Below, "union" means "disjoint-union".

An Lmino looks like this,

        @@
        @

and has these 4 orientations:


        @@      @@       @      @
        @        @      @@      @@

like the 4 corners of a picture-frame.

We now explore how unions of tilable and untilable regions can behave.

================================================================

AN EXAMPLE:


                              +------------+------------+
                              |AAAAAAAAAAAA|AAAAAAAAAAAA|
Region A has 3 cells          |AAAAAAAAAAAA|AAAAAAAAAAAA|
  and /is/ Lmino tilable,     |AAAAAAAAAAAA|AAAAAAAAAAAA|
  using a single Lmino.       |AAAAAAAAAAAA|AAAAAAAAAAAA|
                              +------------+------------+
                              |AAAAAAAAAAAA|
                              |AAAAAAAAAAAA|
                              |AAAAAAAAAAAA|
                              |AAAAAAAAAAAA|
                              +------------+





Region B, below, has 9 cells [I have marked one of them with "o" and the
  rest with "B"].  To see that region B is not tilable, consider 
  the Lmino covering cell-o.
  This Lmino /disconnects/ B into two subregions, one with
  2-cells (untilable, since 2 is not divisible by 3); hence region B
  is untilable.


    +------------+------------+            +------------+------------+
    |BBBBBBBBBBBB|BBBBBBBBBBBB|            |BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|            |BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|            |BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|            |BBBBBBBBBBBB|BBBBBBBBBBBB|
    +------------+------------+------------+------------+------------+
    |BBBBBBBBBBBB|BBBBBBBBBBBB|oooooooooooo|BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|oooooooooooo|BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|oooooooooooo|BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|oooooooooooo|BBBBBBBBBBBB|BBBBBBBBBBBB|
    +------------+------------+------------+------------+------------+





Let's glue A to B to get region U, below.  ("U" is for Union).
Region U has
                3 + 9  =  12

cells.  So if 
U were tilable, then

                12/3  =  4

Lminos would be used.


                              +------------+------------+
  Region U:                   |AAAAAAAAAAAA|AAAAAAAAAAAA|
                              |AAAAAAAAAAAA|AAAAAAAAAAAA|
                              |AAAAAAAAAAAA|AAAAAAAAAAAA|
                              |AAAAAAAAAAAA|AAAAAAAAAAAA|
    +------------+------------+------------+------------+------------+
    |BBBBBBBBBBBB|BBBBBBBBBBBB|AAAAAAAAAAAA|BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|AAAAAAAAAAAA|BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|AAAAAAAAAAAA|BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|AAAAAAAAAAAA|BBBBBBBBBBBB|BBBBBBBBBBBB|
    +------------+------------+------------+------------+------------+
    |BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|
    |BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|BBBBBBBBBBBB|
    +------------+------------+------------+------------+------------+




Indeed, region U /is/ tilable.  E.g


                              +------------+------------+
                              |@@@@@@@@@@@@|@@@@@@@@@@@@|
                              |@@@@@@@@@@@@|@@@@@@@@@@@@|
                              |@@@@@@@@@@@@|@@@@@@@@@@@@|
                              |@@@@@@@@@@@@|@@@@@@@@@@@@|
    +------------+------------+------------+------------+------------+
    |,,,,,,,,,,,,|,,,,,,,,,,,,|************|@@@@@@@@@@@@|............|
    |,,,,,,,,,,,,|,,,,,,,,,,,,|************|@@@@@@@@@@@@|............|
    |,,,,,,,,,,,,|,,,,,,,,,,,,|************|@@@@@@@@@@@@|............|
    |,,,,,,,,,,,,|,,,,,,,,,,,,|************|@@@@@@@@@@@@|............|
    +------------+------------+------------+------------+------------+
    |,,,,,,,,,,,,|************|************|............|............|
    |,,,,,,,,,,,,|************|************|............|............|
    |,,,,,,,,,,,,|************|************|............|............|
    |,,,,,,,,,,,,|************|************|............|............|
    +------------+------------+------------+------------+------------+


================================================================

The Upshot:

  The union of tilable-region with an untilable MIGHT be tilable, or might
be untilable (examples of this latter behaviour are easy to find).

  Also, the union of two Untilable might be tilable or untilable.  (Again,
examples are easy to find.  TAKE A MINUTE to find such examples -yes, I'm
talking to YOU, buddy!)

  The only general fact we have here is that the union of two Tilable
regions is necessarily tilable; this, simply by unioning the tilings.


                                                           -