NO 15 [C#] 峰回路转的格子
原题：
Starting in the top left corner of a 22 grid, there are 6 routes (without backtracking) to the bottom right corner.
![NO 15 [C#] 峰回路转的格子](http://img.e-com-net.com/image/product/15b536eb4485423fb734cbf78c1daa32.png)
How many routes are there through a 2020 grid?
翻译：
/*
* 解释：
* 2乘2的正方形，左上角到右下角是这样6种走法。
*
* 那么20*20的正方形呢？
*/
解题思路：
代码
1
///
<summary>
2 /// 解题思路：
3 /// 1， 我把它想象成了一个网格图。一个只有出度，没有或者不计算入度的图。
4 /// 这样的话，用邻接矩阵描述这个图。二维表构造，每行意义如下：
5 ///
6 /// [当前节点] -〉[向右的节点]-〉[向下的节点]
7 ///
8 /// 然后初始化这个邻接矩阵表即可。
9 ///
10 /// 2， 当然也可以用数学公式来求这个答案。
11 ///
12 /// 公式是： (长+宽)！/长！* 宽！
13 ///
14 /// 3， 请看以下这个结构
15 ///
16 /// 0 1 1
17 /// 1 2 3
18 /// 1 3 6
19 ///
20 /// 6就是答案，在扩大一下这个正方形
21 ///
22 /// 0 1 1 1 1
23 /// 1 2 3 4 5
24 /// 1 3 6 10 15
25 /// 1 4 10 20 35
26 /// 1 5 15 35 70
27 ///
28 /// 说白了，这个就是一个杨辉三角。每一个元素的数字表示的意思就是从0到这个点的可能性。
29 /// </summary>
2 /// 解题思路：
3 /// 1， 我把它想象成了一个网格图。一个只有出度，没有或者不计算入度的图。
4 /// 这样的话，用邻接矩阵描述这个图。二维表构造，每行意义如下：
5 ///
6 /// [当前节点] -〉[向右的节点]-〉[向下的节点]
7 ///
8 /// 然后初始化这个邻接矩阵表即可。
9 ///
10 /// 2， 当然也可以用数学公式来求这个答案。
11 ///
12 /// 公式是： (长+宽)！/长！* 宽！
13 ///
14 /// 3， 请看以下这个结构
15 ///
16 /// 0 1 1
17 /// 1 2 3
18 /// 1 3 6
19 ///
20 /// 6就是答案，在扩大一下这个正方形
21 ///
22 /// 0 1 1 1 1
23 /// 1 2 3 4 5
24 /// 1 3 6 10 15
25 /// 1 4 10 20 35
26 /// 1 5 15 35 70
27 ///
28 /// 说白了，这个就是一个杨辉三角。每一个元素的数字表示的意思就是从0到这个点的可能性。
29 /// </summary>
代码：
代码
1
class
GridRoutes
2 {
3 private int SideLength; // 单边长度
4 private int ArrayLength; // 存储每个点的矩阵长度
5 private int [][] Arr; // 存储矩阵
6 private ulong Count; // 路径总数
7 public ulong Result
8 {
9 get { return this .Count; }
10 }
11 public GridRoutes( int length)
12 {
13 this .Count = 0 ;
14 this .SideLength = length;
15 this .ArrayLength = this .SideLength * this .SideLength;
16 this .Arr = new int [ this .ArrayLength][];
17 #region 初始化
18 for ( int i = 0 ; i < this .Arr.GetLength( 0 ); i ++ )
19 {
20 this .Arr[i] = new int [ 3 ];
21 }
22 #endregion
23 #region 4个顶点
24 // 起点
25 this .Arr[ 0 ][ 0 ] = 0 ;
26 this .Arr[ 0 ][ 1 ] = 1 ;
27 this .Arr[ 0 ][ 2 ] = this .SideLength;
28 // 终点
29 this .Arr[ this .ArrayLength - 1 ][ 0 ] = this .ArrayLength - 1 ;
30 this .Arr[ this .ArrayLength - 1 ][ 1 ] = - 1 ;
31 this .Arr[ this .ArrayLength - 1 ][ 2 ] = - 1 ;
32 // 右上
33 this .Arr[ this .SideLength - 1 ][ 0 ] = this .SideLength - 1 ;
34 this .Arr[ this .SideLength - 1 ][ 1 ] = - 1 ;
35 this .Arr[ this .SideLength - 1 ][ 2 ] = ( this .SideLength - 1 ) + ( this .SideLength - 1 ) + 1 ;
36 // 左下
37 this .Arr[ this .SideLength * ( this .SideLength - 1 )][ 0 ] = this .SideLength * ( this .SideLength - 1 );
38 this .Arr[ this .SideLength * ( this .SideLength - 1 )][ 1 ] = this .SideLength * ( this .SideLength - 1 ) + 1 ;
