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, here's a plan and an implementation of the C program to count the total number of unholy 8x8 matrices: **Plan:** 1. Define a struct to represent an 8x8 matrix using an unsigned long long int (64-bit), where each bit represents a cell in the matrix. 2. Implement a function `is_unholy` that takes a matrix and checks if it is unholy or not. This function should use bit operations to check if there are any isolated 1s in the matrix. 3. Implement a recursive function `count_unholy` that counts the number of unholy matrices. This function should take the number of remaining rows as an argument and enumerate all possible combinations of the next two rows (2^16 possibilities) and recursively call itself with the updated matrix and the remaining number of rows. 4. In the main function, initialize the base case for the recursive function by calling `count_unholy` with 8 rows remaining. **Implementation:** 

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

typedef struct {
    uint64_t bits;
} Matrix;

int is_unholy(Matrix matrix) {
    uint64_t isolated_mask = 0x8080808080808080;
    uint64_t neighbors_mask = 0x7E7E7E7E7E7E7E7E;

    uint64_t isolated_ones = matrix.bits & isolated_mask;
    uint64_t neighbors = (matrix.bits & neighbors_mask) << 1 | (matrix.bits & neighbors_mask) >> 1 |
                         (matrix.bits & neighbors_mask) << 8 | (matrix.bits & neighbors_mask) >> 8;

    return (isolated_ones | neighbors) == matrix.bits;
}

int count_unholy(int rows_remaining) {
    int count = 0;
    if (rows_remaining <= 0) {
        return 1;
    }

    for (uint64_t next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        Matrix matrix = {next_rows};
        if (is_unholy(matrix)) {
            count += count_unholy(rows_remaining - 2);
        }
    }

    return count;
}

int main() {
    int total_unholy = count_unholy(8);
    printf("Total number of unholy 8x8 matrices: %d\n", total_unholy);
    return 0;
}
**Explanation:** 1. The `Matrix` struct represents an 8x8 matrix using a 64-bit unsigned integer, where each bit corresponds to a cell in the matrix. 2. The `is_unholy` function checks if a given matrix is unholy or not. It first creates a mask `isolated_mask` that identifies all positions where a 1 would be isolated. Then, it creates another mask `neighbors_mask` that identifies all positions that have at least one neighbor. It shifts the matrix bits left and right by 1 and 8 positions to get the neighboring bits and combines them using bitwise OR. Finally, it checks if the isolated ones and the neighbors cover the entire matrix, which means there are no isolated ones. 3. The `count_unholy` function is a recursive function that counts the number of unholy matrices. It takes the number of remaining rows as an argument. If there are no remaining rows, it returns 1 (base case). Otherwise, it enumerates all possible combinations of the next two rows (2^16 possibilities) and recursively calls itself with the updated matrix and the remaining number of rows. If the current matrix is unholy (checked using `is_unholy`), it adds the result of the recursive call to the count. 4. In the `main` function, we initialize the base case for the recursive function by calling `count_unholy` with 8 rows remaining, and print the total count of unholy matrices. This program uses bit arithmetic and dynamic programming to efficiently count the number of unholy 8x8 matrices. Note that this implementation assumes that the matrix is represented row-wise, with the least significant bit representing the top-left cell and the most significant bit representing the bottom-right cell.

Extract Code

I extracted the following code from that output: 

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

typedef struct {
    uint64_t bits;
} Matrix;

int is_unholy(Matrix matrix) {
    uint64_t isolated_mask = 0x8080808080808080;
    uint64_t neighbors_mask = 0x7E7E7E7E7E7E7E7E;

    uint64_t isolated_ones = matrix.bits & isolated_mask;
    uint64_t neighbors = (matrix.bits & neighbors_mask) << 1 | (matrix.bits & neighbors_mask) >> 1 |
                         (matrix.bits & neighbors_mask) << 8 | (matrix.bits & neighbors_mask) >> 8;

    return (isolated_ones | neighbors) == matrix.bits;
}

int count_unholy(int rows_remaining) {
    int count = 0;
    if (rows_remaining <= 0) {
        return 1;
    }

    for (uint64_t next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        Matrix matrix = {next_rows};
        if (is_unholy(matrix)) {
            count += count_unholy(rows_remaining - 2);
        }
    }

    return count;
}

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

Run Code Interpreter

Running the following program: 

