Initial Query

A binary 8x8 matrix is 'unholy' if there are no isolated 1s. A 1 is isolated if there isn't another 1 in any of the 4 direct cardinal directions. Write a C program that counts the total number of unholy 8x8 matricies.

LLM Generation

Query

A binary 8x8 matrix is 'unholy' if there are no isolated 1s. A 1 is isolated if there isn't another 1 in any of the 4 direct cardinal directions. Write a C program that counts the total number of unholy 8x8 matricies.

Output

To solve this problem, we need to count the number of 8x8 binary matrices where no 1 is isolated. A 1 is considered isolated if it does not have another 1 in any of the four direct cardinal directions (up, down, left, right). Given the size of the matrix, a brute-force approach is infeasible, so we use dynamic programming (DP) with precomputed validity checks to efficiently count valid matrices. ### Approach 1. **Precomputation**: - **Validity Check**: For each possible pair of rows (current and next), determine if all 1s in the next row have horizontal or vertical neighbors. - **Required Next Row**: For each row, compute the positions that must be set to 1 in the next row to avoid isolated 1s. 2. **Dynamic Programming**: - **Initialization**: Start with all possible configurations for the first row. - **Transition**: For each row, transition to the next row's configurations that are valid and meet the required positions. - **Final Count**: Sum up all valid configurations for the last row. ### Solution Code 

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

#define ROWS 8
#define COLS 8

uint8_t valid[256][256];
uint8_t required_next_table[256];

void precompute() {
    // Precompute valid transitions between current and next row
    for (int current = 0; current < 256; current++) {
        for (int next = 0; next < 256; next++) {
            valid[current][next] = 1;
            for (int j = 0; j < COLS; j++) {
                if ((next >> j) & 1) {
                    int has_horizontal = 0;
                    if (j > 0 && ((next >> (j - 1)) & 1)) has_horizontal = 1;
                    if (j < COLS - 1 && ((next >> (j + 1)) & 1)) has_horizontal = 1;
                    int has_vertical = (current >> j) & 1;
                    if (!has_horizontal && !has_vertical) {
                        valid[current][next] = 0;
                        break;
                    }
                }
            }
        }
    }

    // Precompute required_next for each current row
    for (int current = 0; current < 256; current++) {
        uint8_t required = 0;
        for (int j = 0; j < COLS; j++) {
            if ((current >> j) & 1) {
                int left = (j > 0) ? ((current >> (j - 1)) & 1) : 0;
                int right = (j < COLS - 1) ? ((current >> (j + 1)) & 1) : 0;
                if (!left && !right) {
                    required |= (1 << j);
                }
            }
        }
        required_next_table[current] = required;
    }
}

int main() {
    precompute();

    uint64_t dp[ROWS][256] = {0};

    // Initialize row 0
    for (int current = 0; current < 256; current++) {
        dp[0][current] = 1;
    }

    // Process row 0 to row 1
    for (int current = 0; current < 256; current++) {
        uint8_t required = required_next_table[current];
        for (int next = 0; next < 256; next++) {
            if ((next & required) == required && valid[current][next]) {
                dp[1][next] += dp[0][current];
            }
        }
    }

    // Process rows 1 to 6
    for (int row = 1; row < ROWS - 1; row++) {
        for (int current = 0; current < 256; current++) {
            if (dp[row][current] == 0) continue;
            for (int next = 0; next < 256; next++) {
                if (valid[current][next]) {
                    dp[row + 1][next] += dp[row][current];
                }
            }
        }
    }

    // Sum all valid configurations for the last row (row 7)
    uint64_t total = 0;
    for (int current = 0; current < 256; current++) {
        total += dp[ROWS - 1][current];
    }

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

    return 0;
}
### Explanation - **Precomputation**: We precompute the validity of transitions between rows and the required positions for the next row to avoid isolated 1s. - **Dynamic Programming Initialization**: We start with all possible configurations for the first row. - **Transitions**: For each row, we transition to valid configurations of the next row, ensuring no isolated 1s. - **Final Count**: Sum all valid configurations of the last row, ensuring all 1s are properly adjacent. This approach efficiently counts valid matrices using dynamic programming and precomputation, avoiding the infeasibility of brute-force enumeration.

