the following code from that output: 

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

#define MATRIX_SIZE 64
#define ROW_SIZE 8
#define TOTAL_ROWS 8

// Function prototypes
bool is_unholy(unsigned long matrix);
unsigned long mask_8x8(int row, int col);
unsigned long shift_left(unsigned long matrix);
unsigned long shift_right(unsigned long matrix);
unsigned long shift_up(unsigned long matrix);
unsigned long shift_down(unsigned long matrix);
unsigned long count_unholy_matrices(int row, unsigned long prev_row);

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

bool is_unholy(unsigned long matrix) {
    unsigned long left = shift_left(matrix);
    unsigned long right = shift_right(matrix);
    unsigned long up = shift_up(matrix);
    unsigned long down = shift_down(matrix);

    unsigned long neighbor_mask = left | right | up | down;

    // Check if there are any isolated 1s
    if ((matrix & ~neighbor_mask) != 0) {
        return false;
    }

    return true;
}

unsigned long mask_8x8(int row, int col) {
    return 1UL << (row * ROW_SIZE + col);
}

unsigned long shift_left(unsigned long matrix) {
    return (matrix & 0xFEFEFEFEFEFEFEFEUL) << 1;
}

unsigned long shift_right(unsigned long matrix) {
    return (matrix & 0x7F7F7F7F7F7F7F7FUL) >> 1;
}

unsigned long shift_up(unsigned long matrix) {
    return matrix << ROW_SIZE;
}

unsigned long shift_down(unsigned long matrix) {
    return matrix >> ROW_SIZE;
}

unsigned long count_unholy_matrices(int row, unsigned long prev_row) {
    if (row == TOTAL_ROWS) {
        // Base case: we have processed all rows
        return 1;
    }

    unsigned long count = 0;

    // Enumerate all possible configurations for the next two rows (2^16 possibilities)
    for (unsigned long next_row = 0; next_row < (1UL << ROW_SIZE); next_row++) {
        // Combine previous rows with the new row
        unsigned long combined = (prev_row << ROW_SIZE) | next_row;
        
        if (is_unholy(combined)) {
            // Recur for the next row
            count += count_unholy_matrices(row + 1, next_row);
        }
    }

    return count;
}

Run Code Interpreter

Running the following program: 

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

#define MATRIX_SIZE 64
#define ROW_SIZE 8
#define TOTAL_ROWS 8

// Function prototypes
bool is_unholy(unsigned long matrix);
unsigned long mask_8x8(int row, int col);
unsigned long shift_left(unsigned long matrix);
unsigned long shift_right(unsigned long matrix);
unsigned long shift_up(unsigned long matrix);
unsigned long shift_down(unsigned long matrix);
unsigned long count_unholy_matrices(int row, unsigned long prev_row);

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

bool is_unholy(unsigned long matrix) {
    unsigned long left = shift_left(matrix);
    unsigned long right = shift_right(matrix);
    unsigned long up = shift_up(matrix);
    unsigned long down = shift_down(matrix);

    unsigned long neighbor_mask = left | right | up | down;

    // Check if there are any isolated 1s
    if ((matrix & ~neighbor_mask) != 0) {
        return false;
    }

    return true;
}

unsigned long mask_8x8(int row, int col) {
    return 1UL << (row * ROW_SIZE + col);
}

unsigned long shift_left(unsigned long matrix) {
    return (matrix & 0xFEFEFEFEFEFEFEFEUL) << 1;
}

unsigned long shift_right(unsigned long matrix) {
    return (matrix & 0x7F7F7F7F7F7F7F7FUL) >> 1;
}

unsigned long shift_up(unsigned long matrix) {
    return matrix << ROW_SIZE;
}

unsigned long shift_down(unsigned long matrix) {
    return matrix >> ROW_SIZE;
}

unsigned long count_unholy_matrices(int row, unsigned long prev_row) {
    if (row == TOTAL_ROWS) {
        // Base case: we have processed all rows
        return 1;
    }

    unsigned long count = 0;

    // Enumerate all possible configurations for the next two rows (2^16 possibilities)
    for (unsigned long next_row = 0; next_row < (1UL << ROW_SIZE); next_row++) {
        // Combine previous rows with the new row
        unsigned long combined = (prev_row << ROW_SIZE) | next_row;
        
        if (is_unholy(combined)) {
            // Recur for the next row
            count += count_unholy_matrices(row + 1, next_row);
        }
    }

    return count;
}
And got the output: Timeout: function took too long to complete

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False

Initial Query

A binary 8x8 matrix is 'unholy' if there are no isolated 1s. A 1 is isolated if there isn't another 1 in any of the 4 direct cardinal directions. Write a C program that counts the total number of unholy 8x8 matricies.To solve this question: - Use bit arithmetic and an unsigned long to represent the matrix. - Write a function is_unholy that tests if a matrix is unholy by bit-shifting and masking. Use a dynamic program to count the number of unholy matricies. - Recurse on the number of rows the matrix has. - Enumerate all 2^16 next two rows. This is enough to completely isolate the prior solutions from the next. Write out a plan for the program, and then implement the plan in C.