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

typedef struct {
    uint64_t bits;
} Matrix;

int is_unholy(Matrix matrix) {
    uint64_t isolated_mask = 0x8080808080808080;
    uint64_t neighbors_mask = 0x7E7E7E7E7E7E7E7E;

    uint64_t isolated_ones = matrix.bits & isolated_mask;
    uint64_t neighbors = (matrix.bits & neighbors_mask) << 1 | (matrix.bits & neighbors_mask) >> 1 |
                         (matrix.bits & neighbors_mask) << 8 | (matrix.bits & neighbors_mask) >> 8;

    return (isolated_ones | neighbors) == matrix.bits;
}

int count_unholy(int rows_remaining) {
    int count = 0;
    if (rows_remaining <= 0) {
        return 1;
    }

    for (uint64_t next_rows = 0; next_rows < (1ULL << 16); next_rows++) {
        Matrix matrix = {next_rows};
        if (is_unholy(matrix)) {
            count += count_unholy(rows_remaining - 2);
        }
    }

    return count;
}

int main() {
    int total_unholy = count_unholy(8);
    printf("Total number of unholy 8x8 matrices: %d\n", total_unholy);
    return 0;
}
And got the output: Total number of unholy 8x8 matrices: 256

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, here's a plan for the program: 1. **Define the data structures** - Use an `unsigned long` to represent an 8x8 binary matrix, where each bit corresponds to a cell in the matrix. - Define a constant `ROWS` as 8 to represent the number of rows in the matrix. - Define a constant `COLS` as 8 to represent the number of columns in the matrix. 2. **Implement helper functions** - `is_unholy(unsigned long matrix)`: This function checks if a given matrix is unholy by iterating over each bit and checking if there is a neighboring 1 in any of the four cardinal directions. - `get_neighbors(int row, int col)`: This function returns a bitmask representing the neighboring cells in the four cardinal directions for a given cell at `(row, col)`. 3. **Implement the dynamic programming function** - `count_unholy(int rows_left)`: This function recursively counts the number of unholy matrices with `rows_left` rows remaining. - Base case: If `rows_left` is 0, return 1 (since an empty matrix is considered unholy). - Enumerate all possible combinations of the next two rows (`2^16` combinations). - For each combination, check if it is unholy by calling `is_unholy` and recursively count the number of unholy matrices with `rows_left - 2` remaining. - Return the sum of counts for all combinations. 4. **Main function** - Call `count_unholy(ROWS)` to get the total count of unholy 8x8 matrices. - Print the result. Here's the implementation in C: 

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#include <stdio.h>

#define ROWS 8
#define COLS 8

unsigned long get_neighbors(int row, int col) {
    unsigned long neighbors = 0;
    if (row > 0)
        neighbors |= 1UL << (row - 1) * COLS + col;
    if (row < ROWS - 1)
        neighbors |= 1UL << (row + 1) * COLS + col;
    if (col > 0)
        neighbors |= 1UL << row * COLS + col - 1;
    if (col < COLS - 1)
        neighbors |= 1UL << row * COLS + col + 1;
    return neighbors;
}

int is_unholy(unsigned long matrix) {
    for (int row = 0; row < ROWS; row++) {
        for (int col = 0; col < COLS; col++) {
            unsigned long bit = 1UL << row * COLS + col;
            if ((matrix & bit) && !(matrix & get_neighbors(row, col)))
                return 0;
        }
    }
    return 1;
}

int count_unholy(int rows_left) {
    if (rows_left == 0)
        return 1;

    int count = 0;
    unsigned long mask = (1UL << (2 * COLS)) - 1;
    for (unsigned long rows = 0; rows <= mask; rows++) {
        if (is_unholy(rows)) {
            count += count_unholy(rows_left - 2);
        }
    }
    return count;
}

