我有一个问题，它是Josephus问题的变体。如下所述：
有m张牌的数字从1到m，每张牌都有一个唯一的数字。卡片被发给坐在一个圆圈里的n个人。请注意m>=n。
然后我们选择一个人"；A"；谁坐在"；p〃；就像约瑟夫斯的问题一样。下一步我们跳过"；k〃；p右边的人，而k是该人标记的卡的号码"；A"；，我们做同样的事情，直到圈子里只剩下一个人。
问题是给n个人和m张牌，我们可以选择n张牌并将其分配给n个人吗，以使无论从哪个位置开始(排除第一个位置)，最后存活的人总是圈中的第一个人。
例如，m=n=5，唯一的解决方案是(4，1，5，3，2)。
我认为这个问题是一个np完全问题，但我不能证明它。有人知道找到多项式时间解或证明它是np困难的吗？
---示例解决方案-
2: [ 1, 2]
2: [ 2, 1]
3: [ 1, 3, 2]
3: [ 3, 1, 2]
4: [ 4, 1, 3, 2]
5: [ 4, 1, 5, 3, 2]
7: [ 5, 7, 3, 1, 6, 4, 2]
9: [ 2, 7, 3, 9, 1, 6, 8, 5, 4]
9: [ 3, 1, 2, 7, 6, 5, 9, 4, 8]
9: [ 3, 5, 1, 8, 9, 6, 7, 4, 2]
9: [ 3, 9, 2, 7, 6, 1, 5, 4, 8]
9: [ 6, 1, 8, 3, 7, 9, 4, 5, 2]
10: [ 3, 5, 6, 10, 1, 9, 8, 7, 4, 2]
10: [ 4, 5, 2, 8, 7, 10, 6, 1, 9, 3]
10: [ 5, 1, 9, 2, 10, 3, 7, 6, 8, 4]
10: [ 6, 3, 1, 10, 9, 8, 7, 4, 5, 2]
10: [ 8, 5, 9, 10, 1, 7, 2, 6, 4, 3]
10: [10, 5, 2, 1, 8, 7, 6, 9, 3, 4]
11: [ 2, 1, 10, 11, 9, 3, 7, 5, 6, 8, 4]
11: [ 3, 7, 11, 10, 9, 8, 1, 6, 5, 4, 2]
11: [ 3, 11, 10, 9, 8, 1, 7, 2, 4, 5, 6]
11: [ 4, 1, 10, 2, 9, 8, 7, 5, 11, 3, 6]
11: [ 4, 2, 7, 11, 5, 1, 10, 9, 6, 3, 8]
11: [ 4, 7, 2, 3, 1, 10, 9, 6, 11, 5, 8]
11: [ 4, 7, 3, 9, 11, 10, 1, 8, 6, 5, 2]
11: [ 4, 11, 7, 2, 1, 10, 9, 6, 5, 3, 8]
11: [ 5, 11, 3, 9, 8, 7, 6, 1, 10, 4, 2]
11: [ 6, 1, 10, 2, 9, 8, 7, 5, 11, 3, 4]
11: [ 6, 2, 7, 11, 5, 1, 10, 9, 4, 3, 8]
11: [ 6, 11, 1, 3, 10, 2, 7, 5, 4, 9, 8]
11: [ 9, 5, 3, 1, 10, 2, 8, 7, 11, 6, 4]
12: [ 1, 7, 11, 10, 4, 9, 2, 12, 6, 5, 8, 3]
12: [ 3, 7, 12, 2, 11, 10, 9, 1, 6, 5, 4, 8]
12: [ 3, 8, 11, 2, 12, 9, 1, 7, 5, 10, 4, 6]
12: [ 4, 2, 5, 1, 11, 10, 9, 8, 12, 7, 3, 6]
12: [ 4, 3, 7, 6, 1, 11, 10, 9, 8, 12, 5, 2]
12: [ 5, 1, 6, 11, 9, 2, 10, 7, 12, 8, 3, 4]
12: [ 5, 2, 3, 12, 9, 10, 7, 6, 1, 11, 4, 8]
12: [ 5, 7, 12, 2, 10, 9, 8, 11, 1, 4, 6, 3]
12: [ 7, 1, 2, 3, 5, 9, 10, 8, 11, 6, 12, 4]
12: [ 8, 7, 1, 11, 9, 3, 5, 10, 6, 4, 12, 2]
12: [ 8, 7, 11, 10, 12, 3, 1, 9, 6, 5, 4, 2]
12: [12, 3, 11, 5, 1, 10, 8, 7, 6, 4, 9, 2]
12: [12, 7, 11, 1, 9, 3, 2, 10, 6, 5, 4, 8]
13: [ 2, 1, 4, 7, 11, 6, 3, 10, 13, 5, 8, 12, 9]
13: [ 2, 5, 13, 12, 4, 11, 3, 1, 9, 7, 8, 6, 10]
13: [ 2, 13, 12, 11, 3, 1, 9, 4, 8, 7, 10, 5, 6]
