This is the problem of placing N chess queens on an N × N chessboard so that queens threaten each other sequentially. That is queen[x] threatens queen[x+1] for x = 0 to n-1, and queen[n] threatens queen[0]. So each queen is threatened by two and only two queens. Note the chain of threats is a hamiltonian cycle.
The difference with the classic version is that here each queen must be threatened by two queens and only two queens, and therefore each queen must threaten two queens and only two queens. Therefore this configuration must produce a hamiltonian cycle with N vertices and 2 * N edges. As an example consider the following configuration of queens:
+---+---+---+---+---+---+---+---+
| | | 1 | | | | | |
+---+---+---+---+---+---+---+---+
| | 0 | | | | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| | | 2 | 3 | | | | |
+---+---+---+---+---+---+---+---+
| | | | | 4 | 5 | | |
+---+---+---+---+---+---+---+---+
| | | | | | | 6 | 7 |
+---+---+---+---+---+---+---+---+
The question is how to find all the solutions to this problem?
P.S. Queens can threaten through other queens. i.e. threats to a particular queen can not be obstructed by an intervening queen.