int main() {
    int count = count_unholy(ROWS);
    printf("Total number of unholy 8x8 matrices: %d\n", count);
    return 0;
}
This program first defines the helper functions `get_neighbors` and `is_unholy`. The `get_neighbors` function returns a bitmask representing the neighboring cells in the four cardinal directions for a given cell at `(row, col)`. The `is_unholy` function checks if a given matrix is unholy by iterating over each bit and checking if there is a neighboring 1 in any of the four cardinal directions. The `count_unholy` function implements the dynamic programming approach. It recursively counts the number of unholy matrices with `rows_left` rows remaining. The base case is when `rows_left` is 0, in which case it returns 1 (since an empty matrix is considered unholy). For other cases, it enumerates all possible combinations of the next two rows (`2^16` combinations) using a bitmask. For each combination, it checks if it is unholy by calling `is_unholy` and recursively counts the number of unholy matrices with `rows_left - 2` remaining. The function returns the sum of counts for all combinations. Finally, the `main` function calls `count_unholy(ROWS)` to get the total count of unholy 8x8 matrices and prints the result. Note: This implementation assumes that the matrix is represented as a row-major bit vector, where the least significant bit corresponds to the cell at `(0, 0)`, and the most significant bit corresponds to the cell at `(7, 7)`.

Extract Code

I extracted the following code from that output: 

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#include <stdio.h>

#define ROWS 8
#define COLS 8

unsigned long get_neighbors(int row, int col) {
    unsigned long neighbors = 0;
    if (row > 0)
        neighbors |= 1UL << (row - 1) * COLS + col;
    if (row < ROWS - 1)
        neighbors |= 1UL << (row + 1) * COLS + col;
    if (col > 0)
        neighbors |= 1UL << row * COLS + col - 1;
    if (col < COLS - 1)
        neighbors |= 1UL << row * COLS + col + 1;
    return neighbors;
}

int is_unholy(unsigned long matrix) {
    for (int row = 0; row < ROWS; row++) {
        for (int col = 0; col < COLS; col++) {
            unsigned long bit = 1UL << row * COLS + col;
            if ((matrix & bit) && !(matrix & get_neighbors(row, col)))
                return 0;
        }
    }
    return 1;
}

int count_unholy(int rows_left) {
    if (rows_left == 0)
        return 1;

    int count = 0;
    unsigned long mask = (1UL << (2 * COLS)) - 1;
    for (unsigned long rows = 0; rows <= mask; rows++) {
        if (is_unholy(rows)) {
            count += count_unholy(rows_left - 2);
        }
    }
    return count;
}

int main() {
    int count = count_unholy(ROWS);
    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>

#define ROWS 8
#define COLS 8

unsigned long get_neighbors(int row, int col) {
    unsigned long neighbors = 0;
    if (row > 0)
        neighbors |= 1UL << (row - 1) * COLS + col;
    if (row < ROWS - 1)
        neighbors |= 1UL << (row + 1) * COLS + col;
    if (col > 0)
        neighbors |= 1UL << row * COLS + col - 1;
    if (col < COLS - 1)
        neighbors |= 1UL << row * COLS + col + 1;
    return neighbors;
}

int is_unholy(unsigned long matrix) {
    for (int row = 0; row < ROWS; row++) {
        for (int col = 0; col < COLS; col++) {
            unsigned long bit = 1UL << row * COLS + col;
            if ((matrix & bit) && !(matrix & get_neighbors(row, col)))
                return 0;
        }
    }
    return 1;
}

int count_unholy(int rows_left) {
    if (rows_left == 0)
        return 1;

    int count = 0;
    unsigned long mask = (1UL << (2 * COLS)) - 1;
    for (unsigned long rows = 0; rows <= mask; rows++) {
        if (is_unholy(rows)) {
            count += count_unholy(rows_left - 2);
        }
    }
    return count;
}

int main() {
    int count = count_unholy(ROWS);
    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

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, here's a plan for the program: 1. **Representing the Matrix** - Define a type `matrix_t` as an `unsigned long` to represent an 8x8 binary matrix. - Implement a function `set_bit(matrix_t *m, int row, int col, int val)` to set a bit in the matrix. - Implement a function `get_bit(matrix_t m, int row, int col)` to get the value of a bit in the matrix. 2. **Checking for Unholy Matrix** - Implement a function `is_unholy(matrix_t m)` that checks if a matrix is unholy. - In this function, iterate over each bit in the matrix. - For each set bit (1), check if there are any other set bits in the four cardinal directions. - If a set bit is found to be isolated, return 0 (not unholy). - If all set bits are not isolated, return 1 (unholy). 3. **Counting Unholy Matrices** - Implement a function `count_unholy(int rows)` that recursively counts the number of unholy matrices with `rows` rows. - Base case: If `rows` is 0, return 1 (the empty matrix is considered unholy). - Recursive case: - Initialize a counter variable `count` to 0. - Iterate over all possible combinations of the next two rows (2^16 combinations). - For each combination, recursively count the number of unholy matrices with `rows - 2` rows and add it to `count`. - Return `count`. 4. **Main Function** - In the `main()` function, call `count_unholy(8)` to get the total number of unholy 8x8 matrices. - Print the result. Here's the implementation in C: 

