八皇后问题 C#版本算法 回溯法
2012-05-02 14:13
531 查看
本文中的代码在文章最后提供下载链接。
关于八皇后的问题,自己百度。回溯法的概念也可以自己维基百科中查看。
回溯法无非就是遍历解空间,如果合适的话就继续,不合适的话就放弃当前节点。可以采用递归实现。八皇后问题的经典算法就是采用回溯法。网上也有许多版本的解法,但是C#版本的似乎很少。笔者在此奉献上。
此处八皇后的算法的伪代码如下:
在第row行放置皇后(row)
{
for 第row行的格子
{
把当前位置加入作为候选解加入解决方案中
If 目前解决方案没有问题
{
再下一行放皇后,递归调用PlaceQueen(row+1);
如果是最后一行放完了,则成功,打印棋盘
}
else
将位置(row,col)上的皇后取走,尝试下一列位置 //回溯
}
伪代码的算法能理解之后,就可以写代码了。
对于棋盘,可以使用二维数组实现,但是本例中使用的是一个List序列,序列中有64个格子的信息。每个格子的定义如下。
//格子类。坐标x,y
初始化格子的代码如下:
这样的话lstAllPosition就存放了64个格子了。由于采用了C#作为实现的语言,因此尽情的使用他的一些优点,比如采用Linq和拉姆达表达式来选定特定的行的元素。此处,需要有C#的一些知识。
对于放置皇后的实现。本人写了2种方法,2中方法本质都是一样的,只不过理解起来稍微有些不同。
第一种方法中规中矩。
看如下代码和注释:
第二种方法也很好理解,只是把solution.Remove(item);删除元素的语句写到外面,表示,无论怎样,都会把刚加入的元素删除。
但是在删除元素之前,由于递归调用的结果,在某些情况下solution列的元素加满到8个时候,还没删除元素。这是就直接打印出结果,然后再删除最后一个元素。很容易推断,到最后一个元素放入solution的时候,肯定是递归嵌套了8层了。
其中 验证当前集合是否有效的函数如下:
//验证有效性
打印棋盘的方法如下,使用了Linq和拉姆达表达式。
//打印棋盘
结果如下:共92个解。
![](http://img1.51cto.com/attachment/201205/2/2359144_1335939184JLkc.png)
所有的解集如下:
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1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
-----2-------
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
-----3-------
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
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0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
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1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
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0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
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0 1 0 0 0 0 0 0
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1 0 0 0 0 0 0 0
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0 1 0 0 0 0 0 0
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0 1 0 0 0 0 0 0
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0 0 0 1 0 0 0 0
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0 1 0 0 0 0 0 0
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0 1 0 0 0 0 0 0
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0 1 0 0 0 0 0 0
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0 1 0 0 0 0 0 0
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0 0 1 0 0 0 0 0
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0 0 0 0 0 0 1 0
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0 0 0 0 0 1 0 0
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0 0 1 0 0 0 0 0
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0 0 1 0 0 0 0 0
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0 0 1 0 0 0 0 0
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-----21-------
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0 0 1 0 0 0 0 0
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0 0 1 0 0 0 0 0
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0 0 0 0 0 0 1 0
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-----24-------
0 0 1 0 0 0 0 0
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0 0 1 0 0 0 0 0
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0 0 1 0 0 0 0 0
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0 0 1 0 0 0 0 0
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0 0 0 1 0 0 0 0
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-----30-------
0 0 0 1 0 0 0 0
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0 0 0 1 0 0 0 0
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0 0 0 1 0 0 0 0
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0 0 0 1 0 0 0 0
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-----34-------
0 0 0 1 0 0 0 0
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0 0 0 1 0 0 0 0
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-----36-------
0 0 0 1 0 0 0 0
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0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
-----86-------
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
-----87-------
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
-----88-------
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
-----89-------
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
-----90-------
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
-----91-------
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
-----92-------
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
本文出自 “一只博客” 博客,请务必保留此出处http://cnn237111.blog.51cto.com/2359144/850282
关于八皇后的问题,自己百度。回溯法的概念也可以自己维基百科中查看。
回溯法无非就是遍历解空间,如果合适的话就继续,不合适的话就放弃当前节点。可以采用递归实现。八皇后问题的经典算法就是采用回溯法。网上也有许多版本的解法,但是C#版本的似乎很少。笔者在此奉献上。
此处八皇后的算法的伪代码如下:
在第row行放置皇后(row)
{
for 第row行的格子
{
把当前位置加入作为候选解加入解决方案中
If 目前解决方案没有问题
{
再下一行放皇后,递归调用PlaceQueen(row+1);
如果是最后一行放完了,则成功,打印棋盘
}
else
将位置(row,col)上的皇后取走,尝试下一列位置 //回溯
}
伪代码的算法能理解之后,就可以写代码了。
对于棋盘,可以使用二维数组实现,但是本例中使用的是一个List序列,序列中有64个格子的信息。每个格子的定义如下。
//格子类。坐标x,y
public class position { public int x { set; get; } public int y { set; get; } public override string ToString() { return string.Format("[{0},{1}]", x, y); } }
初始化格子的代码如下:
#region 初始化格子 for (int i = 1; i <= dimension; i++) { for (int j = 1; j <= dimension; j++) { lstAllPosition.Add(new position { x = i, y = j }); } } #endregion
这样的话lstAllPosition就存放了64个格子了。由于采用了C#作为实现的语言,因此尽情的使用他的一些优点,比如采用Linq和拉姆达表达式来选定特定的行的元素。此处,需要有C#的一些知识。
对于放置皇后的实现。本人写了2种方法,2中方法本质都是一样的,只不过理解起来稍微有些不同。
第一种方法中规中矩。
看如下代码和注释:
static void PlaceQueen(int row) { #region 实现方法一 //当前行的元素 var currentrow = lstAllPosition.Where(m => m.x == row).ToList(); for (int index = 0; index < currentrow.Count;index++ ) { //加入一个元素 solution.Add(currentrow[index]); //如果有效的话,继续下面的递归 if (Valid(solution)) { //如果row是最后一行了,说明已经满足。打印。 if (row == dimension) { print(solution); //找到一组解之后,删除该元素,继续下一组解的查找 solution.Remove(solution.Last()); } else { //递归调用下一行 PlaceQueen(row + 1); } } else { solution.Remove(currentrow[index]); } //如果该行的元素每个都被试过了,但是还是没能运行到print(solution)方法中去,则该方法即将运行结束,结束前,把最后一个元素也删掉。 //由于该方法是递归的,因此结束之后,会继续重新下一个元素。 if (index == dimension-1 && solution.Count > 0) { solution.Remove(solution.Last()); } } #endregion }
第二种方法也很好理解,只是把solution.Remove(item);删除元素的语句写到外面,表示,无论怎样,都会把刚加入的元素删除。