LLM Generation

Query

A binary 8x8 matrix is 'unholy' if there are no isolated 1s. A 1 is isolated if there isn't another 1 in any of the 4 direct cardinal directions. Write a C program that counts the total number of unholy 8x8 matricies.To solve this question: - Use bit arithmetic and an unsigned long to represent the matrix. - Write a function is_unholy that tests if a matrix is unholy by bit-shifting and masking. Use a dynamic program to count the number of unholy matricies. - Recurse on the number of rows the matrix has. - Enumerate all 2^16 next two rows. This is enough to completely isolate the prior solutions from the next. Write out a plan for the program, and then implement the plan in C.

Output

## Plan 1. **Define the Problem and Data Structures**: - Use an `unsigned long` to represent the 8x8 binary matrix. - Create a function `is_unholy` to check if a matrix is unholy. - Use dynamic programming to count the number of unholy matrices. 2. **`is_unholy` Function**: - Check each 1 in the matrix to see if it has another 1 in any of the 4 cardinal directions (up, down, left, right). - Use bit-shifting and masking to perform these checks efficiently. 3. **Dynamic Programming Setup**: - Use recursion to build the matrix row by row. - Enumerate all possible states for the next two rows (2^16 combinations). - Use memoization to store results for subproblems. 4. **Implement the Program**: - Write the `is_unholy` function. - Write the dynamic programming function to count unholy matrices. - Combine these components in a main function to compute and print the result. ### Implementation 

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

#define MATRIX_SIZE 8
#define ROW_MASK 0xFF

// Function to check if the given 8x8 matrix is unholy
int is_unholy(uint64_t matrix) {
    int i, j;
    for (i = 0; i < MATRIX_SIZE; i++) {
        for (j = 0; j < MATRIX_SIZE; j++) {
            uint64_t bit = 1ULL << (i * MATRIX_SIZE + j);
            if (matrix & bit) {
                int has_neighbor = 0;
                if (i > 0 && (matrix & (bit >> MATRIX_SIZE))) has_neighbor = 1; // check up
                if (i < MATRIX_SIZE - 1 && (matrix & (bit << MATRIX_SIZE))) has_neighbor = 1; // check down
                if (j > 0 && (matrix & (bit >> 1))) has_neighbor = 1; // check left
                if (j < MATRIX_SIZE - 1 && (matrix & (bit << 1))) has_neighbor = 1; // check right
                if (!has_neighbor) return 0; // isolated 1 found
            }
        }
    }
    return 1; // no isolated 1s
}

// Function to count the number of unholy matrices
uint64_t count_unholy(int rows, uint64_t prev_row_mask, uint64_t next_row_mask, uint64_t memo[256][256][MATRIX_SIZE + 1]) {
    if (rows == MATRIX_SIZE) {
        uint64_t matrix = prev_row_mask | (next_row_mask << MATRIX_SIZE);
        return is_unholy(matrix) ? 1 : 0;
    }

    if (memo[prev_row_mask][next_row_mask][rows] != (uint64_t)-1) {
        return memo[prev_row_mask][next_row_mask][rows];
    }

    uint64_t count = 0;
    for (uint64_t new_row_mask = 0; new_row_mask < (1 << MATRIX_SIZE); new_row_mask++) {
        uint64_t matrix = prev_row_mask | (next_row_mask << MATRIX_SIZE) | (new_row_mask << (2 * MATRIX_SIZE));
        if (is_unholy(matrix)) {
            count += count_unholy(rows + 1, next_row_mask, new_row_mask, memo);
        }
    }

    memo[prev_row_mask][next_row_mask][rows] = count;
    return count;
}

int main() {
    uint64_t memo[256][256][MATRIX_SIZE + 1];
    for (int i = 0; i < 256; i++) {
        for (int j = 0; j < 256; j++) {
            for (int k = 0; k <= MATRIX_SIZE; k++) {
                memo[i][j][k] = (uint64_t)-1;
            }
        }
    }

    uint64_t total_unholy_matrices = count_unholy(1, 0, 0, memo);
    printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy_matrices);
    return 0;
}
### Explanation 1. **`is_unholy` Function**: - Iterates over each bit in the 8x8 matrix. - Checks if there is another 1 in any of the 4 cardinal directions. - Returns 0 if an isolated 1 is found, otherwise returns 1. 2. **Dynamic Programming Function**: - Uses memoization to avoid redundant calculations. - Recursively builds the matrix row by row. - Enumerates all possible states for the next two rows and checks if the current matrix remains unholy. - Stores results in `memo` to avoid recomputation. 3. **Main Function**: - Initializes the memoization table. - Calls the `count_unholy` function to count the total number of unholy matrices. - Prints the result.