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#include <stdio.h>

typedef unsigned long matrix_t;

void set_bit(matrix_t *m, int row, int col, int val) {
    int index = row * 8 + col;
    if (val)
        *m |= (1UL << index);
    else
        *m &= ~(1UL << index);
}

int get_bit(matrix_t m, int row, int col) {
    int index = row * 8 + col;
    return (m >> index) & 1;
}

int is_unholy(matrix_t m) {
    for (int i = 0; i < 64; i++) {
        int row = i / 8;
        int col = i % 8;
        if (get_bit(m, row, col)) {
            if ((row > 0 && get_bit(m, row - 1, col)) ||
                (row < 7 && get_bit(m, row + 1, col)) ||
                (col > 0 && get_bit(m, row, col - 1)) ||
                (col < 7 && get_bit(m, row, col + 1)))
                continue;
            return 0;
        }
    }
    return 1;
}

int count_unholy(int rows) {
    if (rows == 0)
        return 1;

    int count = 0;
    for (matrix_t next_rows = 0; next_rows < (1UL << 16); next_rows++) {
        count += count_unholy(rows - 2);
    }
    return count;
}

int main() {
    int num_unholy = count_unholy(8);
    printf("Number of unholy 8x8 matrices: %d\n", num_unholy);
    return 0;
}
Note that this implementation has a time complexity of O(2^16 * 2^(8*6)), which is extremely inefficient for larger matrices. For an 8x8 matrix, it should be reasonably fast, but for larger matrices, you may need to use more advanced techniques like dynamic programming or bit-level parallelism to improve the performance.

Extract Code

I extracted the following code from that output: 

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#include <stdio.h>

typedef unsigned long matrix_t;

void set_bit(matrix_t *m, int row, int col, int val) {
    int index = row * 8 + col;
    if (val)
        *m |= (1UL << index);
    else
        *m &= ~(1UL << index);
}

int get_bit(matrix_t m, int row, int col) {
    int index = row * 8 + col;
    return (m >> index) & 1;
}

int is_unholy(matrix_t m) {
    for (int i = 0; i < 64; i++) {
        int row = i / 8;
        int col = i % 8;
        if (get_bit(m, row, col)) {
            if ((row > 0 && get_bit(m, row - 1, col)) ||
                (row < 7 && get_bit(m, row + 1, col)) ||
                (col > 0 && get_bit(m, row, col - 1)) ||
                (col < 7 && get_bit(m, row, col + 1)))
                continue;
            return 0;
        }
    }
    return 1;
}

int count_unholy(int rows) {
    if (rows == 0)
        return 1;

    int count = 0;
    for (matrix_t next_rows = 0; next_rows < (1UL << 16); next_rows++) {
        count += count_unholy(rows - 2);
    }
    return count;
}

int main() {
    int num_unholy = count_unholy(8);
    printf("Number of unholy 8x8 matrices: %d\n", num_unholy);
    return 0;
}

Run Code Interpreter

Running the following program: 

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#include <stdio.h>

typedef unsigned long matrix_t;

void set_bit(matrix_t *m, int row, int col, int val) {
    int index = row * 8 + col;
    if (val)
        *m |= (1UL << index);
    else
        *m &= ~(1UL << index);
}

int get_bit(matrix_t m, int row, int col) {
    int index = row * 8 + col;
    return (m >> index) & 1;
}

int is_unholy(matrix_t m) {
    for (int i = 0; i < 64; i++) {
        int row = i / 8;
        int col = i % 8;
        if (get_bit(m, row, col)) {
            if ((row > 0 && get_bit(m, row - 1, col)) ||
                (row < 7 && get_bit(m, row + 1, col)) ||
                (col > 0 && get_bit(m, row, col - 1)) ||
                (col < 7 && get_bit(m, row, col + 1)))
                continue;
            return 0;
        }
    }
    return 1;
}

int count_unholy(int rows) {
    if (rows == 0)
        return 1;

    int count = 0;
    for (matrix_t next_rows = 0; next_rows < (1UL << 16); next_rows++) {
        count += count_unholy(rows - 2);
    }
    return count;
}

int main() {
    int num_unholy = count_unholy(8);
    printf("Number of unholy 8x8 matrices: %d\n", num_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

