int NumSpaces(const apmatrix<char> & crossWord)
// precondition: crossWord contains only uppercase letters or spaces    
//               # of rows = crossWord.numrows(),
//               # of columns = crossWord.numcols();    
// postcondition: returns the number of positions in crossWords
//                that contain the space character ' '    
{
    int j,k;
    int rows = crossWord.numrows();
    int cols = crossWord.numcols();
    int spaces = 0;
    
    for(j=0; j < rows; j++)
    {
	for(k=0; k < cols; k++)
	{
	    if (crossWord[j][k] == ' ')
	    {
		spaces++;
	    }
	}
    }
    return spaces;
}