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

To solve this problem, we need to count the number of 8x8 binary matrices that are 'unholy', meaning no 1 in the matrix is isolated. A 1 is isolated if it does not have another 1 in any of the four cardinal directions (up, down, left, right). The solution uses dynamic programming (DP) with bitwise operations to efficiently check the validity of each row configuration. ### Approach 1. **Bitwise Representation**: Each row of the matrix is represented as an 8-bit integer, allowing efficient bitwise operations. 2. **Dynamic Programming**: Use a DP approach to build the matrix row by row, keeping track of valid configurations of the last two rows. This helps in efficiently checking the validity of each new row added. 3. **Validity Checks**: For each row added, check if all 1s in the previous rows have adjacent 1s either in the same row or in the neighboring rows (previous and next rows). ### Solution Code 

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

typedef uint8_t row_t;

bool is_row_valid_with_next(row_t row, row_t next_row) {
    for (int j = 0; j < 8; j++) {
        if (row & (1 << j)) {
            bool down = (next_row & (1 << j)) != 0;
            bool left = (j < 7) && (row & (1 << (j + 1))) != 0;
            bool right = (j > 0) && (row & (1 << (j - 1))) != 0;
            if (!(down || left || right)) {
                return false;
            }
        }
    }
    return true;
}

bool is_prev_valid(row_t prev_prev, row_t prev, row_t next_row) {
    for (int j = 0; j < 8; j++) {
        if (prev & (1 << j)) {
            bool up = (prev_prev & (1 << j)) != 0;
            bool down = (next_row & (1 << j)) != 0;
            bool left = (j < 7) && (prev & (1 << (j + 1))) != 0;
            bool right = (j > 0) && (prev & (1 << (j - 1))) != 0;
            if (!(up || down || left || right)) {
                return false;
            }
        }
    }
    return true;
}

bool is_last_row_valid(row_t last_row, row_t prev_row) {
    for (int j = 0; j < 8; j++) {
        if (last_row & (1 << j)) {
            bool up = (prev_row & (1 << j)) != 0;
            bool left = (j < 7) && (last_row & (1 << (j + 1))) != 0;
            bool right = (j > 0) && (last_row & (1 << (j - 1))) != 0;
            if (!(up || left || right)) {
                return false;
            }
        }
    }
    return true;
}

int main() {
    uint64_t dp_prev[256][256] = {0}; // Previous DP state
    uint64_t dp_current[256][256] = {0}; // Current DP state

    // Initialize for k=2 (rows 0 and 1)
    for (row_t r0 = 0; r0 < 256; ++r0) {
        for (row_t r1 = 0; r1 < 256; ++r1) {
            if (is_row_valid_with_next(r0, r1)) {
                dp_prev[r0][r1] = 1;
            }
        }
    }

    // Process steps for k=3 to k=7 (adding rows 2 to 6)
    for (int step = 3; step <= 7; ++step) {
        // Reset current DP to zero
        for (row_t i = 0; i < 256; ++i) {
            for (row_t j = 0; j < 256; ++j) {
                dp_current[i][j] = 0;
            }
        }

        // Iterate through all possible previous states
        for (row_t prev_prev = 0; prev_prev < 256; ++prev_prev) {
            for (row_t prev = 0; prev < 256; ++prev) {
                uint64_t count = dp_prev[prev_prev][prev];
                if (count == 0) continue;

                // Try all possible next rows
                for (row_t next_row = 0; next_row < 256; ++next_row) {
                    if (is_prev_valid(prev_prev, prev, next_row)) {
                        dp_current[prev][next_row] += count;
                    }
                }
            }
        }

        // Swap dp_prev and dp_current for the next iteration
        uint64_t (*temp)[256] = dp_prev;
        dp_prev = dp_current;
        dp_current = temp;
    }