39 this .Arr[ this .SideLength * ( this .SideLength - 1 )][ 2 ] = - 1 ;
40 #endregion
41 #region 去顶点的各条边
42 // up
43 for ( int i = 1 ; i < this .SideLength - 1 ; i ++ )
44 {
45 this .Arr[i][ 0 ] = i;
46 this .Arr[i][ 1 ] = i + 1 ;
47 this .Arr[i][ 2 ] = i + this .SideLength;
48 }
49 // down
50 for ( int i = this .SideLength * ( this .SideLength - 1 ) + 1 ; i < this .ArrayLength - 1 ; i ++ )
51 {
52 this .Arr[i][ 0 ] = i;
53 this .Arr[i][ 1 ] = i + 1 ;
54 this .Arr[i][ 2 ] = - 1 ;
55 }
56 // left
57 for ( int i = this .SideLength; i < this .SideLength * ( this .SideLength - 1 ); i = i + this .SideLength)
58 {
59 this .Arr[i][ 0 ] = i;
60 this .Arr[i][ 1 ] = i + 1 ;
61 this .Arr[i][ 2 ] = i + this .SideLength;
62 }
63 // right
64 for ( int i = this .SideLength * 2 - 1 ; i < this .ArrayLength - 1 ; i = i + this .SideLength)
65 {
66 this .Arr[i][ 0 ] = i;
67 this .Arr[i][ 1 ] = - 1 ;
68 this .Arr[i][ 2 ] = i + this .SideLength;
69 }
70 #endregion
71 #region 中间所有点
72 for ( int i = 2 ; i < this .SideLength; i ++ ) // 行数
73 {
74 for ( int j = 2 ; j < this .SideLength; j ++ ) // 列数
75 {
76 int tmp = (j - 1 ) + this .SideLength * (i - 1 );
77 this .Arr[tmp][ 0 ] = tmp;
78 this .Arr[tmp][ 1 ] = tmp + 1 ;
79 this .Arr[tmp][ 2 ] = tmp + this .SideLength;
80 }
81 }
82 #endregion
83 #region 显示数组矩阵内容
84 for ( int i = 0 ; i < Arr.GetLength( 0 ); i ++ )
85 {
86 Console.WriteLine( string .Format( " 矩阵序号 {0:}, 往左 {1, 2}, 往右 {2, 2} " , Arr[i][ 0 ], Arr[i][ 1 ], Arr[i][ 2 ]));
87 }
88 #endregion
89 }
90 /// <summary>
91 /// 寻路
92 /// </summary>
93 /// <param name="index"> 当前点 </param>
94 /// <param name="end"> 目标点 </param>
95 public void Go( int index, int end)
96 {
97 if ( this .Arr[index][ 0 ] == end)
98 {
99 // TODO
100 this .Count = this .Count + 1 ;
101 }
102 else
103 {
104 if ( this .Arr[index][ 1 ] != - 1 )
105 {
106 Go( this .Arr[index][ 1 ], end);
107 }
108 if ( this .Arr[index][ 2 ] != - 1 )
109 {
110 Go( this .Arr[index][ 2 ], end);
111 }
112 }
113 }
114 }
2 {
3 private int SideLength; // 单边长度
4 private int ArrayLength; // 存储每个点的矩阵长度
5 private int [][] Arr; // 存储矩阵
6 private ulong Count; // 路径总数
7 public ulong Result
8 {
9 get { return this .Count; }
10 }
11 public GridRoutes( int length)
12 {
13 this .Count = 0 ;
14 this .SideLength = length;
15 this .ArrayLength = this .SideLength * this .SideLength;
16 this .Arr = new int [ this .ArrayLength][];
17 #region 初始化
18 for ( int i = 0 ; i < this .Arr.GetLength( 0 ); i ++ )
19 {
20 this .Arr[i] = new int [ 3 ];
21 }
22 #endregion
23 #region 4个顶点
24 // 起点
25 this .Arr[ 0 ][ 0 ] = 0 ;
26 this .Arr[ 0 ][ 1 ] = 1 ;
