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Project Euler: Problem 11

Problem 11 is fairly straightforward to brute-force, although my solution uses more code than is strictly necessary. Since one of the things I enjoy most about Python is its support for functional programming concepts, I went with a reduce-based solution.

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    """Solves Problem 11 from Project Euler."""

    import operator

    OFFSET = 3
    GRID = [
        [8, 2, 22, 97, 38, 15, 0, 40, 0, 75, 4, 5, 7, 78, 52, 12, 50, 77, 91, 8],
        [49, 49, 99, 40, 17, 81, 18, 57, 60, 87, 17, 40, 98, 43, 69, 48, 4, 56, 62, 0],
        [81, 49, 31, 73, 55, 79, 14, 29, 93, 71, 40, 67, 53, 88, 30, 3, 49, 13, 36, 65],
        [52, 70, 95, 23, 4, 60, 11, 42, 69, 24, 68, 56, 1, 32, 56, 71, 37, 2, 36, 91],
        [22, 31, 16, 71, 51, 67, 63, 89, 41, 92, 36, 54, 22, 40, 40, 28, 66, 33, 13, 80],
        [24, 47, 32, 60, 99, 3, 45, 2, 44, 75, 33, 53, 78, 36, 84, 20, 35, 17, 12, 50],
        [32, 98, 81, 28, 64, 23, 67, 10, 26, 38, 40, 67, 59, 54, 70, 66, 18, 38, 64, 70],
        [67, 26, 20, 68, 2, 62, 12, 20, 95, 63, 94, 39, 63, 8, 40, 91, 66, 49, 94, 21],
        [24, 55, 58, 5, 66, 73, 99, 26, 97, 17, 78, 78, 96, 83, 14, 88, 34, 89, 63, 72],
        [21, 36, 23, 9, 75, 0, 76, 44, 20, 45, 35, 14, 0, 61, 33, 97, 34, 31, 33, 95],
        [78, 17, 53, 28, 22, 75, 31, 67, 15, 94, 3, 80, 4, 62, 16, 14, 9, 53, 56, 92],
        [16, 39, 5, 42, 96, 35, 31, 47, 55, 58, 88, 24, 0, 17, 54, 24, 36, 29, 85, 57],
        [86, 56, 0, 48, 35, 71, 89, 7, 5, 44, 44, 37, 44, 60, 21, 58, 51, 54, 17, 58],
        [19, 80, 81, 68, 5, 94, 47, 69, 28, 73, 92, 13, 86, 52, 17, 77, 4, 89, 55, 40],
        [4, 52, 8, 83, 97, 35, 99, 16, 7, 97, 57, 32, 16, 26, 26, 79, 33, 27, 98, 66],
        [88, 36, 68, 87, 57, 62, 20, 72, 3, 46, 33, 67, 46, 55, 12, 32, 63, 93, 53, 69],
        [4, 42, 16, 73, 38, 25, 39, 11, 24, 94, 72, 18, 8, 46, 29, 32, 40, 62, 76, 36],
        [20, 69, 36, 41, 72, 30, 23, 88, 34, 62, 99, 69, 82, 67, 59, 85, 74, 4, 36, 16],
        [20, 73, 35, 29, 78, 31, 90, 1, 74, 31, 49, 71, 48, 86, 81, 16, 23, 57, 5, 54],
        [1, 70, 54, 71, 83, 51, 54, 69, 16, 92, 33, 48, 61, 43, 52, 1, 89, 19, 67, 48]
    ]
    INVALID_COORDS_TUPLE = 0,

    def _build_up(row_index, col_index):
        """Build the sequence going straight up from row_index."""
        if row_index < OFFSET:
            return INVALID_COORDS_TUPLE

        return (GRID[row_index][col_index],
                GRID[row_index - 1][col_index],
                GRID[row_index - 2][col_index],
                GRID[row_index - 3][col_index])

    def _build_down(row_index, col_index):
        """Build the sequence going straight down from row_index."""
        if row_index + OFFSET >= len(GRID):
            return INVALID_COORDS_TUPLE

        return (GRID[row_index][col_index],
                GRID[row_index + 1][col_index],
                GRID[row_index + 2][col_index],
                GRID[row_index + 3][col_index])

    def _build_left(row_index, col_index):
        """Build the sequence going straight left from col_index."""
        if col_index < OFFSET:
            return INVALID_COORDS_TUPLE

        return (GRID[row_index][col_index],
                GRID[row_index][col_index - 1],
                GRID[row_index][col_index - 2],
                GRID[row_index][col_index - 3])

    def _build_right(row_index, col_index):
        """Build the sequence going straight right from row_index."""
        if col_index + OFFSET >= len(GRID[row_index]):
            return INVALID_COORDS_TUPLE

        return (GRID[row_index][col_index],
                GRID[row_index][col_index + 1],
                GRID[row_index][col_index + 2],
                GRID[row_index][col_index + 3])

    def _build_up_right(row_index, col_index):
        """Build the sequence going up and to the right from \
        (row_index, col_index)."""
        if row_index < OFFSET:
            return INVALID_COORDS_TUPLE
        if col_index + OFFSET >= len(GRID[row_index]):
            return INVALID_COORDS_TUPLE

