Computer Science Department Programming Methods Laboratory Foundations of Programming
Exercises 3
29/03/2001

Your task is to solve the eight-queens puzzle in Funnel. The eight-queens puzzle asks how to place eight queens on a chessboard so that no queen is in check from any other; i.e. no two queens are in the same row, column, or diagonal.

One possible solution is illustrated in the following figure:

One way to solve the puzzle is to work across the board, placing a queen in each column. Once we have placed k - 1 queens, we must place the kth queen in a position where it does not check any of the queens already on the board. This approach can be formulated as a recursive algorithm: Assume that we have already generated the sequence of all possible ways to place k - 1 queens in the first k - 1 columns of the board. For each of these ways, generate an extended set of positions by placing a queen in each row of the kth column. Now filter these, keeping only the positions for which the queen in the kth column is safe with respect to the other queens. This produces the sequence of all qays to place k queens in the first k columns. By continuing this process, we will produce not only one solution, but all solutions to the puzzle. This solution can be implemented with a function `queens`, which returns a sequence of all solutions to the problem of placing n queens on an n x n chessboard.

```def queens(n) = {
val rows = List.enum(1, n)
def queenCols(k) = {
if (k == 0)
List.make(empty)
else
queenCols(k - 1)
Complete this program by implementing the representation for sets of board positions, including the function `adjoin`, which adjoins a new row/column position to a set of positions, and `empty`, which represents an empty set of positions. You must also write the function `safe`, which determines for a set of positions, whether the queen in the kth column is safe with respect to the others.
```[Source: Abelson, Sussman "Structure and Interpretation of Computer Programs"]```