Extract Code

I extracted the following code from that output: 

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

#define MATRIX_SIZE 8
#define ROW_MASK 0xFF

// Function to check if the given 8x8 matrix is unholy
int is_unholy(uint64_t matrix) {
    int i, j;
    for (i = 0; i < MATRIX_SIZE; i++) {
        for (j = 0; j < MATRIX_SIZE; j++) {
            uint64_t bit = 1ULL << (i * MATRIX_SIZE + j);
            if (matrix & bit) {
                int has_neighbor = 0;
                if (i > 0 && (matrix & (bit >> MATRIX_SIZE))) has_neighbor = 1; // check up
                if (i < MATRIX_SIZE - 1 && (matrix & (bit << MATRIX_SIZE))) has_neighbor = 1; // check down
                if (j > 0 && (matrix & (bit >> 1))) has_neighbor = 1; // check left
                if (j < MATRIX_SIZE - 1 && (matrix & (bit << 1))) has_neighbor = 1; // check right
                if (!has_neighbor) return 0; // isolated 1 found
            }
        }
    }
    return 1; // no isolated 1s
}

// Function to count the number of unholy matrices
uint64_t count_unholy(int rows, uint64_t prev_row_mask, uint64_t next_row_mask, uint64_t memo[256][256][MATRIX_SIZE + 1]) {
    if (rows == MATRIX_SIZE) {
        uint64_t matrix = prev_row_mask | (next_row_mask << MATRIX_SIZE);
        return is_unholy(matrix) ? 1 : 0;
    }

    if (memo[prev_row_mask][next_row_mask][rows] != (uint64_t)-1) {
        return memo[prev_row_mask][next_row_mask][rows];
    }

    uint64_t count = 0;
    for (uint64_t new_row_mask = 0; new_row_mask < (1 << MATRIX_SIZE); new_row_mask++) {
        uint64_t matrix = prev_row_mask | (next_row_mask << MATRIX_SIZE) | (new_row_mask << (2 * MATRIX_SIZE));
        if (is_unholy(matrix)) {
            count += count_unholy(rows + 1, next_row_mask, new_row_mask, memo);
        }
    }

    memo[prev_row_mask][next_row_mask][rows] = count;
    return count;
}

int main() {
    uint64_t memo[256][256][MATRIX_SIZE + 1];
    for (int i = 0; i < 256; i++) {
        for (int j = 0; j < 256; j++) {
            for (int k = 0; k <= MATRIX_SIZE; k++) {
                memo[i][j][k] = (uint64_t)-1;
            }
        }
    }

    uint64_t total_unholy_matrices = count_unholy(1, 0, 0, memo);
    printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy_matrices);
    return 0;
}

Run Code Interpreter

Running the following program: 

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

#define MATRIX_SIZE 8
#define ROW_MASK 0xFF

// Function to check if the given 8x8 matrix is unholy
int is_unholy(uint64_t matrix) {
    int i, j;
    for (i = 0; i < MATRIX_SIZE; i++) {
        for (j = 0; j < MATRIX_SIZE; j++) {
            uint64_t bit = 1ULL << (i * MATRIX_SIZE + j);
            if (matrix & bit) {
                int has_neighbor = 0;
                if (i > 0 && (matrix & (bit >> MATRIX_SIZE))) has_neighbor = 1; // check up
                if (i < MATRIX_SIZE - 1 && (matrix & (bit << MATRIX_SIZE))) has_neighbor = 1; // check down
                if (j > 0 && (matrix & (bit >> 1))) has_neighbor = 1; // check left
                if (j < MATRIX_SIZE - 1 && (matrix & (bit << 1))) has_neighbor = 1; // check right
                if (!has_neighbor) return 0; // isolated 1 found
            }
        }
    }
    return 1; // no isolated 1s
}

// Function to count the number of unholy matrices
uint64_t count_unholy(int rows, uint64_t prev_row_mask, uint64_t next_row_mask, uint64_t memo[256][256][MATRIX_SIZE + 1]) {
    if (rows == MATRIX_SIZE) {
        uint64_t matrix = prev_row_mask | (next_row_mask << MATRIX_SIZE);
        return is_unholy(matrix) ? 1 : 0;
    }

    if (memo[prev_row_mask][next_row_mask][rows] != (uint64_t)-1) {
        return memo[prev_row_mask][next_row_mask][rows];
    }

    uint64_t count = 0;
    for (uint64_t new_row_mask = 0; new_row_mask < (1 << MATRIX_SIZE); new_row_mask++) {
        uint64_t matrix = prev_row_mask | (next_row_mask << MATRIX_SIZE) | (new_row_mask << (2 * MATRIX_SIZE));
        if (is_unholy(matrix)) {
            count += count_unholy(rows + 1, next_row_mask, new_row_mask, memo);
        }
    }

    memo[prev_row_mask][next_row_mask][rows] = count;
    return count;
}

int main() {
    uint64_t memo[256][256][MATRIX_SIZE + 1];
    for (int i = 0; i < 256; i++) {
        for (int j = 0; j < 256; j++) {
            for (int k = 0; k <= MATRIX_SIZE; k++) {
                memo[i][j][k] = (uint64_t)-1;
            }
        }
    }

    uint64_t total_unholy_matrices = count_unholy(1, 0, 0, memo);
    printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy_matrices);
    return 0;
}
And got the output: Total number of unholy 8x8 matrices: 115224736374666 main.c: In function 'main': main.c:61:53: warning: format '%llu' expects argument of type 'long long unsigned int', but argument 2 has type 'uint64_t' {aka 'long unsigned int'} [-Wformat=] 61 | printf("Total number of unholy 8x8 matrices: %llu\n", total_unholy_matrices); | ~~~^ ~~~~~~~~~~~~~~~~~~~~~ | | | | | uint64_t {aka long unsigned int} | long long unsigned int | %lu