13: [ 3, 5, 2, 1, 12, 9, 11, 10, 7, 6, 13, 4, 8]
13: [ 3, 5, 13, 1, 11, 2, 9, 8, 7, 12, 6, 4, 10]
13: [ 4, 13, 3, 1, 12, 11, 10, 9, 7, 2, 5, 6, 8]
13: [ 6, 4, 3, 1, 10, 11, 13, 5, 9, 12, 7, 8, 2]
13: [ 6, 4, 13, 7, 5, 1, 12, 11, 10, 9, 8, 3, 2]
13: [ 6, 7, 3, 13, 12, 11, 10, 2, 1, 9, 5, 4, 8]
13: [ 6, 7, 13, 11, 2, 10, 9, 1, 8, 12, 5, 3, 4]
13: [ 6, 11, 7, 13, 1, 10, 2, 12, 9, 8, 5, 4, 3]
13: [ 7, 3, 2, 1, 11, 10, 9, 8, 13, 5, 12, 4, 6]
13: [ 7, 5, 13, 3, 10, 11, 2, 9, 1, 6, 8, 4, 12]
13: [ 7, 5, 13, 3, 11, 2, 9, 8, 1, 6, 12, 4, 10]
13: [ 7, 5, 13, 3, 11, 12, 2, 1, 9, 8, 6, 4, 10]
13: [ 7, 9, 1, 11, 3, 13, 2, 10, 12, 6, 5, 4, 8]
13: [ 8, 3, 5, 11, 13, 9, 10, 7, 1, 6, 4, 12, 2]
13: [ 8, 3, 13, 1, 5, 11, 10, 9, 12, 7, 6, 4, 2]
13: [ 9, 3, 13, 2, 10, 4, 1, 7, 6, 5, 12, 11, 8]
13: [ 9, 4, 7, 5, 1, 11, 13, 10, 12, 8, 6, 3, 2]
13: [ 9, 5, 4, 13, 2, 11, 8, 10, 1, 7, 12, 3, 6]
13: [ 9, 5, 13, 4, 11, 1, 8, 3, 7, 12, 6, 10, 2]
13: [10, 4, 3, 5, 13, 1, 9, 11, 7, 6, 8, 12, 2]
13: [11, 2, 7, 3, 12, 1, 10, 9, 6, 5, 13, 4, 8]
13: [11, 13, 5, 2, 10, 9, 8, 7, 1, 6, 4, 3, 12]
13: [11, 13, 7, 1, 12, 9, 2, 3, 10, 5, 4, 6, 8]
13: [12, 1, 3, 5, 11, 13, 4, 10, 9, 8, 7, 6, 2]
13: [12, 7, 13, 3, 11, 1, 9, 8, 6, 5, 10, 4, 2]
13: [12, 13, 7, 11, 2, 5, 1, 9, 10, 6, 4, 3, 8]
13: [13, 3, 1, 12, 11, 2, 9, 10, 7, 6, 4, 5, 8]
13: [13, 3, 7, 1, 5, 12, 4, 10, 9, 8, 11, 6, 2]
14: [ 3, 5, 13, 14, 1, 12, 11, 10, 9, 8, 7, 6, 4, 2]
14: [ 3, 9, 1, 13, 11, 10, 2, 4, 7, 14, 6, 8, 5, 12]
14: [ 3, 14, 4, 12, 11, 1, 9, 8, 2, 13, 7, 5, 10, 6]
14: [ 4, 11, 1, 13, 7, 10, 12, 2, 14, 9, 8, 5, 6, 3]
14: [ 4, 14, 2, 5, 13, 1, 12, 11, 7, 6, 10, 9, 3, 8]
14: [ 5, 7, 1, 13, 12, 11, 10, 2, 9, 8, 14, 6, 4, 3]
14: [ 6, 3, 14, 5, 11, 13, 2, 12, 9, 1, 7, 4, 8, 10]
14: [ 6, 14, 1, 12, 5, 13, 2, 11, 9, 7, 8, 4, 3, 10]
14: [ 7, 5, 13, 12, 1, 11, 4, 10, 2, 14, 9, 8, 6, 3]
14: [ 7, 11, 5, 13, 1, 3, 2, 4, 10, 9, 14, 6, 8, 12]
14: [ 7, 14, 1, 13, 2, 5, 11, 12, 10, 9, 8, 4, 3, 6]
14: [ 8, 7, 5, 13, 2, 11, 3, 9, 10, 12, 1, 14, 4, 6]
14: [11, 2, 10, 5, 8, 7, 9, 1, 13, 14, 12, 4, 3, 6]
14: [11, 3, 14, 2, 13, 1, 10, 8, 9, 7, 5, 12, 4, 6]
