Problem 18 - Maximum Path Sum I

Problem

This problem comes from Project Euler 18 ⤴

Problem

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.

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

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

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

It’s easier to solve the problem by starting at the bottom and working towards the top. This allows us to have just one number at the end instead of a row of numbers.

They say “a picture is worth a thousand words,” so here’s an animation of what the code will do.

The code looks at two adjacent numbers in a lower row and adds whichever is greater to the number above them. You may have to watch the animation a few times to get it, but this explains what’s going on far better than any words I can say.

Code

# Project Euler: Problem 18
# Maximum path sum I

a = [
    [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]
]

rows = len(a)-1
for x in range(rows, -1, -1):
    columns = len(a[x])-1
    for y in range(columns):
        if(a[x][y] > a[x][y+1]):
            a[x-1][y] += a[x][y]
        else:
            a[x-1][y] += a[x][y+1]

print(a[0])

Note

I use x and y in the code above for coordinates for the 2D array a in the form a[x][y].

• x is a row such as a[2] = [17, 47, 82] and,
  
• y is a column of that row such as a[2][0] = 17.