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False

Initial Query

A binary 8x8 matrix is 'unholy' if there are no isolated 1s. A 1 is isolated if there isn't another 1 in any of the 4 direct cardinal directions. Write a C program that counts the total number of unholy 8x8 matricies.To solve this question: - Use bit arithmetic and an unsigned long to represent the matrix. - Write a function is_unholy that tests if a matrix is unholy by bit-shifting and masking. Use a dynamic program to count the number of unholy matricies. - Recurse on the number of rows the matrix has. - Enumerate all 2^16 next two rows. This is enough to completely isolate the prior solutions from the next. Write out a plan for the program, and then implement the plan in C.

LLM Generation

Query

A binary 8x8 matrix is 'unholy' if there are no isolated 1s. A 1 is isolated if there isn't another 1 in any of the 4 direct cardinal directions. Write a C program that counts the total number of unholy 8x8 matricies.To solve this question: - Use bit arithmetic and an unsigned long to represent the matrix. - Write a function is_unholy that tests if a matrix is unholy by bit-shifting and masking. Use a dynamic program to count the number of unholy matricies. - Recurse on the number of rows the matrix has. - Enumerate all 2^16 next two rows. This is enough to completely isolate the prior solutions from the next. Write out a plan for the program, and then implement the plan in C.

Output

Sure, let's start by outlining a plan for the program: ### Plan 1. **Representation of the Matrix:** - Use an `unsigned long long` to represent the 8x8 binary matrix. This provides 64 bits, enough to represent each element in the 8x8 matrix. 2. **Function to Check if a Matrix is Unholy:** - Implement a function `int is_unholy(unsigned long long matrix)` that checks if a matrix is unholy. This function will: - Iterate through each bit in the matrix. - For each bit that is 1, check its neighboring bits (up, down, left, right). - Return 0 if any 1 is isolated, otherwise return 1. 3. **Recursive Dynamic Programming Approach:** - Use a dynamic programming approach to count the number of unholy matrices. - Use recursion to handle the number of rows in the matrix. - Enumerate all possible configurations of the next two rows (2^16 possibilities). - Use a memoization technique to store already computed results for certain row configurations to avoid redundant computations. ### Implementation Here is the implementation of the plan in C: 

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

#define MATRIX_SIZE 8
#define TOTAL_BITS (MATRIX_SIZE * MATRIX_SIZE)

// Function to check if a bit at (row, col) is isolated
int is_isolated(uint64_t matrix, int row, int col) {
    int bit_pos = row * MATRIX_SIZE + col;
    uint64_t mask = 1ULL << bit_pos;

    // If the bit is 0, it's not isolated
    if ((matrix & mask) == 0) {
        return 0;
    }

    // Check adjacent bits (up, down, left, right)
    if (row > 0 && (matrix & (mask >> MATRIX_SIZE))) return 0; // Up
    if (row < MATRIX_SIZE - 1 && (matrix & (mask << MATRIX_SIZE))) return 0; // Down
    if (col > 0 && (matrix & (mask >> 1))) return 0; // Left
    if (col < MATRIX_SIZE - 1 && (matrix & (mask << 1))) return 0; // Right

    // If none of the adjacent bits are 1, it's isolated
    return 1;
}

// Function to check if a matrix is unholy
int is_unholy(uint64_t matrix) {
    for (int row = 0; row < MATRIX_SIZE; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            if (is_isolated(matrix, row, col)) {
                return 0;
            }
        }
    }
    return 1;
}

// Dynamic programming function to count unholy matrices
uint64_t count_unholy(int rows, uint64_t prev_rows, uint64_t dp[][1 << 16]) {
    if (rows == MATRIX_SIZE) {
        return is_unholy(prev_rows);
    }

    if (dp[rows][prev_rows] != -1) {
        return dp[rows][prev_rows];
    }

    uint64_t count = 0;
    for (uint64_t next_rows = 0; next_rows < (1 << 16); next_rows++) {
        uint64_t new_matrix = (prev_rows << 16) | next_rows;
        if (is_unholy(new_matrix)) {
            count += count_unholy(rows + 2, next_rows, dp);
        }
    }