    // After processing all rows up to 7, check the validity of the 8th row
    uint64_t total = 0;
    for (row_t prev = 0; prev < 256; ++prev) {
        for (row_t curr = 0; curr < 256; ++curr) {
            if (dp_prev[prev][curr] > 0 && is_last_row_valid(curr, prev)) {
                total += dp_prev[prev][curr];
            }
        }
    }

    printf("%llu\n", total);

    return 0;
}
### Explanation 1. **Initialization**: The DP table is initialized for the first two rows, checking if each configuration of the first row is valid given the second row. 2. **Dynamic Programming Transitions**: For each subsequent row, the DP state is updated by considering all possible next rows and checking if the current row configuration remains valid. 3. **Final Check**: After processing all rows up to the 7th row, the final check ensures the last row is valid without a subsequent row, summing all valid configurations to get the total count of unholy matrices. This approach efficiently uses bitwise operations and dynamic programming to handle the large state space, ensuring that all valid configurations are counted without explicitly generating all possible matrices.

Extract Code

I extracted the following code from that output: 

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

typedef uint8_t row_t;

bool is_row_valid_with_next(row_t row, row_t next_row) {
    for (int j = 0; j < 8; j++) {
        if (row & (1 << j)) {
            bool down = (next_row & (1 << j)) != 0;
            bool left = (j < 7) && (row & (1 << (j + 1))) != 0;
            bool right = (j > 0) && (row & (1 << (j - 1))) != 0;
            if (!(down || left || right)) {
                return false;
            }
        }
    }
    return true;
}

bool is_prev_valid(row_t prev_prev, row_t prev, row_t next_row) {
    for (int j = 0; j < 8; j++) {
        if (prev & (1 << j)) {
            bool up = (prev_prev & (1 << j)) != 0;
            bool down = (next_row & (1 << j)) != 0;
            bool left = (j < 7) && (prev & (1 << (j + 1))) != 0;
            bool right = (j > 0) && (prev & (1 << (j - 1))) != 0;
            if (!(up || down || left || right)) {
                return false;
            }
        }
    }
    return true;
}

bool is_last_row_valid(row_t last_row, row_t prev_row) {
    for (int j = 0; j < 8; j++) {
        if (last_row & (1 << j)) {
            bool up = (prev_row & (1 << j)) != 0;
            bool left = (j < 7) && (last_row & (1 << (j + 1))) != 0;
            bool right = (j > 0) && (last_row & (1 << (j - 1))) != 0;
            if (!(up || left || right)) {
                return false;
            }
        }
    }
    return true;
}

int main() {
    uint64_t dp_prev[256][256] = {0}; // Previous DP state
    uint64_t dp_current[256][256] = {0}; // Current DP state

    // Initialize for k=2 (rows 0 and 1)
    for (row_t r0 = 0; r0 < 256; ++r0) {
        for (row_t r1 = 0; r1 < 256; ++r1) {
            if (is_row_valid_with_next(r0, r1)) {
                dp_prev[r0][r1] = 1;
            }
        }
    }

    // Process steps for k=3 to k=7 (adding rows 2 to 6)
    for (int step = 3; step <= 7; ++step) {
        // Reset current DP to zero
        for (row_t i = 0; i < 256; ++i) {
            for (row_t j = 0; j < 256; ++j) {
                dp_current[i][j] = 0;
            }
        }

        // Iterate through all possible previous states
        for (row_t prev_prev = 0; prev_prev < 256; ++prev_prev) {
            for (row_t prev = 0; prev < 256; ++prev) {
                uint64_t count = dp_prev[prev_prev][prev];
                if (count == 0) continue;

                // Try all possible next rows
                for (row_t next_row = 0; next_row < 256; ++next_row) {
                    if (is_prev_valid(prev_prev, prev, next_row)) {
                        dp_current[prev][next_row] += count;
                    }
                }
            }
        }

        // Swap dp_prev and dp_current for the next iteration
        uint64_t (*temp)[256] = dp_prev;
        dp_prev = dp_current;
        dp_current = temp;
    }

    // After processing all rows up to 7, check the validity of the 8th row
    uint64_t total = 0;
    for (row_t prev = 0; prev < 256; ++prev) {
        for (row_t curr = 0; curr < 256; ++curr) {
            if (dp_prev[prev][curr] > 0 && is_last_row_valid(curr, prev)) {
                total += dp_prev[prev][curr];
            }
        }
    }

    printf("%llu\n", total);

    return 0;
}

Run Code Interpreter

Running the following program: 