27 this .Arr[ 0 ][ 2 ] = this .SideLength;
28 // 终点
29 this .Arr[ this .ArrayLength - 1 ][ 0 ] = this .ArrayLength - 1 ;
30 this .Arr[ this .ArrayLength - 1 ][ 1 ] = - 1 ;
31 this .Arr[ this .ArrayLength - 1 ][ 2 ] = - 1 ;
32 // 右上
33 this .Arr[ this .SideLength - 1 ][ 0 ] = this .SideLength - 1 ;
34 this .Arr[ this .SideLength - 1 ][ 1 ] = - 1 ;
35 this .Arr[ this .SideLength - 1 ][ 2 ] = ( this .SideLength - 1 ) + ( this .SideLength - 1 ) + 1 ;
36 // 左下
37 this .Arr[ this .SideLength * ( this .SideLength - 1 )][ 0 ] = this .SideLength * ( this .SideLength - 1 );
38 this .Arr[ this .SideLength * ( this .SideLength - 1 )][ 1 ] = this .SideLength * ( this .SideLength - 1 ) + 1 ;
39 this .Arr[ this .SideLength * ( this .SideLength - 1 )][ 2 ] = - 1 ;
40 #endregion
41 #region 去顶点的各条边
42 // up
43 for ( int i = 1 ; i < this .SideLength - 1 ; i ++ )
44 {
45 this .Arr[i][ 0 ] = i;
46 this .Arr[i][ 1 ] = i + 1 ;
47 this .Arr[i][ 2 ] = i + this .SideLength;
48 }
49 // down
50 for ( int i = this .SideLength * ( this .SideLength - 1 ) + 1 ; i < this .ArrayLength - 1 ; i ++ )
51 {
52 this .Arr[i][ 0 ] = i;
53 this .Arr[i][ 1 ] = i + 1 ;
54 this .Arr[i][ 2 ] = - 1 ;
55 }
56 // left
57 for ( int i = this .SideLength; i < this .SideLength * ( this .SideLength - 1 ); i = i + this .SideLength)
58 {
59 this .Arr[i][ 0 ] = i;
60 this .Arr[i][ 1 ] = i + 1 ;
61 this .Arr[i][ 2 ] = i + this .SideLength;
62 }
63 // right
64 for ( int i = this .SideLength * 2 - 1 ; i < this .ArrayLength - 1 ; i = i + this .SideLength)
65 {
66 this .Arr[i][ 0 ] = i;
67 this .Arr[i][ 1 ] = - 1 ;
68 this .Arr[i][ 2 ] = i + this .SideLength;
69 }
70 #endregion
71 #region 中间所有点
72 for ( int i = 2 ; i < this .SideLength; i ++ ) // 行数
73 {
74 for ( int j = 2 ; j < this .SideLength; j ++ ) // 列数
75 {
76 int tmp = (j - 1 ) + this .SideLength * (i - 1 );
77 this .Arr[tmp][ 0 ] = tmp;
78 this .Arr[tmp][ 1 ] = tmp + 1 ;
79 this .Arr[tmp][ 2 ] = tmp + this .SideLength;
80 }
81 }
82 #endregion
83 #region 显示数组矩阵内容
84 for ( int i = 0 ; i < Arr.GetLength( 0 ); i ++ )
85 {
86 Console.WriteLine( string .Format( " 矩阵序号 {0:}, 往左 {1, 2}, 往右 {2, 2} " , Arr[i][ 0 ], Arr[i][ 1 ], Arr[i][ 2 ]));
87 }
88 #endregion
89 }
90 /// <summary>
91 /// 寻路
92 /// </summary>
93 /// <param name="index"> 当前点 </param>
94 /// <param name="end"> 目标点 </param>
95 public void Go( int index, int end)
96 {
97 if ( this .Arr[index][ 0 ] == end)
98 {
99 // TODO
100 this .Count = this .Count + 1 ;
101 }
102 else
103 {
104 if ( this .Arr[index][ 1 ] != - 1 )
105 {
106 Go( this .Arr[index][ 1 ], end);
107 }
108 if ( this .Arr[index][ 2 ] != - 1 )
109 {
110 Go( this .Arr[index][ 2 ], end);
111 }
112 }
113 }
114 }