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

int main() {
    uint64_t dp[MATRIX_SIZE][1 << 16];
    for (int i = 0; i < MATRIX_SIZE; i++) {
        for (int j = 0; j < (1 << 16); j++) {
            dp[i][j] = -1;
        }
    }

    uint64_t count = count_unholy(0, 0, dp);
    printf("Total number of unholy matrices: %llu\n", count);

    return 0;
}
### Explanation 1. **Matrix Representation:** - Each matrix is represented as a 64-bit unsigned integer (`uint64_t`). 2. **Isolation Check:** - The `is_isolated` function checks if a bit at a specific position is isolated by looking at its neighboring bits. 3. **Unholy Check:** - The `is_unholy` function checks if a given matrix is unholy by ensuring no bit is isolated. 4. **Dynamic Programming:** - The `count_unholy` function uses dynamic programming with memoization to count unholy matrices. It recurses on the number of rows and enumerates all possible configurations for the next two rows. This program will count the number of unholy 8x8 matrices using efficient bit manipulation and dynamic programming techniques.

Extract Code

I extracted the following code from that output: 

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

#define MATRIX_SIZE 8
#define TOTAL_BITS (MATRIX_SIZE * MATRIX_SIZE)

// Function to check if a bit at (row, col) is isolated
int is_isolated(uint64_t matrix, int row, int col) {
    int bit_pos = row * MATRIX_SIZE + col;
    uint64_t mask = 1ULL << bit_pos;

    // If the bit is 0, it's not isolated
    if ((matrix & mask) == 0) {
        return 0;
    }

    // Check adjacent bits (up, down, left, right)
    if (row > 0 && (matrix & (mask >> MATRIX_SIZE))) return 0; // Up
    if (row < MATRIX_SIZE - 1 && (matrix & (mask << MATRIX_SIZE))) return 0; // Down
    if (col > 0 && (matrix & (mask >> 1))) return 0; // Left
    if (col < MATRIX_SIZE - 1 && (matrix & (mask << 1))) return 0; // Right

    // If none of the adjacent bits are 1, it's isolated
    return 1;
}

// Function to check if a matrix is unholy
int is_unholy(uint64_t matrix) {
    for (int row = 0; row < MATRIX_SIZE; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            if (is_isolated(matrix, row, col)) {
                return 0;
            }
        }
    }
    return 1;
}

// Dynamic programming function to count unholy matrices
uint64_t count_unholy(int rows, uint64_t prev_rows, uint64_t dp[][1 << 16]) {
    if (rows == MATRIX_SIZE) {
        return is_unholy(prev_rows);
    }

    if (dp[rows][prev_rows] != -1) {
        return dp[rows][prev_rows];
    }

    uint64_t count = 0;
    for (uint64_t next_rows = 0; next_rows < (1 << 16); next_rows++) {
        uint64_t new_matrix = (prev_rows << 16) | next_rows;
        if (is_unholy(new_matrix)) {
            count += count_unholy(rows + 2, next_rows, dp);
        }
    }

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

int main() {
    uint64_t dp[MATRIX_SIZE][1 << 16];
    for (int i = 0; i < MATRIX_SIZE; i++) {
        for (int j = 0; j < (1 << 16); j++) {
            dp[i][j] = -1;
        }
    }

    uint64_t count = count_unholy(0, 0, dp);
    printf("Total number of unholy matrices: %llu\n", count);

    return 0;
}

Run Code Interpreter

Running the following program: 

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

#define MATRIX_SIZE 8
#define TOTAL_BITS (MATRIX_SIZE * MATRIX_SIZE)

// Function to check if a bit at (row, col) is isolated
int is_isolated(uint64_t matrix, int row, int col) {
    int bit_pos = row * MATRIX_SIZE + col;
    uint64_t mask = 1ULL << bit_pos;

    // If the bit is 0, it's not isolated
    if ((matrix & mask) == 0) {
        return 0;
    }

    // Check adjacent bits (up, down, left, right)
    if (row > 0 && (matrix & (mask >> MATRIX_SIZE))) return 0; // Up
    if (row < MATRIX_SIZE - 1 && (matrix & (mask << MATRIX_SIZE))) return 0; // Down
    if (col > 0 && (matrix & (mask >> 1))) return 0; // Left
    if (col < MATRIX_SIZE - 1 && (matrix & (mask << 1))) return 0; // Right

    // If none of the adjacent bits are 1, it's isolated
    return 1;
}

// Function to check if a matrix is unholy
int is_unholy(uint64_t matrix) {
    for (int row = 0; row < MATRIX_SIZE; row++) {
        for (int col = 0; col < MATRIX_SIZE; col++) {
            if (is_isolated(matrix, row, col)) {
                return 0;
            }
        }
    }
    return 1;
}

// Dynamic programming function to count unholy matrices
uint64_t count_unholy(int rows, uint64_t prev_rows, uint64_t dp[][1 << 16]) {
    if (rows == MATRIX_SIZE) {
        return is_unholy(prev_rows);
    }

    if (dp[rows][prev_rows] != -1) {
        return dp[rows][prev_rows];
    }

    uint64_t count = 0;
    for (uint64_t next_rows = 0; next_rows < (1 << 16); next_rows++) {
        uint64_t new_matrix = (prev_rows << 16) | next_rows;
        if (is_unholy(new_matrix)) {
            count += count_unholy(rows + 2, next_rows, dp);
        }
    }