14: [11, 5, 3, 14, 2, 1, 13, 10, 8, 7, 6, 12, 4, 9]
14: [11, 14, 5, 3, 13, 1, 10, 2, 9, 4, 7, 8, 12, 6]
14: [12, 1, 14, 3, 13, 4, 10, 9, 2, 7, 6, 5, 11, 8]
14: [12, 11, 7, 5, 13, 3, 2, 14, 1, 9, 8, 4, 6, 10]
14: [12, 14, 7, 13, 6, 5, 11, 1, 10, 9, 8, 4, 3, 2]
14: [13, 1, 7, 2, 11, 3, 9, 14, 8, 6, 5, 10, 4, 12]
14: [13, 11, 3, 1, 4, 2, 7, 10, 9, 6, 14, 12, 5, 8]
14: [14, 1, 13, 3, 11, 5, 10, 9, 2, 6, 8, 7, 4, 12]
14: [14, 5, 1, 13, 12, 2, 11, 3, 7, 9, 6, 8, 4, 10]
---可能有助于数学解决方案---我注意到，从长度9开始，每个长度至少有一个解有一个递减1的长整数序列。
9: [3, 1, 2, 7, 6, 5, 9, 4, 8]
10: [6, 3, 1, 10, 9, 8, 7, 4, 5, 2]
11: [3, 7, 11, 10, 9, 8, 1, 6, 5, 4, 2]
11: [3, 11, 10, 9, 8, 1, 7, 2, 4, 5, 6]
11: [5, 11, 3, 9, 8, 7, 6, 1, 10, 4, 2]
12: [4, 2, 5, 1, 11, 10, 9, 8, 12, 7, 3, 6]
12: [4, 3, 7, 6, 1, 11, 10, 9, 8, 12, 5, 2]
13: [6, 4, 13, 7, 5, 1, 12, 11, 10, 9, 8, 3, 2]
14: [3, 5, 13, 14, 1, 12, 11, 10, 9, 8, 7, 6, 4, 2]
我注意到，对于我测试的每一个长度，除了非常小的，至少有一个解决方案包含相对长的下降数字。到目前为止，这个答案只考虑m=n。以下是几个例子；请注意，超额是n-run_len:
n = 3, run_len = 2, excess = 1: [1] + [3-2] + []
n = 4, run_len = 2, excess = 2: [4, 1] + [3-2] + []
n = 5, run_len = 2, excess = 3: [4, 1, 5] + [3-2] + []
n = 6, no solution
n = 7, run_len = 1, excess = 6: [5] + [7-7] + [3, 1, 6, 4, 2]
n = 8, no solution
n = 9, run_len = 3, excess = 6: [3, 1, 2] + [7-5] + [9, 4, 8]
n = 10, run_len = 4, excess = 6: [6, 3, 1] + [10-7] + [4, 5, 2]
n = 11, run_len = 4, excess = 7: [3, 7] + [11-8] + [1, 6, 5, 4, 2]
n = 12, run_len = 4, excess = 8: [4, 2, 5, 1] + [11-8] + [12, 7, 3, 6]
n = 13, run_len = 5, excess = 8: [6, 4, 13, 7, 5, 1] + [12-8] + [3, 2]
n = 14, run_len = 7, excess = 7: [3, 5, 13, 14, 1] + [12-6] + [4, 2]
n = 15, run_len = 8, excess = 7: [3, 15, 2] + [13-6] + [1, 5, 4, 14]
n = 16, run_len = 6, excess = 10: [6, 3, 1, 10] + [16-11] + [2, 9, 7, 4, 5, 8]
n = 17, run_len = 8, excess = 9: [2, 5, 17, 15, 14, 1] + [13-6] + [4, 3, 16]
n = 18, run_len = 10, excess = 8: [6, 3, 17, 18, 1] + [16-7] + [5, 4, 2]
n = 19, run_len = 10, excess = 9: [4, 19, 3, 17, 18, 1] + [16-7] + [5, 6, 2]
n = 20, no solution found with run_length >= 10