Extract Code

I extracted the following code from that output: 

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

#define ROWS 8
#define COLS 8

uint8_t valid[256][256];
uint8_t required_next_table[256];

void precompute() {
    // Precompute valid transitions between current and next row
    for (int current = 0; current < 256; current++) {
        for (int next = 0; next < 256; next++) {
            valid[current][next] = 1;
            for (int j = 0; j < COLS; j++) {
                if ((next >> j) & 1) {
                    int has_horizontal = 0;
                    if (j > 0 && ((next >> (j - 1)) & 1)) has_horizontal = 1;
                    if (j < COLS - 1 && ((next >> (j + 1)) & 1)) has_horizontal = 1;
                    int has_vertical = (current >> j) & 1;
                    if (!has_horizontal && !has_vertical) {
                        valid[current][next] = 0;
                        break;
                    }
                }
            }
        }
    }

    // Precompute required_next for each current row
    for (int current = 0; current < 256; current++) {
        uint8_t required = 0;
        for (int j = 0; j < COLS; j++) {
            if ((current >> j) & 1) {
                int left = (j > 0) ? ((current >> (j - 1)) & 1) : 0;
                int right = (j < COLS - 1) ? ((current >> (j + 1)) & 1) : 0;
                if (!left && !right) {
                    required |= (1 << j);
                }
            }
        }
        required_next_table[current] = required;
    }
}

int main() {
    precompute();

    uint64_t dp[ROWS][256] = {0};

    // Initialize row 0
    for (int current = 0; current < 256; current++) {
        dp[0][current] = 1;
    }

    // Process row 0 to row 1
    for (int current = 0; current < 256; current++) {
        uint8_t required = required_next_table[current];
        for (int next = 0; next < 256; next++) {
            if ((next & required) == required && valid[current][next]) {
                dp[1][next] += dp[0][current];
            }
        }
    }

    // Process rows 1 to 6
    for (int row = 1; row < ROWS - 1; row++) {
        for (int current = 0; current < 256; current++) {
            if (dp[row][current] == 0) continue;
            for (int next = 0; next < 256; next++) {
                if (valid[current][next]) {
                    dp[row + 1][next] += dp[row][current];
                }
            }
        }
    }

    // Sum all valid configurations for the last row (row 7)
    uint64_t total = 0;
    for (int current = 0; current < 256; current++) {
        total += dp[ROWS - 1][current];
    }

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

    return 0;
}

Run Code Interpreter

Running the following program: 

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

#define ROWS 8
#define COLS 8

uint8_t valid[256][256];
uint8_t required_next_table[256];

void precompute() {
    // Precompute valid transitions between current and next row
    for (int current = 0; current < 256; current++) {
        for (int next = 0; next < 256; next++) {
            valid[current][next] = 1;
            for (int j = 0; j < COLS; j++) {
                if ((next >> j) & 1) {
                    int has_horizontal = 0;
                    if (j > 0 && ((next >> (j - 1)) & 1)) has_horizontal = 1;
                    if (j < COLS - 1 && ((next >> (j + 1)) & 1)) has_horizontal = 1;
                    int has_vertical = (current >> j) & 1;
                    if (!has_horizontal && !has_vertical) {
                        valid[current][next] = 0;
                        break;
                    }
                }
            }
        }
    }

    // Precompute required_next for each current row
    for (int current = 0; current < 256; current++) {
        uint8_t required = 0;
        for (int j = 0; j < COLS; j++) {
            if ((current >> j) & 1) {
                int left = (j > 0) ? ((current >> (j - 1)) & 1) : 0;
                int right = (j < COLS - 1) ? ((current >> (j + 1)) & 1) : 0;
                if (!left && !right) {
                    required |= (1 << j);
                }
            }
        }
        required_next_table[current] = required;
    }
}

int main() {
    precompute();

    uint64_t dp[ROWS][256] = {0};

    // Initialize row 0
    for (int current = 0; current < 256; current++) {
        dp[0][current] = 1;
    }

    // Process row 0 to row 1
    for (int current = 0; current < 256; current++) {
        uint8_t required = required_next_table[current];
        for (int next = 0; next < 256; next++) {
            if ((next & required) == required && valid[current][next]) {
                dp[1][next] += dp[0][current];
            }
        }
    }

    // Process rows 1 to 6
    for (int row = 1; row < ROWS - 1; row++) {
        for (int current = 0; current < 256; current++) {
            if (dp[row][current] == 0) continue;
            for (int next = 0; next < 256; next++) {
                if (valid[current][next]) {
                    dp[row + 1][next] += dp[row][current];
                }
            }
        }
    }

    // Sum all valid configurations for the last row (row 7)
    uint64_t total = 0;
    for (int current = 0; current < 256; current++) {
        total += dp[ROWS - 1][current];
    }

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

    return 0;
}
And got the output: 132797167652221230 main.c: In function 'main': main.c:85:16: warning: format '%llu' expects argument of type 'long long unsigned int', but argument 2 has type 'uint64_t' {aka 'long unsigned int'} [-Wformat=] 85 | printf("%llu\n", total); | ~~~^ ~~~~~ | | | | | uint64_t {aka long unsigned int} | long long unsigned int | %lu

Substring Evaluation

Testing if the previous output contains the string 1121231537486377866: False