Robert Eisele
Engineer, Systems Architect and DBA

Project Euler 18: Maximum path sum I

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)

Solution

In this task, the optimal solution, which can be seen as a dynamic programming solution, is much easier than the trivial brute-force-algorithm. To come up with the solution, every line must be added to any successor line, from bottom to top. Since every cell has two predecessors, we take the maximum of both. With this principle, the solution can be read off the top cell of the triangle. Implemented, this looks like:

var triangle = [
  [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],
];

function solution(t) {
  for (i = t.length - 2; i >= 0; i--)
    for (j = 0; j < i; j++)
      t[i][j]+= Math.max(t[i + 1][j], t[i + 1][j + 1]);
  return triangle[0][0];
}
solution(triangle);
    

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