        return (GRID[row_index][col_index],
                GRID[row_index - 1][col_index + 1],
                GRID[row_index - 2][col_index + 2],
                GRID[row_index - 3][col_index + 3])

    def _build_down_right(row_index, col_index):
        """Build the sequence going down and to the right from \
        (row_index, col_index)."""
        if row_index + OFFSET >= len(GRID):
            return INVALID_COORDS_TUPLE
        if col_index + OFFSET >= len(GRID[row_index]):
            return INVALID_COORDS_TUPLE

        return (GRID[row_index][col_index],
                GRID[row_index + 1][col_index + 1],
                GRID[row_index + 2][col_index + 2],
                GRID[row_index + 3][col_index + 3])

    def _build_down_left(row_index, col_index):
        """Build the sequence going down and to the left from \
        (row_index, col_index)."""
        if row_index + OFFSET >= len(GRID):
            return INVALID_COORDS_TUPLE
        if col_index < OFFSET:
            return INVALID_COORDS_TUPLE

        return (GRID[row_index][col_index],
                GRID[row_index + 1][col_index - 1],
                GRID[row_index + 2][col_index - 2],
                GRID[row_index + 3][col_index - 3])

    def _build_up_left(row_index, col_index):
        """Build the sequence going down and to the left from \
        (row_index, col_index)."""
        if row_index < OFFSET:
            return INVALID_COORDS_TUPLE
        if col_index < OFFSET:
            return INVALID_COORDS_TUPLE

        return (GRID[row_index][col_index],
                GRID[row_index - 1][col_index - 1],
                GRID[row_index - 2][col_index - 2],
                GRID[row_index - 3][col_index - 3])

    def _build_all_possible_sequences():
        """Build a set of all possible 4-element sequences."""
        sequences = set()
        for row in range(len(GRID)):
            for col in range(len(GRID[row])):
                sequences.add(_build_up(row, col))
                sequences.add(_build_down(row, col))
                sequences.add(_build_left(row, col))
                sequences.add(_build_right(row, col))
                sequences.add(_build_up_right(row, col))
                sequences.add(_build_down_right(row, col))
                sequences.add(_build_down_left(row, col))
                sequences.add(_build_up_left(row, col))
        return sequences

    def problem_11():
        """Find the largest product of four adjacent elements."""
        products = [reduce(operator.mul, sequence)
                    for sequence in _build_all_possible_sequences()]
        return max(products)

    if __name__ == '__main__':
        print problem_11()

Of course, this isn’t the best fit for a functional solution. For one thing, it wastes an awful lot of memory. The more imperative approach below runs in less than half the time on my box, and it’s (a little) less code, to boot. The lesson, as always: KISS.

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    """Solves Problem 11 from Project Euler."""

    OFFSET = 3
    GRID = [
        [8, 2, 22, 97, 38, 15, 0, 40, 0, 75, 4, 5, 7, 78, 52, 12, 50, 77, 91, 8],
        [49, 49, 99, 40, 17, 81, 18, 57, 60, 87, 17, 40, 98, 43, 69, 48, 4, 56, 62, 0],
        [81, 49, 31, 73, 55, 79, 14, 29, 93, 71, 40, 67, 53, 88, 30, 3, 49, 13, 36, 65],
        [52, 70, 95, 23, 4, 60, 11, 42, 69, 24, 68, 56, 1, 32, 56, 71, 37, 2, 36, 91],
        [22, 31, 16, 71, 51, 67, 63, 89, 41, 92, 36, 54, 22, 40, 40, 28, 66, 33, 13, 80],
        [24, 47, 32, 60, 99, 3, 45, 2, 44, 75, 33, 53, 78, 36, 84, 20, 35, 17, 12, 50],
        [32, 98, 81, 28, 64, 23, 67, 10, 26, 38, 40, 67, 59, 54, 70, 66, 18, 38, 64, 70],
        [67, 26, 20, 68, 2, 62, 12, 20, 95, 63, 94, 39, 63, 8, 40, 91, 66, 49, 94, 21],
        [24, 55, 58, 5, 66, 73, 99, 26, 97, 17, 78, 78, 96, 83, 14, 88, 34, 89, 63, 72],
        [21, 36, 23, 9, 75, 0, 76, 44, 20, 45, 35, 14, 0, 61, 33, 97, 34, 31, 33, 95],
        [78, 17, 53, 28, 22, 75, 31, 67, 15, 94, 3, 80, 4, 62, 16, 14, 9, 53, 56, 92],
        [16, 39, 5, 42, 96, 35, 31, 47, 55, 58, 88, 24, 0, 17, 54, 24, 36, 29, 85, 57],
        [86, 56, 0, 48, 35, 71, 89, 7, 5, 44, 44, 37, 44, 60, 21, 58, 51, 54, 17, 58],
        [19, 80, 81, 68, 5, 94, 47, 69, 28, 73, 92, 13, 86, 52, 17, 77, 4, 89, 55, 40],
        [4, 52, 8, 83, 97, 35, 99, 16, 7, 97, 57, 32, 16, 26, 26, 79, 33, 27, 98, 66],
        [88, 36, 68, 87, 57, 62, 20, 72, 3, 46, 33, 67, 46, 55, 12, 32, 63, 93, 53, 69],
        [4, 42, 16, 73, 38, 25, 39, 11, 24, 94, 72, 18, 8, 46, 29, 32, 40, 62, 76, 36],
        [20, 69, 36, 41, 72, 30, 23, 88, 34, 62, 99, 69, 82, 67, 59, 85, 74, 4, 36, 16],
        [20, 73, 35, 29, 78, 31, 90, 1, 74, 31, 49, 71, 48, 86, 81, 16, 23, 57, 5, 54],
        [1, 70, 54, 71, 83, 51, 54, 69, 16, 92, 33, 48, 61, 43, 52, 1, 89, 19, 67, 48]
    ]
    INVALID_COORDS_PRODUCT = 0