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

typedef uint8_t row_t;

bool is_row_valid_with_next(row_t row, row_t next_row) {
    for (int j = 0; j < 8; j++) {
        if (row & (1 << j)) {
            bool down = (next_row & (1 << j)) != 0;
            bool left = (j < 7) && (row & (1 << (j + 1))) != 0;
            bool right = (j > 0) && (row & (1 << (j - 1))) != 0;
            if (!(down || left || right)) {
                return false;
            }
        }
    }
    return true;
}

bool is_prev_valid(row_t prev_prev, row_t prev, row_t next_row) {
    for (int j = 0; j < 8; j++) {
        if (prev & (1 << j)) {
            bool up = (prev_prev & (1 << j)) != 0;
            bool down = (next_row & (1 << j)) != 0;
            bool left = (j < 7) && (prev & (1 << (j + 1))) != 0;
            bool right = (j > 0) && (prev & (1 << (j - 1))) != 0;
            if (!(up || down || left || right)) {
                return false;
            }
        }
    }
    return true;
}

bool is_last_row_valid(row_t last_row, row_t prev_row) {
    for (int j = 0; j < 8; j++) {
        if (last_row & (1 << j)) {
            bool up = (prev_row & (1 << j)) != 0;
            bool left = (j < 7) && (last_row & (1 << (j + 1))) != 0;
            bool right = (j > 0) && (last_row & (1 << (j - 1))) != 0;
            if (!(up || left || right)) {
                return false;
            }
        }
    }
    return true;
}

int main() {
    uint64_t dp_prev[256][256] = {0}; // Previous DP state
    uint64_t dp_current[256][256] = {0}; // Current DP state

    // Initialize for k=2 (rows 0 and 1)
    for (row_t r0 = 0; r0 < 256; ++r0) {
        for (row_t r1 = 0; r1 < 256; ++r1) {
            if (is_row_valid_with_next(r0, r1)) {
                dp_prev[r0][r1] = 1;
            }
        }
    }

    // Process steps for k=3 to k=7 (adding rows 2 to 6)
    for (int step = 3; step <= 7; ++step) {
        // Reset current DP to zero
        for (row_t i = 0; i < 256; ++i) {
            for (row_t j = 0; j < 256; ++j) {
                dp_current[i][j] = 0;
            }
        }

        // Iterate through all possible previous states
        for (row_t prev_prev = 0; prev_prev < 256; ++prev_prev) {
            for (row_t prev = 0; prev < 256; ++prev) {
                uint64_t count = dp_prev[prev_prev][prev];
                if (count == 0) continue;

                // Try all possible next rows
                for (row_t next_row = 0; next_row < 256; ++next_row) {
                    if (is_prev_valid(prev_prev, prev, next_row)) {
                        dp_current[prev][next_row] += count;
                    }
                }
            }
        }

        // Swap dp_prev and dp_current for the next iteration
        uint64_t (*temp)[256] = dp_prev;
        dp_prev = dp_current;
        dp_current = temp;
    }

    // After processing all rows up to 7, check the validity of the 8th row
    uint64_t total = 0;
    for (row_t prev = 0; prev < 256; ++prev) {
        for (row_t curr = 0; curr < 256; ++curr) {
            if (dp_prev[prev][curr] > 0 && is_last_row_valid(curr, prev)) {
                total += dp_prev[prev][curr];
            }
        }
    }

    printf("%llu\n", total);

    return 0;
}
And got the output: main.c: In function 'main': main.c:90:17: error: assignment to expression with array type 90 | dp_prev = dp_current; | ^ main.c:91:20: error: assignment to expression with array type 91 | dp_current = temp; | ^ main.c:104:16: warning: format '%llu' expects argument of type 'long long unsigned int', but argument 2 has type 'uint64_t' {aka 'long unsigned int'} [-Wformat=] 104 | printf("%llu\n", total); | ~~~^ ~~~~~ | | | | | uint64_t {aka long unsigned int} | long long unsigned int | %lu main.sh: line 2: ./a.out: No such file or directory

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False