# Statement

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)

# Solution

For this type of problem we have to use dynamic programming in bottom-up perspective,

if we don't use that approach we won't be able to deal with the amount of different branches.

What we have to do is consider if we just have 2 rows. We would pick the highest from the lowest

row to add it to the top one. If we have 3 rows we can process the 2 lower ones and convert them

in the best choices and then we would have the first case again.

Then, how do we do that, for each pair of consecutive values of the lowest row we pick the highest

and add it to 'parent' cell. We repeat the exact algorithm for each row until we get to the 2nd.

When we finish that, we'll have the maximum value in the head of the pyramid.

if __name__ == '__main__': triangle = [ [75], [95, 64], [17, 47, 82], [18, 35, 87, 10], [20, 4, 82, 47, 65], [19, 1, 23, 75, 3, 34], [88, 2, 77, 73, 7, 63, 67], [99, 65, 4, 28, 6, 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, 4, 68, 89, 53, 67, 30, 73, 16, 69, 87, 40, 31], [4, 62, 98, 27, 23, 9, 70, 98, 73, 93, 38, 53, 60, 4, 23] ] for index in range(len(triangle) - 1, 0, -1): for intern_index in range(0, len(triangle[index]) - 1): if triangle[index][intern_index] > triangle[index][intern_index + 1]: triangle[index - 1][intern_index] += triangle[index][intern_index] else: triangle[index - 1][intern_index] += triangle[index][intern_index + 1] print("The result is:", triangle[0][0])

The Python file is available for download here.