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 😉

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