但是在删除元素之前,由于递归调用的结果,在某些情况下solution列的元素加满到8个时候,还没删除元素。这是就直接打印出结果,然后再删除最后一个元素。很容易推断,到最后一个元素放入solution的时候,肯定是递归嵌套了8层了。
static void PlaceQueen(int row) { #region 实现方法二 foreach (var item in lstAllPosition.Where(m => m.x == row).ToList()) { //Console.WriteLine("-------"); solution.Add(item); //solution.ForEach(m => Console.Write(m.ToString())); //Console.WriteLine(); //如果有效的话,继续下面的递归 if (Valid(solution)) { //如果row是最后一行了,说明已经满足。打印。 if (row == dimension) { print(solution); } else { //递归调用下一行 PlaceQueen(row + 1); } } //Console.WriteLine("***"); //把当前元素去除,尝试下一个元素。此语句,每一轮都会执行,除非该轮的调用, //每次递归调用PlaceQueen时Valid(solution)都是true,也就是说找到符合条件的结果集,打印完后。 //还是会继续删除最厚放入的元素,继续尝试下一个元素是否匹配。 solution.Remove(item); //solution.ForEach(m => Console.Write(m.ToString())); //Console.WriteLine(); //Console.WriteLine("-------"); } #endregion }
其中 验证当前集合是否有效的函数如下:
//验证有效性
static bool Valid(List<position> solve) { var item2 = solve.Last(); foreach (var item1 in solve)//把最后加入的元素后之前每一个元素做比较。 { if (item1 == item2) { break; } if (item1.y == item2.y)//y轴相同 { return false; } else if (Math.Abs(decimal.Parse((item1.y - item2.y).ToString()) / (item1.x - item2.x)) == 1)//斜线 { return false; } } return true; }
打印棋盘的方法如下,使用了Linq和拉姆达表达式。
//打印棋盘
static void print(List<position> solve) { Console.ForegroundColor = ConsoleColor.Gray; Console.WriteLine("-----" + (count++) + "-------"); for (int i = 1; i <= dimension; i++) { foreach (var item in lstAllPosition.Where(m => m.x == i).ToList()) { if (solve.Contains(item)) { Console.ForegroundColor = ConsoleColor.Yellow; Console.Write("1 "); } else { Console.ForegroundColor = ConsoleColor.Gray; Console.Write("0 "); } } Console.WriteLine(); } }
结果如下:共92个解。
![](http://img1.51cto.com/attachment/201205/2/2359144_1335939184JLkc.png)
所有的解集如下:
-----1-------
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
-----2-------
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
-----3-------
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
-----4-------
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----5-------
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
-----6-------
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
-----7-------
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----8-------
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
-----9-------
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
-----10-------
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
-----11-------
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----12-------
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
-----13-------
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
-----14-------
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
-----15-------
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
-----16-------
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
-----17-------
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
-----18-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
-----19-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
-----20-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
-----21-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
-----22-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
-----23-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
-----24-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
-----25-------
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
-----26-------
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
-----27-------
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
-----28-------
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
-----29-------
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
-----30-------
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
-----31-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
-----32-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
-----33-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
-----34-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----35-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
-----36-------
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
-----37-------
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
-----38-------
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
-----39-------
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
-----40-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----41-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
-----42-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
-----43-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
-----44-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
-----45-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----46-------
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
-----47-------
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
-----48-------
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
-----49-------
0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
-----50-------
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
-----51-------
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
-----52-------
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0
-----53-------
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
-----54-------
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
-----55-------
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
-----56-------
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
-----57-------
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
-----58-------
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
-----59-------
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
-----60-------
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1
-----61-------
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
-----62-------
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
-----63-------
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
-----64-------
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0
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