define clean [ matrix m ]
  matrix m2 = matrix_of_zeroes(m.dimensions)
  for point p (m.allpoints)
    if m.value(p) and not m2.value(p)
      if not try_solution(m, m2, [ p ])
        m.value(p) = 0
  # by now m and m2 are the same

define try_solution [ matrix m, matrix m2, array of points plist ]
  int row = first_row_not_already_tried(plist)
  if not defined row
    # all rows tried, so this is a solution
    for point p (@plist)
      m2.value(p) = 1
    return true
  for int col (grep column_not_in_plist(plist, $_), m.allcols)
    point p = point_at(row, col)

    if m.value(p)
      if try_solution(m, m2, [ @plist, p ])
        return true
  return false