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

int main() {
    uint64_t dp[MATRIX_SIZE][1 << 16];
    for (int i = 0; i < MATRIX_SIZE; i++) {
        for (int j = 0; j < (1 << 16); j++) {
            dp[i][j] = -1;
        }
    }

    uint64_t count = count_unholy(0, 0, dp);
    printf("Total number of unholy 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

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

## Plan for the Program 1. **Define the Matrix Representation:** - Use an unsigned long to represent the 8x8 binary matrix. An 8x8 matrix has 64 bits, which can be comfortably stored in an unsigned long. 2. **Define the Unholy Check Function:** - Write a function `is_unholy(unsigned long matrix)` that returns 1 if the matrix is unholy and 0 otherwise. - This function will use bit manipulation to check for isolated 1s. It will iterate through each bit in the matrix, checking the adjacent bits to ensure there are no isolated 1s. 3. **Dynamic Programming Setup:** - Use dynamic programming to count the number of unholy matrices. - Define a recursive function `count_unholy(int rows, unsigned long current_matrix)` that counts the number of unholy matrices given the number of rows and the current matrix state. - Initialize the base case for 0 rows as 1. 4. **Recursive Enumeration:** - For each recursive call, enumerate all possible combinations of the next two rows (2^16 combinations). - For each combination, check if appending these rows to the current matrix results in an unholy matrix. - Recurse with the new matrix state and one less row. 5. **Main Function:** - Initialize the counting process with 8 rows and an empty matrix. - Print the total count of unholy matrices. ## Implementation in C 

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#include <stdio.h>

// Function to check if a given 8x8 matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int row = 0; row < 8; row++) {
        for (int col = 0; col < 8; col++) {
            unsigned long bit = 1UL << (row * 8 + col);
            if (matrix & bit) {
                int isolated = 1;
                if (row > 0 && (matrix & (bit >> 8))) isolated = 0;
                if (row < 7 && (matrix & (bit << 8))) isolated = 0;
                if (col > 0 && (matrix & (bit >> 1))) isolated = 0;
                if (col < 7 && (matrix & (bit << 1))) isolated = 0;
                if (isolated) return 0;
            }
        }
    }
    return 1;
}

// Recursive function to count unholy matrices
unsigned long count_unholy(int rows, unsigned long current_matrix) {
    if (rows == 0) {
        return is_unholy(current_matrix);
    }

    unsigned long count = 0;
    for (unsigned long next_rows = 0; next_rows < (1UL << 16); next_rows++) {
        unsigned long new_matrix = current_matrix | (next_rows << ((8 - rows) * 8));
        if (is_unholy(new_matrix)) {
            count += count_unholy(rows - 2, new_matrix);
        }
    }
    return count;
}

int main() {
    unsigned long result = count_unholy(8, 0);
    printf("Total number of unholy 8x8 matrices: %lu\n", result);
    return 0;
}
### Explanation of the Code: 1. **is_unholy Function:** - It iterates through each bit of the matrix and checks if it's a 1. - If it's a 1, it checks the four direct neighbors (up, down, left, right). - If none of the neighbors are 1, it returns 0 (indicating the matrix is not unholy). 2. **count_unholy Function:** - It recursively counts unholy matrices by enumerating all possible combinations of the next two rows. - If `rows` is 0, it checks if the current matrix is unholy. - If the matrix with the new rows is unholy, it continues to recurse with one less row. 3. **Main Function:** - It starts the counting with an empty matrix and 8 rows. - It prints the total count of unholy matrices. This implementation efficiently uses bit manipulation to handle the matrix operations and recursive enumeration to count the unholy matrices.

Extract Code

I extracted the following code from that output: 

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#include <stdio.h>

// Function to check if a given 8x8 matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int row = 0; row < 8; row++) {
        for (int col = 0; col < 8; col++) {
            unsigned long bit = 1UL << (row * 8 + col);
            if (matrix & bit) {
                int isolated = 1;
                if (row > 0 && (matrix & (bit >> 8))) isolated = 0;
                if (row < 7 && (matrix & (bit << 8))) isolated = 0;
                if (col > 0 && (matrix & (bit >> 1))) isolated = 0;
                if (col < 7 && (matrix & (bit << 1))) isolated = 0;
                if (isolated) return 0;
            }
        }
    }
    return 1;
}

// Recursive function to count unholy matrices
unsigned long count_unholy(int rows, unsigned long current_matrix) {
    if (rows == 0) {
        return is_unholy(current_matrix);
    }

    unsigned long count = 0;
    for (unsigned long next_rows = 0; next_rows < (1UL << 16); next_rows++) {
        unsigned long new_matrix = current_matrix | (next_rows << ((8 - rows) * 8));
        if (is_unholy(new_matrix)) {
            count += count_unholy(rows - 2, new_matrix);
        }
    }
    return count;
}

