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Problem 18


Problem 18


Maximum path sum I

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)


最大路径和 I

从如下数字三角形的顶端出发,不断移动到下一行与其相邻的数直至到达底部,所能得到的最大路径和是$23$。

3
7 4
2 4 6
8 5 9 3

如上图,最大路径和为$3 + 7 + 4 + 9 = 23$。

从如下数字三角形的顶端出发到达底部,求所能得到的最大路径和。

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

注意: 在这个问题中,由于只有$16384$条路径,通过穷举所有的路径来解决问题是可行的。然而,对于第67题,虽然是一道相同类型的题目,但是其中的数字三角形将拥有一百行,就不再能够通过暴力枚举的方法来解决,而需要一个更加聪明的办法!;o)