# Euler Problem 18¶

By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.

                        3
                       7 4
                      2 4 6
                     8 5 9 3

That is, 3 + 7 + 4 + 9 = 23.

Find the maximum total from top to bottom of the triangle below:

    75
    95 64
    17 47 82
    18 35 87 10
    20 04 82 47 65
    19 01 23 75 03 34
    88 02 77 73 07 63 67
    99 65 04 28 06 16 70 92
    41 41 26 56 83 40 80 70 33
    41 48 72 33 47 32 37 16 94 29
    53 71 44 65 25 43 91 52 97 51 14
    70 11 33 28 77 73 17 78 39 68 17 57
    91 71 52 38 17 14 91 43 58 50 27 29 48
    63 66 04 68 89 53 67 30 73 16 69 87 40 31
    04 62 98 27 23 09 70 98 73 93 38 53 60 04 23

NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route. However, Problem 67, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)

This is incredibly simple we simply from bottom to top choosing the higher value for each mini-tree.

For example,

      95  64
    17  47  82

will become:

     142 146

In [1]:

    t = """
    75
    95 64
    17 47 82
    18 35 87 10
    20 04 82 47 65
    19 01 23 75 03 34
    88 02 77 73 07 63 67
    99 65 04 28 06 16 70 92
    41 41 26 56 83 40 80 70 33
    41 48 72 33 47 32 37 16 94 29
    53 71 44 65 25 43 91 52 97 51 14
    70 11 33 28 77 73 17 78 39 68 17 57
    91 71 52 38 17 14 91 43 58 50 27 29 48
    63 66 04 68 89 53 67 30 73 16 69 87 40 31
    04 62 98 27 23 09 70 98 73 93 38 53 60 04 23
    """

In [2]:

    def reduce_rows(xs, ys):
        """ xs is lower row and ys is upper row """
        assert(len(xs) == len(ys) + 1)
        return [max([xs[i] + ys[i], xs[i + 1] + ys[i]]) for i in range(len(ys))]

    assert(reduce_rows([17, 47, 82], [95, 64]) == [142, 146])

Okay, now all we have to do is the parsing and then a simple fold.

In [3]:

    xss = [list(map(int, xs.split())) for xs in t.split("\n") if xs]
    xss.reverse()
    from functools import reduce
    s = reduce(reduce_rows, xss[1:], xss[0])[0]
    assert(s == 1074)

Okay, let's put this into a nice function an then solve problem 67 right away.

In [4]:

    def find_greatest_path_sum_in_triangle_string(ts):
        from functools import reduce
        xss = [list(map(int, xs.split())) for xs in ts.split("\n") if xs]
        xss.reverse()
        r = lambda xs, ys: [max([xs[i] + ys[i], xs[i + 1] + ys[i]]) for i in range(len(ys))]
        return reduce(r, xss[1:], xss[0])[0]

    print(find_greatest_path_sum_in_triangle_string(t))

    1074