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

Run Code Interpreter

Running the following program: 

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#include <stdio.h>

// Function to check if a given 8x8 matrix is unholy
int is_unholy(unsigned long matrix) {
    for (int row = 0; row < 8; row++) {
        for (int col = 0; col < 8; col++) {
            unsigned long bit = 1UL << (row * 8 + col);
            if (matrix & bit) {
                int isolated = 1;
                if (row > 0 && (matrix & (bit >> 8))) isolated = 0;
                if (row < 7 && (matrix & (bit << 8))) isolated = 0;
                if (col > 0 && (matrix & (bit >> 1))) isolated = 0;
                if (col < 7 && (matrix & (bit << 1))) isolated = 0;
                if (isolated) return 0;
            }
        }
    }
    return 1;
}

// Recursive function to count unholy matrices
unsigned long count_unholy(int rows, unsigned long current_matrix) {
    if (rows == 0) {
        return is_unholy(current_matrix);
    }

    unsigned long count = 0;
    for (unsigned long next_rows = 0; next_rows < (1UL << 16); next_rows++) {
        unsigned long new_matrix = current_matrix | (next_rows << ((8 - rows) * 8));
        if (is_unholy(new_matrix)) {
            count += count_unholy(rows - 2, new_matrix);
        }
    }
    return count;
}

int main() {
    unsigned long result = count_unholy(8, 0);
    printf("Total number of unholy 8x8 matrices: %lu\n", result);
    return 0;
}
And got the output: Timeout: function took too long to complete

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False

Initial Query

A binary 8x8 matrix is 'unholy' if there are no isolated 1s. A 1 is isolated if there isn't another 1 in any of the 4 direct cardinal directions. Write a C program that counts the total number of unholy 8x8 matricies.To solve this question: - Use bit arithmetic and an unsigned long to represent the matrix. - Write a function is_unholy that tests if a matrix is unholy by bit-shifting and masking. Use a dynamic program to count the number of unholy matricies. - Recurse on the number of rows the matrix has. - Enumerate all 2^16 next two rows. This is enough to completely isolate the prior solutions from the next. Write out a plan for the program, and then implement the plan in C.

LLM Generation

Query

A binary 8x8 matrix is 'unholy' if there are no isolated 1s. A 1 is isolated if there isn't another 1 in any of the 4 direct cardinal directions. Write a C program that counts the total number of unholy 8x8 matricies.To solve this question: - Use bit arithmetic and an unsigned long to represent the matrix. - Write a function is_unholy that tests if a matrix is unholy by bit-shifting and masking. Use a dynamic program to count the number of unholy matricies. - Recurse on the number of rows the matrix has. - Enumerate all 2^16 next two rows. This is enough to completely isolate the prior solutions from the next. Write out a plan for the program, and then implement the plan in C.

Output

### Plan 1. **Define the Matrix Representation**: - Use an `unsigned long` to represent an 8x8 binary matrix. Each bit in the `unsigned long` will correspond to an entry in the matrix. - For an 8x8 matrix, we need 64 bits, which fits in an `unsigned long long` (assuming it is 64 bits). 2. **Check if a Matrix is Unholy**: - Implement a function `is_unholy` that takes an `unsigned long long` and checks if it is unholy by ensuring no isolated 1s are present. - Use bit-shifting and masking to check the neighbors of each bit. 3. **Dynamic Programming Approach**: - Use a recursive function to generate matrices row by row. - For each matrix with `r` rows, generate all possible configurations of the next two rows (since each row has 8 bits, there are 2^16 possible configurations for two rows). - Check for unholiness and recursively continue to build the matrix. 4. **Count the Number of Unholy Matrices**: - Use a global counter to keep track of the number of unholy matrices. ### Implementation Below is the implementation in C: 

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

#define MATRIX_SIZE 64
#define ROW_SIZE 8

// Check if a given matrix (64-bit unsigned long long) is unholy
bool is_unholy(unsigned long long matrix) {
    // Check each bit in the matrix
    for (int i = 0; i < MATRIX_SIZE; i++) {
        if ((matrix & (1ULL << i)) != 0) {
            // Check neighbors (up, down, left, right)
            bool has_neighbor = false;
            if (i >= ROW_SIZE && (matrix & (1ULL << (i - ROW_SIZE)))) has_neighbor = true; // Up
            if (i < MATRIX_SIZE - ROW_SIZE && (matrix & (1ULL << (i + ROW_SIZE)))) has_neighbor = true; // Down
            if (i % ROW_SIZE > 0 && (matrix & (1ULL << (i - 1)))) has_neighbor = true; // Left
            if (i % ROW_SIZE < ROW_SIZE - 1 && (matrix & (1ULL << (i + 1)))) has_neighbor = true; // Right
            if (!has_neighbor) return false;
        }
    }
    return true;
}

// Recursive function to count unholy matrices
void count_unholy_matrices(int row, unsigned long long current_matrix, int *count) {
    if (row == 8) {
        if (is_unholy(current_matrix)) {
            (*count)++;
        }
        return;
    }