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, here's a plan for the program: 1. **Define Constants and Data Structures** - Define constants for the matrix size (8x8) and the number of bits required to represent the matrix (64 bits). - Define an `unsigned long long` data type to represent the matrix (64 bits). 2. **Write the `is_unholy` Function** - Implement a function `is_unholy` that takes an `unsigned long long` matrix and returns `1` if the matrix is unholy (no isolated 1s), and `0` otherwise. - Use bit-shifting and masking operations to check for isolated 1s in the matrix. 3. **Implement the Dynamic Programming Solution** - Define a function `count_unholy_matrices` that takes the number of rows `n` and returns the count of unholy matrices with `n` rows. - Base case: If `n` is 0, return 1 (empty matrix is considered unholy). - Recursive case: - Enumerate all possible combinations of the next two rows (2^16 combinations). - For each combination, check if it is compatible with the previous rows (no isolated 1s at the boundary). - If compatible, recursively count the number of unholy matrices for the remaining rows and add it to the total count. - Use memoization (cache the results) to avoid redundant computations. 4. **Main Function** - In the `main` function, call the `count_unholy_matrices` function with `n=8` and print the result. Here's the C implementation: 

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

#define MATRIX_SIZE 8
#define MATRIX_BITS (MATRIX_SIZE * MATRIX_SIZE)

// Cache for memoization
unsigned long long memo[MATRIX_SIZE + 1];

// Check if a matrix is unholy
int is_unholy(uint64_t matrix) {
    uint64_t mask = 0x8000000000000000;
    for (int i = 0; i < MATRIX_BITS; i++) {
        if (matrix & mask) {
            // Check if there are neighbors in cardinal directions
            uint64_t neighbors = (matrix & (mask << 1)) | (matrix & (mask >> 1)) |
                                 (matrix & (mask << MATRIX_SIZE)) | (matrix & (mask >> MATRIX_SIZE));
            if (!neighbors)
                return 0; // Isolated 1 found, matrix is not unholy
        }
        mask >>= 1;
    }
    return 1; // No isolated 1s found, matrix is unholy
}

// Count the number of unholy matrices with n rows
unsigned long long count_unholy_matrices(int n) {
    if (n == 0)
        return 1; // Empty matrix is considered unholy

    if (memo[n] != 0)
        return memo[n]; // Return cached result

    unsigned long long count = 0;
    uint64_t mask = (1ULL << (2 * MATRIX_SIZE)) - 1; // Mask for the next two rows

    for (uint64_t rows = 0; rows <= mask; rows++) {
        if (is_unholy(rows)) {
            count += count_unholy_matrices(n - 2); // Recurse for the remaining rows
        }
    }

    memo[n] = count; // Cache the result
    return count;
}

int main() {
    unsigned long long count = count_unholy_matrices(MATRIX_SIZE);
    printf("Number of unholy 8x8 matrices: %llu\n", count);
    return 0;
}
This program uses bit operations and dynamic programming to efficiently count the number of unholy 8x8 matrices. The `is_unholy` function checks if a matrix is unholy by iterating over all bits and checking for isolated 1s. The `count_unholy_matrices` function recursively enumerates all possible combinations of the next two rows and checks if they are compatible with the previous rows. The results are cached using memoization to avoid redundant computations. Note that this solution has a time complexity of O(2^16 * 8) and a space complexity of O(8), which is reasonable for small matrix sizes like 8x8. However, for larger matrix sizes, the time and space complexity may become impractical.