n = 21, run_len = 14, excess = 7: [3, 21, 2] + [19-6] + [1, 5, 4, 20]
n = 22, run_len = 14, excess = 8: [22, 3, 2, 1] + [20-7] + [5, 21, 4, 6]
n = 23, run_len = 14, excess = 9: [7, 1, 23, 3] + [21-8] + [6, 5, 22, 4, 2]
n = 24, run_len = 16, excess = 8: [6, 5, 24, 2] + [22-7] + [3, 1, 23, 4]
n = 25, run_len = 17, excess = 8: [25, 3, 2, 1] + [23-7] + [5, 24, 4, 6]
n = 26, run_len = 17, excess = 9: [26, 3, 25, 2, 1] + [23-7] + [5, 24, 4, 6]
n = 27, run_len = 20, excess = 7: [3, 27, 2] + [25-6] + [1, 5, 4, 26]
n = 28, run_len = 18, excess = 10: [28, 1, 27, 2, 3] + [25-8] + [6, 5, 7, 4, 26]
n = 29, run_len = 20, excess = 9: [2, 5, 29, 27, 26, 1] + [25-6] + [4, 3, 28]
n = 30, run_len = 23, excess = 7: [30, 5, 2, 1] + [28-6] + [29, 3, 4]
n = 31, run_len = 24, excess = 7: [5, 31, 3] + [29-6] + [1, 30, 4, 2]
n = 32, run_len = 23, excess = 9: [7, 32, 31, 2, 1] + [30-8] + [5, 4, 3, 6]
n = 33, run_len = 26, excess = 7: [3, 33, 2] + [31-6] + [1, 5, 4, 32]
n = 34, run_len = 27, excess = 7: [3, 5, 33, 34, 1] + [32-6] + [4, 2]
n = 35, run_len = 27, excess = 8: [5, 35, 3, 33, 34, 1] + [32-6] + [4, 2]
n = 36, run_len = 26, excess = 10: [35, 7, 3, 1, 36, 2] + [34-9] + [6, 5, 4, 8]
n = 37, run_len = 29, excess = 8: [6, 5, 2, 1] + [35-7] + [36, 37, 3, 4]
n = 38, run_len = 29, excess = 9: [3, 7, 37, 38, 1] + [36-8] + [6, 4, 5, 2]
n = 39, run_len = 32, excess = 7: [3, 39, 2] + [37-6] + [1, 5, 4, 38]
n = 40, run_len = 31, excess = 9: [5, 2, 1] + [38-8] + [3, 7, 40, 4, 6, 39]
n = 41, run_len = 33, excess = 8: [3, 5, 1, 40, 2] + [38-6] + [41, 39, 4]
n = 42, run_len = 33, excess = 9: [42, 3, 41, 2, 1] + [39-7] + [5, 4, 40, 6]
n = 43, run_len = 34, excess = 9: [6, 5, 7, 43, 1] + [41-8] + [42, 4, 3, 2]
n = 44, run_len = 35, excess = 9: [5, 3, 2, 1] + [42-8] + [43, 7, 4, 44, 6]
n = 45, run_len = 38, excess = 7: [3, 45, 2] + [43-6] + [1, 5, 4, 44]
n = 50, run_len = 43, excess = 7: [50, 5, 2, 1] + [48-6] + [49, 3, 4]
n = 100, run_len = 91, excess = 9: [5, 2, 1] + [98-8] + [3, 7, 100, 4, 6, 99]
n = 201, run_len = 194, excess = 7: [3, 201, 2] + [199-6] + [1, 5, 4, 200]
上表中缺少20，因为游程长度最多为10，并且计算需要很长时间。我测试过的没有一个更大的值相对于n有这么小的最大运行长度。
