Which one of the following queen positions  provides a solution for  the 4-queens problem?a.(4,2,3,1)b.(2,3,1,4)c.(4,3,2,1)d.(3,1,4,2)

The 4-queens problem is a classic chess problem where the task is to place four queens on a 4x4 chessboard such that no two queens threaten each other. This means that no two queens can be in the same row, column, or diagonal.

The options given are in the form of a sequence of numbers, each representing the column position of a queen in each row from top to bottom.

Let's evaluate each option:

a. (4,2,3,1): The queens are in the 4th column of the 1st row, 2nd column of the 2nd row, 3rd column of the 3rd row, and 1st column of the 4th row. This configuration does not have any queens threatening each other.

b. (2,3,1,4): The queens are in the 2nd column of the 1st row, 3rd column of the 2nd row, 1st column of the 3rd row, and 4th column of the 4th row. The queen in the 1st row and the queen in the 3rd row are on the same diagonal, so they threaten each other.

c. (4,3,2,1): The queens are in the 4th column of the 1st row, 3rd column of the 2nd row, 2nd column of the 3rd row, and 1st column of the 4th row. The queens are all on the same diagonal, so they all threaten each other.

d. (3,1,4,2): The queens are in the 3rd column of the 1st row, 1st column of the 2nd row, 4th column of the 3rd row, and 2nd column of the 4th row. This configuration does not have any queens threatening each other.

So, the correct answer is both a. (4,2,3,1) and d. (3,1,4,2) as they are the only configurations where no two queens threaten each other.