Problem 18：Maximum path sum I

原题链接：http://projecteuler.net/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)

题目大意是：

从下面的三角形的顶端开始，向下面一行的相邻数字移动，从顶端到底端的最大总和为23.

3

7 4

2 4 6

8 5 9 3

也就是 3 + 7 + 4 + 9 = 23.

找出从以下三角形的顶端走到底端的最大总和：

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

解法1：

参考自：http://www.mathblog.dk/project-euler-18/

简单地说，就是从下自上，找出那一层里的每两个数的较大者，并加到这两个数对应的上一层的那个数里。例如对于上面的小三角，第4层是8,5,9，3，而8与5这两个相邻的数，8比5大，8然后加到8,5对应的上一层的那个数2上，5和9，9和3，这两对也是如此，则小三角简化成如下所示：

3

7 4

10 13 15

一直简化到只剩下最顶层。

php代码：

$str=<<<EOF
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
EOF;
$lines = explode("\n",$str);
$numbers = array();
foreach($lines as $line){
$numbers[]=explode(" ",trim($line));
}
$line_num = count($numbers);

for($i=$line_num - 1;$i>=0;$i--){
$len = count($numbers[$i]);
for($j=0;$j<$len;$j++){
$a = $numbers[$i][$j]+$numbers[$i+1][$j];
$b = $numbers[$i][$j]+$numbers[$i+1][$j+1];
$numbers[$i][$j] = max($a,$b);
}
}
var_export($numbers[0]);

注：题目的中文翻译源自http://pe.spiritzhang.com