我通过检查从n-1递减的游程长度，以及游程&周围元素。这大大减少了搜索空间。
对于给定的n，如果n的任何解中的最大游程是长度n-k，那么它将在O(k！*n)中找到。虽然这看起来很可怕，但如果k有一个恒定的上界(例如，对于所有足够大的n，k<=某个阈值)，那么这实际上是O(n)在上面的例子中，过量就是我所说的k。我还没有找到任何大于10的，但我还没有一个n=20的解决方案。如果它有溶液，那么它的过量将超过10。
更新：这里有很多模式。
如果n mod 6是3并且n>=9，则[3，n，2，[n-2，n-3，…，6]，1，5，4，n-1]有效。
如果n mod 12是5并且n>＝17则[2，5，n，n-2，n-3，1，[n-4，n-5，…，6]，4，3，n-1]有效。
如果n mod 20是10，则[n，5，2，1，[n-2，n-3，…，6]，n-1，3，4]是有效的。
如果n mod 60是7、11、31或47，则[5、n、3、[n-2、n-3、…、6]、1、n-1、4、2]有效。
如果n mod 60是6或18并且n>=18则[6，3，n-1，n，1，[n-2，n-3，…，7]，5，4，2]有效。
如果n mod 60是1、22、25或52并且n>＝22则[n，3，2，1]，[n-2，n-3，…，7]，5，n-1，4，6]有效。
如果n mod 60为23，则[7，1，n，3，[n-2，n-3，…，8]，6，5，n-1，4，2]有效。
如果n mod 60是14或34，则[3，5，n-1，n，1，[n-2，n-3，…，6]，4，2]是有效的。
如果n mod 60是24，则[6，5，n，2，[n-2，n-1，…，7]，3，1，n-1和4]是有效的
如果n mod 60是2、6、26、42并且n>＝26则[n，3，n-1，2，1，[n-3，n-4，…，7]，5，n-2，4，6]有效。
如果n mod 60是16或28，则[n，1，n-1，2，3，[n-3，n-4，…，8]，6，5，7，4，n-2]有效。
如果n mod 60是32，那么[7，n，n-1，2，1，[n-2，n-3，…，8]，5，4，3，6]是有效的。
如果n mod 60为35或47，则[5，n，3，n-2，n-1，[n-3，n-4，…，6]，4，2]有效。
如果n mod 60为37，则[6，5，2，1，[n-2，n-1，…，7]，n-1、n，3，4]
如果n mod 60为38，则[3，7，n-1，n，1]+[n-2，n-3，…，8]+[6，4，5，2]
如果n mod 60为40，则[5，2，1，[n-2，n-3，…，8]，3，7，n，4，6，n-1]为有效
如果n mod 60为0并且n>=60则[3，5，n，2，[n-2，n-3，…，7]，1，6，n-1，4]为有效
如果n mod 60是7、19或31并且n>＝19则[4，n，3，n-2，n-1，[n-3，n-4，…，7]，5，6，2]是有效的
如果n mod 60是23、38或43，则[7、3、n、1、[n-2、n-3、…、8]、6、5、n-1、4、2]是有效的解
如果n mod 60是14或44并且n>=74则[3，5，n-1，n，1，[n-3，n-4，…，6]，n-2，4，2]有效。
如果n mod 60是1或49并且n>＝49则[3，5，n，1，[n-2，n-3，…，7]，2，n-1，4，6]有效。
如果n mod 60是6、18、30、42或54并且n>=18则[n，3，n-1，2，1，[n-3，n-4，…，7]，5，4，n-2，6]有效。
如果n mod 60是10、18、38或58并且n>＝18则[n-1，7，5，n，1，[n-2，n-3，…，8]，2，6，4，3]有效。
当前为n mod 60求解的是以下任何值：
0, 1, 2, 3, 5, 6, 7, 9,
10, 11, 14, 15, 16, 17, 18, 19,
21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 37, 38, 39,
40, 41, 42, 43, 44, 45, 47, 49,
50, 51, 52, 53, 54, 57, 58
此外，
如果n mod 42为31，则[n，3，2，1，[n-2，n-3，…，8]，n-1，5，4，7，6]有效。
如果n mod 420为36或396，则[n-1，7，3，1，n，2，[n-2，n-3，…，9]，6，5，4，8]有效。
---n=21的示例，使用上面列出的第一个模式和所有起始索引。
1: [21, 2, 18, 19, 16, 17, 14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 5, 4, 20, 1]
2: [ 2, 18, 21, 16, 19, 14, 17, 12, 15, 10, 13, 8, 11, 6, 9, 5, 1, 4, 20, 7]