    def _calc_up(row_index, col_index):
        """Build the sequence going straight up from row_index."""
        if row_index < OFFSET:
            return INVALID_COORDS_PRODUCT

        return GRID[row_index][col_index] * \
                GRID[row_index - 1][col_index] * \
                GRID[row_index - 2][col_index] * \
                GRID[row_index - 3][col_index]

    def _calc_down(row_index, col_index):
        """Build the sequence going straight down from row_index."""
        if row_index + OFFSET >= len(GRID):
            return INVALID_COORDS_PRODUCT

        return GRID[row_index][col_index] * \
                GRID[row_index + 1][col_index] * \
                GRID[row_index + 2][col_index] * \
                GRID[row_index + 3][col_index]

    def _calc_left(row_index, col_index):
        """Build the sequence going straight left from col_index."""
        if col_index < OFFSET:
            return INVALID_COORDS_PRODUCT

        return GRID[row_index][col_index] * \
                GRID[row_index][col_index - 1] * \
                GRID[row_index][col_index - 2] * \
                GRID[row_index][col_index - 3]

    def _calc_right(row_index, col_index):
        """Build the sequence going straight right from row_index."""
        if col_index + OFFSET >= len(GRID[row_index]):
            return INVALID_COORDS_PRODUCT

        return GRID[row_index][col_index] * \
                GRID[row_index][col_index + 1] * \
                GRID[row_index][col_index + 2] * \
                GRID[row_index][col_index + 3]

    def _calc_up_right(row_index, col_index):
        """Build the sequence going up and to the right from \
        (row_index, col_index)."""
        if row_index < OFFSET:
            return INVALID_COORDS_PRODUCT
        if col_index + OFFSET >= len(GRID[row_index]):
            return INVALID_COORDS_PRODUCT

        return GRID[row_index][col_index] * \
                GRID[row_index - 1][col_index + 1] * \
                GRID[row_index - 2][col_index + 2] * \
                GRID[row_index - 3][col_index + 3]

    def _calc_down_right(row_index, col_index):
        """Build the sequence going down and to the right from \
        (row_index, col_index)."""
        if row_index + OFFSET >= len(GRID):
            return INVALID_COORDS_PRODUCT
        if col_index + OFFSET >= len(GRID[row_index]):
            return INVALID_COORDS_PRODUCT

        return GRID[row_index][col_index] * \
                GRID[row_index + 1][col_index + 1] * \
                GRID[row_index + 2][col_index + 2] * \
                GRID[row_index + 3][col_index + 3]

    def _calc_down_left(row_index, col_index):
        """Build the sequence going down and to the left from \
        (row_index, col_index)."""
        if row_index + OFFSET >= len(GRID):
            return INVALID_COORDS_PRODUCT
        if col_index < OFFSET:
            return INVALID_COORDS_PRODUCT

        return GRID[row_index][col_index] * \
                GRID[row_index + 1][col_index - 1] * \
                GRID[row_index + 2][col_index - 2] * \
                GRID[row_index + 3][col_index - 3]

    def _calc_up_left(row_index, col_index):
        """Build the sequence going down and to the left from \
        (row_index, col_index)."""
        if row_index < OFFSET:
            return INVALID_COORDS_PRODUCT
        if col_index < OFFSET:
            return INVALID_COORDS_PRODUCT

        return GRID[row_index][col_index] * \
                GRID[row_index - 1][col_index - 1] * \
                GRID[row_index - 2][col_index - 2] * \
                GRID[row_index - 3][col_index - 3]

    def _calc_all_possible_sequences():
        """Build a set of all possible 4-element sequences."""
        products = set()
        for row in range(len(GRID)):
            for col in range(len(GRID[row])):
                products.add(_calc_up(row, col))
                products.add(_calc_down(row, col))
                products.add(_calc_left(row, col))
                products.add(_calc_right(row, col))
                products.add(_calc_up_right(row, col))
                products.add(_calc_down_right(row, col))
                products.add(_calc_down_left(row, col))
                products.add(_calc_up_left(row, col))
        return products

    def problem_11():
        """Find the largest product of four adjacent elements."""
        return max(_calc_all_possible_sequences())

    if __name__ == '__main__':
        print problem_11()

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