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 MATRIX_BITS (MATRIX_SIZE * MATRIX_SIZE)

// Cache for memoization
unsigned long long memo[MATRIX_SIZE + 1];

// Check if a matrix is unholy
int is_unholy(uint64_t matrix) {
    uint64_t mask = 0x8000000000000000;
    for (int i = 0; i < MATRIX_BITS; i++) {
        if (matrix & mask) {
            // Check if there are neighbors in cardinal directions
            uint64_t neighbors = (matrix & (mask << 1)) | (matrix & (mask >> 1)) |
                                 (matrix & (mask << MATRIX_SIZE)) | (matrix & (mask >> MATRIX_SIZE));
            if (!neighbors)
                return 0; // Isolated 1 found, matrix is not unholy
        }
        mask >>= 1;
    }
    return 1; // No isolated 1s found, matrix is unholy
}

// Count the number of unholy matrices with n rows
unsigned long long count_unholy_matrices(int n) {
    if (n == 0)
        return 1; // Empty matrix is considered unholy

    if (memo[n] != 0)
        return memo[n]; // Return cached result

    unsigned long long count = 0;
    uint64_t mask = (1ULL << (2 * MATRIX_SIZE)) - 1; // Mask for the next two rows

    for (uint64_t rows = 0; rows <= mask; rows++) {
        if (is_unholy(rows)) {
            count += count_unholy_matrices(n - 2); // Recurse for the remaining rows
        }
    }

    memo[n] = count; // Cache the result
    return count;
}

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

Run Code Interpreter

Running the following program: 

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

#define MATRIX_SIZE 8
#define MATRIX_BITS (MATRIX_SIZE * MATRIX_SIZE)

// Cache for memoization
unsigned long long memo[MATRIX_SIZE + 1];

// Check if a matrix is unholy
int is_unholy(uint64_t matrix) {
    uint64_t mask = 0x8000000000000000;
    for (int i = 0; i < MATRIX_BITS; i++) {
        if (matrix & mask) {
            // Check if there are neighbors in cardinal directions
            uint64_t neighbors = (matrix & (mask << 1)) | (matrix & (mask >> 1)) |
                                 (matrix & (mask << MATRIX_SIZE)) | (matrix & (mask >> MATRIX_SIZE));
            if (!neighbors)
                return 0; // Isolated 1 found, matrix is not unholy
        }
        mask >>= 1;
    }
    return 1; // No isolated 1s found, matrix is unholy
}

// Count the number of unholy matrices with n rows
unsigned long long count_unholy_matrices(int n) {
    if (n == 0)
        return 1; // Empty matrix is considered unholy

    if (memo[n] != 0)
        return memo[n]; // Return cached result

    unsigned long long count = 0;
    uint64_t mask = (1ULL << (2 * MATRIX_SIZE)) - 1; // Mask for the next two rows

    for (uint64_t rows = 0; rows <= mask; rows++) {
        if (is_unholy(rows)) {
            count += count_unholy_matrices(n - 2); // Recurse for the remaining rows
        }
    }

    memo[n] = count; // Cache the result
    return count;
}

int main() {
    unsigned long long count = count_unholy_matrices(MATRIX_SIZE);
    printf("Number of unholy 8x8 matrices: %llu\n", count);
    return 0;
}
And got the output: Number of unholy 8x8 matrices: 347131339034603841

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, here's a plan for the program: 1. **Represent the matrix using an unsigned long integer** - Each bit in the unsigned long represents a cell in the 8x8 matrix. - Bit manipulation operations like shifting and masking can be used to check and modify the matrix. 2. **Define a function `is_unholy` to check if a matrix is unholy** - The function takes an unsigned long representing the matrix as input. - It checks each cell in the matrix to see if there is an isolated 1. - If an isolated 1 is found, the function returns 0 (false). - If no isolated 1 is found, the function returns 1 (true). 3. **Define a dynamic programming function `count_unholy` to count the number of unholy matrices** - The function takes an integer `n` representing the number of rows remaining in the matrix. - Base case: When `n` is 0, return 1 (as an empty matrix is considered unholy). - Recursive case: - Iterate over all possible combinations of the next two rows (2^16 combinations). - For each combination, recursively call `count_unholy` with `n-2` and add the result to a running total. - Return the running total. 4. **Implement the `is_unholy` function** - Use bit manipulation operations to check if a cell is isolated or not. - Check the cells above, below, to the left, and to the right of the current cell. - If any of the neighboring cells have a 1, the current cell is not isolated. 5. **Implement the `count_unholy` function** - Use the dynamic programming approach as described in step 3. - Utilize bit manipulation operations to generate all possible combinations of the next two rows. - Recursively call `count_unholy` for each combination and accumulate the results. 6. **Write the main function** - Call the `count_unholy` function with `n=8` (representing the entire 8x8 matrix). - Print the result. Here's the C implementation of the plan: 