3: [19, 21, 18, 2, 16, 17, 14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 5, 4, 20, 1]
4: [18, 21, 19, 17, 2, 15, 16, 13, 14, 11, 12, 9, 10, 7, 8, 1, 5, 4, 20, 6]
5: [17, 21, 19, 18, 16, 2, 14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 5, 4, 20, 1]
6: [16, 21, 19, 18, 17, 15, 2, 13, 14, 11, 12, 9, 10, 7, 8, 1, 5, 4, 20, 6]
7: [15, 21, 19, 18, 17, 16, 14, 2, 12, 13, 10, 11, 8, 9, 6, 7, 5, 4, 20, 1]
8: [14, 21, 19, 18, 17, 16, 15, 13, 2, 11, 12, 9, 10, 7, 8, 1, 5, 4, 20, 6]
9: [13, 21, 19, 18, 17, 16, 15, 14, 12, 2, 10, 11, 8, 9, 6, 7, 5, 4, 20, 1]
10: [12, 21, 19, 18, 17, 16, 15, 14, 13, 11, 2, 9, 10, 7, 8, 1, 5, 4, 20, 6]
11: [11, 21, 19, 18, 17, 16, 15, 14, 13, 12, 10, 2, 8, 9, 6, 7, 5, 4, 20, 1]
12: [10, 21, 19, 18, 17, 16, 15, 14, 13, 12, 11, 9, 2, 7, 8, 1, 5, 4, 20, 6]
13: [ 9, 21, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 8, 2, 6, 7, 5, 4, 20, 1]
14: [ 8, 21, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 7, 2, 1, 5, 4, 20, 6]
15: [ 7, 21, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 6, 2, 5, 4, 20, 1]
16: [ 6, 21, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 1, 5, 4, 20, 2]
17: [ 1, 5, 2, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 4, 19, 20, 21]
18: [ 5, 2, 18, 19, 16, 17, 14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 1, 20, 21]
19: [ 4, 2, 18, 19, 16, 17, 14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 5, 20, 21, 1]
20: [20, 4, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 1, 5, 21, 2]
你可以观察到递减运行中的元素与该模式适用的所有n值的其他元素之间的关系。这不是一个证明，但你可以将其转化为一个证明。尽管我认为这项工作需要分别为每个模式完成，这超出了我将花时间回答s/O问题的范围。
---我们可以用m>n.---
模式[n-1，n，1，[n-2，n-3，…，3]，n+5]对n有效，mod 4为1并且n>=9.
模式[n，2，1，[n-2，n-3，…，3]，n+4]对n有效，mod 2为0并且n>=6.
有了这两个，再加上我们已经发现的，我们几乎得到了一切。我通过检查有限范围内的单个替换值找到了这些值。
0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53, 54, 56, 57, 58
如果n mod 30是29，那么[3，n，2，[n-2，n-3，…，4]，n-1，n+15)是有效的，给我们n mod 60是59。我们只剩下一个未知：n mod 60是55。
最后！如果n mod 12是7(即n mod 60是7、19、31、43或55)，则[n-1，n，1，[n-2，n-3，…，6]，2，5，3，n+4]对于所有n>＝19.
我们现在有了所有n mod 60的解决方案，在大多数情况下使用m=n，在最坏的情况下使用m=n+15。