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A Polyomino board is a grid of squares that need to be covered by Polyomino puzzle pieces. It is possible for some parts of the grid to be blocked off.

For example in the classical Pentomino puzzle, the board is an 8x8 chessboard, with a 2x2 grid at the center of the board blocked of. This leaves 8x8-2x2 = 64 - 4 = 60 squares to be filled. The 60 remaining squares need to be covered up by fitting all of the 12 pentomino shapes, which are 5 squares each.

The Board.t data structure keeps track of the current state of the board. This means that it provides a way to:

determine which squares are still vacant

for occupied squares, determine which puzzle piece has been placed to cover that square.

count the number of vacant spots (to determine if the puzzle is solved)

A value of type square indicates the state of a square at a given board {x;y} coordinate.

A square can be in either one of the following states:

Blocked: a square that is not meant to be covered. Either the square is not within the bounds of the board, or it is one of the squares on the board that are blocked from the start. (e.g. the 4 center squares in the classic board)

Occupied: a polyomino was placed on the board and it covers the square.

Vacant: no polymino has been placed that covers the square yet.

Each board has a size consisting of a width and height. All non-blocked board coordinates are contained within the range 0..width for the x coordinate and 0..height for the y coordinate.

This function returns the size of the board as a point {x=width;y=height}

Initialize a Polyomino board starting state from a 'string image'. The string image represents the vacant squares by `#` characters and any other sqaures by '.'. Note that the dimensions of the board are implied. So for example the classic pentomino board is initialized like so: