Project Euler Q18, Q67

here's the question

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)

After read the problem, u might wanna try brute force the problem right? if wasn't for that note on the 100 lines problem.

So I do some research on the net and found some method that they already solve the problem.
I try to just read it and understand the solution then come back and write my own lua code.

the general idea of solving this kinda problem is.

1. assume that u already have the answer


2. work our way from bottom not top


3. say on the line just above the bottom line, each member of that line will have only one possible maximum choice, say item number 1 on line 99 will look at item number 1 and 2 on line 100 and choose the maximum out come,


4. use math.max(line[100]item[1], line[100]item[2]) to pick the greater number.


5. combine the greater number with the item on the above line we'll have something like
   1. line[99][item[1]] = line[99][item[1] + math.max(line[100]item[1],line[100][item[2])
      
6. now just create two for loop to get through the line and the item
7. this will 'flaten' the triangle until it has only posible maximum result.

the code will look like this.
-----
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}
}

local myMax = math.max

for i = #triangle, 2, -1 do
for j = 1, #triangle[i-1] do
triangle[i-1][j] = triangle[i-1][j] + (myMax(triangle[i][j],triangle[i][j+1]))
end
end

print (triangle[1][1])
-----

on Question 67 u just replace the triangle data with the new data...
which I just type it in :P with laziness to find new readIn method