Sudoku solver

function [res,s] = boardSolve(board)

% Code that solves sudoku boards recursively with a brute-force algorithm.

%Checking if board is solved. Returns the board if it's been solved.
    if ~ismember(board,0)
        res = board;
        s = 0;
        return

    else
        % making a list of the coordinates of all empty points on the board
        [r,c] = find(board == 0);

        empties = [r,c];

        [x,y] = size(empties);

        possibles = zeros(x,9);

%       Makes a list of possible moves corresponding to each cell in the
%       variable 'empties'.

        for i = 1:x
            for j = 1:9
                possibles(i,j) = validity(board,empties(i,1),empties(i,2),j);
            end
        end


%       Checks if there are any possible moves at all. Returns if there
%       aren't.
        if ~ismember(possibles,1)
            res = board;
            s = 0;
            return
        end

%       Starts a for-loop that goes through every single legal placement
%       of numbers. One number is placed on the board and then this
%       function is called recursively with the new board.

        for i = 1:x
            for j = 1:9
                if possibles(i,j) == 1
                    if ~ismember(board,0)
                        res = board;
                        s = 0;
                        return
                    else
                        board(empties(i,1),empties(i,2)) = j;
                        [res,s] = boardSolve(board);
                    end
                end
            end
        end

    end
end


%Checks the validity of a placement
function valid = validity(board,row,col,val)

    valid = 1;


    %Determines which region of the board a cell is in.
    q = squareFind(row,col);

    if val == 0
        valid = 1;
    elseif ismember(val,board(row,:)) || ismember(val,board(:,col))
        valid = 0;
    elseif ismember(val,board(q(1,:),q(2,:)))
        valid = false;
    end

end

%Determines which region of the board a cell is in.
function mat = squareFind(row,col)

    if row<=3
        if col<=3
            x = 1;
            y = 1;
        elseif (col <=6)
            x = 1;
            y = 4;
        else
            x = 1;
            y = 7;
        end
    elseif row <=6
        if col<=3
            x = 4;
            y = 1;
        elseif (col <=6)
            x = 4;
            y = 4;
        else
            x = 4;
            y = 7;
        end
    else
        if col<=3
            x = 7;
            y = 1;
        elseif (col <=6)
            x = 7;
            y = 4;
        else
            x = 7;
            y = 7;
        end
    end

    [start] = [x,y];


    mat = [start(1):start(1)+2;start(2):start(2)+2];

end