Problems 18 and 67 of Project Euler were the most interesting ones to me, as I *almost* brute forced them and yet without taking twenty billion years as claimed.

Here’s the 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.`

```
```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)`

Here is my analogy to solve this problem,

And here’s my solution in Python

n = [] for i in range(15): n.append(map(int,raw_input().split())) newresult = n.pop(0) for i in range(14): x = n.pop(0) result = list(newresult) # print len(result) newresult =[] t = len(x) for j in range(t): if(j==0): newresult.append(result[0]+x.pop(0)) elif(j==t-1): newresult.append(result[0]+x.pop(0)) else: newresult.append(max((result.pop(0)+x[0]),result[0]+x.pop(0))) print newresult print max(newresult)

I essentially keep a list of all possible values by limiting to the max at each step, isn’t that amazing ðŸ˜‰