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#include <stdio.h>

// Function to check if a matrix is unholy
int is_unholy(unsigned long matrix) {
    unsigned long mask = 0x8040201008040201; // Mask for checking neighbors
    unsigned long isolated_mask = ~(mask | (mask << 1) | (mask >> 1)); // Mask for isolated cells

    // Check if there are any isolated 1s
    if ((matrix & isolated_mask) != 0)
        return 0; // Matrix is not unholy

    return 1; // Matrix is unholy
}

// Dynamic programming function to count the number of unholy matrices
unsigned long count_unholy(int n) {
    if (n == 0)
        return 1; // Base case: an empty matrix is considered unholy

    unsigned long count = 0;

    // Iterate over all possible combinations of the next two rows
    for (unsigned long rows = 0; rows < (1UL << 16); rows++) {
        unsigned long matrix = rows << (64 - 16); // Shift rows to the top
        if (is_unholy(matrix))
            count += count_unholy(n - 2); // Recursively count unholy matrices with n-2 rows
    }

    return count;
}

int main() {
    unsigned long count = count_unholy(8);
    printf("Number of unholy 8x8 matrices: %lu\n", count);
    return 0;
}
This program uses bit manipulation to represent the matrix and checks for isolated 1s using masks. The `count_unholy` function uses dynamic programming to recursively count the number of unholy matrices by enumerating all possible combinations of the next two rows and recursively solving for the remaining rows. Note: This program may take a significant amount of time to run due to the large number of recursive calls required to enumerate all possible matrices.

Extract Code

I extracted the following code from that output: 

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#include <stdio.h>

// Function to check if a matrix is unholy
int is_unholy(unsigned long matrix) {
    unsigned long mask = 0x8040201008040201; // Mask for checking neighbors
    unsigned long isolated_mask = ~(mask | (mask << 1) | (mask >> 1)); // Mask for isolated cells

    // Check if there are any isolated 1s
    if ((matrix & isolated_mask) != 0)
        return 0; // Matrix is not unholy

    return 1; // Matrix is unholy
}

// Dynamic programming function to count the number of unholy matrices
unsigned long count_unholy(int n) {
    if (n == 0)
        return 1; // Base case: an empty matrix is considered unholy

    unsigned long count = 0;

    // Iterate over all possible combinations of the next two rows
    for (unsigned long rows = 0; rows < (1UL << 16); rows++) {
        unsigned long matrix = rows << (64 - 16); // Shift rows to the top
        if (is_unholy(matrix))
            count += count_unholy(n - 2); // Recursively count unholy matrices with n-2 rows
    }

    return count;
}

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

Run Code Interpreter

Running the following program: 

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#include <stdio.h>

// Function to check if a matrix is unholy
int is_unholy(unsigned long matrix) {
    unsigned long mask = 0x8040201008040201; // Mask for checking neighbors
    unsigned long isolated_mask = ~(mask | (mask << 1) | (mask >> 1)); // Mask for isolated cells

    // Check if there are any isolated 1s
    if ((matrix & isolated_mask) != 0)
        return 0; // Matrix is not unholy

    return 1; // Matrix is unholy
}

// Dynamic programming function to count the number of unholy matrices
unsigned long count_unholy(int n) {
    if (n == 0)
        return 1; // Base case: an empty matrix is considered unholy

    unsigned long count = 0;

    // Iterate over all possible combinations of the next two rows
    for (unsigned long rows = 0; rows < (1UL << 16); rows++) {
        unsigned long matrix = rows << (64 - 16); // Shift rows to the top
        if (is_unholy(matrix))
            count += count_unholy(n - 2); // Recursively count unholy matrices with n-2 rows
    }

    return count;
}

int main() {
    unsigned long count = count_unholy(8);
    printf("Number of unholy 8x8 matrices: %lu\n", count);
    return 0;
}
And got the output: Number of unholy 8x8 matrices: 1048576

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False