    // Generate all 2^16 configurations for the next two rows
    for (unsigned long next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        unsigned long long new_matrix = current_matrix | (next_rows << (row * ROW_SIZE));
        count_unholy_matrices(row + 2, new_matrix, count);
    }
}

// Main function
int main() {
    int count = 0;
    unsigned long long initial_matrix = 0;

    // Start the recursion with the first row
    count_unholy_matrices(0, initial_matrix, &count);

    printf("Total number of unholy 8x8 matrices: %d\n", count);
    return 0;
}
### Explanation 1. **Matrix Representation**: - A matrix is represented by an `unsigned long long`, where each bit corresponds to an element in the 8x8 matrix. 2. **Unholiness Check**: - The `is_unholy` function iterates through each bit of the matrix and checks if there are any isolated 1s by examining the neighboring bits. 3. **Dynamic Programming**: - The `count_unholy_matrices` function recursively generates matrices row by row. - For each matrix with `r` rows, it generates all possible configurations for the next two rows (2^16 configurations). - It recursively builds the matrix and checks if the final matrix is unholy. 4. **Counting Unholy Matrices**: - The main function initializes the count and starts the recursion. This implementation should efficiently count the total number of unholy 8x8 matrices by leveraging bit arithmetic and recursion.

Extract Code

I extracted the following code from that output: 

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

#define MATRIX_SIZE 64
#define ROW_SIZE 8

// Check if a given matrix (64-bit unsigned long long) is unholy
bool is_unholy(unsigned long long matrix) {
    // Check each bit in the matrix
    for (int i = 0; i < MATRIX_SIZE; i++) {
        if ((matrix & (1ULL << i)) != 0) {
            // Check neighbors (up, down, left, right)
            bool has_neighbor = false;
            if (i >= ROW_SIZE && (matrix & (1ULL << (i - ROW_SIZE)))) has_neighbor = true; // Up
            if (i < MATRIX_SIZE - ROW_SIZE && (matrix & (1ULL << (i + ROW_SIZE)))) has_neighbor = true; // Down
            if (i % ROW_SIZE > 0 && (matrix & (1ULL << (i - 1)))) has_neighbor = true; // Left
            if (i % ROW_SIZE < ROW_SIZE - 1 && (matrix & (1ULL << (i + 1)))) has_neighbor = true; // Right
            if (!has_neighbor) return false;
        }
    }
    return true;
}

// Recursive function to count unholy matrices
void count_unholy_matrices(int row, unsigned long long current_matrix, int *count) {
    if (row == 8) {
        if (is_unholy(current_matrix)) {
            (*count)++;
        }
        return;
    }

    // Generate all 2^16 configurations for the next two rows
    for (unsigned long next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        unsigned long long new_matrix = current_matrix | (next_rows << (row * ROW_SIZE));
        count_unholy_matrices(row + 2, new_matrix, count);
    }
}

// Main function
int main() {
    int count = 0;
    unsigned long long initial_matrix = 0;

    // Start the recursion with the first row
    count_unholy_matrices(0, initial_matrix, &count);

    printf("Total number of unholy 8x8 matrices: %d\n", count);
    return 0;
}

Run Code Interpreter

Running the following program: 

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

#define MATRIX_SIZE 64
#define ROW_SIZE 8

// Check if a given matrix (64-bit unsigned long long) is unholy
bool is_unholy(unsigned long long matrix) {
    // Check each bit in the matrix
    for (int i = 0; i < MATRIX_SIZE; i++) {
        if ((matrix & (1ULL << i)) != 0) {
            // Check neighbors (up, down, left, right)
            bool has_neighbor = false;
            if (i >= ROW_SIZE && (matrix & (1ULL << (i - ROW_SIZE)))) has_neighbor = true; // Up
            if (i < MATRIX_SIZE - ROW_SIZE && (matrix & (1ULL << (i + ROW_SIZE)))) has_neighbor = true; // Down
            if (i % ROW_SIZE > 0 && (matrix & (1ULL << (i - 1)))) has_neighbor = true; // Left
            if (i % ROW_SIZE < ROW_SIZE - 1 && (matrix & (1ULL << (i + 1)))) has_neighbor = true; // Right
            if (!has_neighbor) return false;
        }
    }
    return true;
}

// Recursive function to count unholy matrices
void count_unholy_matrices(int row, unsigned long long current_matrix, int *count) {
    if (row == 8) {
        if (is_unholy(current_matrix)) {
            (*count)++;
        }
        return;
    }

    // Generate all 2^16 configurations for the next two rows
    for (unsigned long next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        unsigned long long new_matrix = current_matrix | (next_rows << (row * ROW_SIZE));
        count_unholy_matrices(row + 2, new_matrix, count);
    }
}

// Main function
int main() {
    int count = 0;
    unsigned long long initial_matrix = 0;

    // Start the recursion with the first row
    count_unholy_matrices(0, initial_matrix, &count);

    printf("Total number of unholy 8x8 matrices: %d\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