# Problem 18

Published on 2011-07-26This problem asks you find the largest possible sum that can be achieved by adding together the numbers encountered on a path to the bottom of a triangle composed of random 2-digit numbers. It reads:

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

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

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

As this problem is a simplified clone of problem 67, the only difference being the size of the triangle, I'll save my explanation of my code for my next post (which will include a modified version of this code designed for #67). Rather than implementing a brute force solution (totally doable, according to the problem text, as there are "only" 16384 (or 2^{15} / 2) routes down the triangle), I wrote something that could be used for triangles of any size. As soon as I learn file I/O in C#, I'll use the same code to tackle #67.

using System; using System.Collections.Generic; using System.Linq; using System.Text; namespace Problem018 { class Program { static void Main(string[] args) { List<int[]> triangle = new List<int[]>(); triangle.Add(new int[] {75}); triangle.Add(new int[] {95, 64 }); triangle.Add(new int[] {17, 47, 82}); triangle.Add(new int[] {18, 35, 87, 10}); triangle.Add(new int[] {20, 04, 82, 47, 65}); triangle.Add(new int[] {19, 01, 23, 75, 03, 34}); triangle.Add(new int[] {88, 02, 77, 73, 07, 63, 67}); triangle.Add(new int[] {99, 65, 04, 28, 06, 16, 70, 92}); triangle.Add(new int[] {41, 41, 26, 56, 83, 40, 80, 70, 33}); triangle.Add(new int[] {41, 48, 72, 33, 47, 32, 37, 16, 94, 29}); triangle.Add(new int[] {53, 71, 44, 65, 25, 43, 91, 52, 97, 51, 14}); triangle.Add(new int[] {70, 11, 33, 28, 77, 73, 17, 78, 39, 68, 17, 57}); triangle.Add(new int[] {91, 71, 52, 38, 17, 14, 91, 43, 58, 50, 27, 29, 48}); triangle.Add(new int[] {63, 66, 04, 68, 89, 53, 67, 30, 73, 16, 69, 87, 40, 31}); triangle.Add(new int[] {04, 62, 98, 27, 23, 09, 70, 98, 73, 93, 38, 53, 60, 04, 23}); for (int i = triangle.Count() - 2; i >= 0; i--) { for (int j = 0; j < triangle[i].Length; j++) { triangle[i][j] += Math.Max(triangle[i + 1][j], triangle[i + 1][j + 1]); } } Console.WriteLine("The maximum total is {0}", triangle[0][0]); } } }