Newsgroups: rec.puzzles
From: mag@dogwood.cs.uiuc.edu (Tom Magliery)
Subject: Re: o o . o . o o tiles Z*Z  [SPOILER]
Keywords: tiling
Organization: University of Illinois, Dept. of Comp. Sci., Urbana, IL
Date: Fri, 23 Apr 1993 04:07:05 GMT

(Torsten Sillke) writes:

>  Try to tile Z*Z with
>
>    o o . o . o o 
>
>  There are different tilings.


This was much clearer on my graph paper with colored markers than 
it is in ASCII, but here goes anyway.  I'll use "OO O OO" because
it's easier to comprehend.

Start with an infinite "horizontal diagonalish" row, as follows:

             OO O OO
                OO O OO
                   OO O OO
                      OO O OO
                         OO O OO
                            OO O OO
                               OO O OO
                                  OO O OO
                                     OO O OO

Next put another row underneath it (and another beneath that, etc.):

             OO O OO
      OO O OO   OO O OO
         OO O OO   OO O OO
            OO O OO   OO O OO
               OO O OO   OO O OO
                  OO O OO   OO O OO
                     OO O OO   OO O OO
                        OO O OO   OO O OO
                           OO O OO   OO O OO
                              OO O OO

Now observe that the remaining spaces can be filled with vertical
tiles, in diagonalish rows perpendicular to these diagonalish rows.
(They don't look perpendicular in ASCII form, but they are.)  To make
it easier to see, I've replaced all the O's with -'s and put in some
of the vertical tiles with X's (part of three vertical-diagonalish
rows of them).

                  X
                  X  X
                     X  X
                 XX     X
             -- -X--XX
      -- - --   --X-X--XX
         -- - --XXX--X-X--
            -- -X--XXX--X- --
               --X-X--XXX-- - --
               XXX--X-X--   -- - --
               X  XXX--X- --   -- - --
                X X  XXX-- - --   -- - --
               XX  X X     -- - --   -- - --
                  XX  X       -- - --
               X     XX
               X  X
                  X  X
                     X

mag
-- 
Tom Magliery ** Dept of CS ** 1304 W Springfield ** Urbana IL 61801 ** USA