.. -*- coding: utf-8 -*- ================================== Polyominoes: Puzzles & Solutions ================================== :Author: David Goodger :Date: $Date: 2015-02-24 14:21:02 -0600 (Tue, 24 Feb 2015) $ :Revision: $Revision: 600 $ :Web site: http://puzzler.sourceforge.net/ :Copyright: © 1998-2015 by David J. Goodger :License: `GPL 2 <../COPYING.html>`__ .. image:: images/puzzler.png :align: center .. sidebar:: Also see: * `Pentominoes: Puzzles & Solutions `_ * `Hexominoes: Puzzles & Solutions `_ * `Solid Pentominoes: Puzzles & Solutions `_ * `An Introduction to Polyominoes `_ * `Notes on Polyominoes `_ * `Polyform Puzzler: Puzzles & Solutions `_ * `Polyform Puzzler FAQ `_ (`polyform details `__, `numbers of polyforms `__, `interpreting solution files `__) .. contents:: Polyominoes of Order 1 Through 3 ================================ This puzzle uses the 1 monomino, 1 domino, and 2 trominoes, for a total of 9 squares. * .. _polyominoes-123-square: Square: `6 solutions <../solutions/ominoes/polyominoes-123-square.txt>`__ .. image:: images/ominoes/polyominoes-123-square.png A trivial result, but the number of solutions comes in handy in more complex puzzles like the `poly-5 diamond`_. Tetrominoes =========== The 5 tetrominoes are well known from their use in the video game "Tetris". Tetris does not allow the pieces to be flipped (mirror-reflected); its pieces comprise the 7 "one-sided" tetrominoes. Due to a `parity imbalance`_, the 5 free tetrominoes cannot fit into a simple rectangle without introducing holes or other irregularities. .. _parity imbalance: http://en.wikipedia.org/wiki/Tetromino#Tiling_the_rectangle_and_filling_the_box_with_2D_pieces However, if we consider a 5x4 rectangle and join the short sides together to form a cylindrical tube, we can find solutions: * 5x4 tube: `7 solutions <../solutions/ominoes/tetrominoes-5x4-tube.txt>`__ .. image:: images/ominoes/tetrominoes-5x4-tube.png In this solution, the 4x1 "I4" piece spans the join between right and left edges. It appears to be chopped in two above, but if you join the left & right edges, it forms one continuous piece. Here it is, unwrapped: .. image:: images/ominoes/tetrominoes-5x4-tube-unwrapped.png See `this article by Alexandre Owen Muñiz `__ and `a video by Edo Timmermans `__. Polyominoes of Order 1 Through 4 ================================ These puzzles use the 1 monomino, 1 domino, 2 trominoes, and 5 tetrominoes, for a total of 29 squares. * Square-plus (design from Kadon / Kate Jones): `563 solutions <../solutions/ominoes/polyominoes-1234-square-plus.txt>`__ .. image:: images/ominoes/polyominoes-1234-square-plus.png * 7x3-plus (design by Dan Klarskov): `17 solutions <../solutions/ominoes/polyominoes-1234-7x3-plus.txt>`__ .. image:: images/ominoes/polyominoes-1234-7x3-plus.png * Skewered square (design by Dan Klarskov): `1,320 solutions <../solutions/ominoes/polyominoes-1234-skewered-square.txt>`__ .. image:: images/ominoes/polyominoes-1234-skewered-square.png * Skewered 9x3 rectangle (design by Dan Klarskov): `5,249 solutions <../solutions/ominoes/polyominoes-1234-skewered-9x3.txt>`__ .. image:: images/ominoes/polyominoes-1234-skewered-9x3.png * Skewered 7x3 rectangle: `747 solutions <../solutions/ominoes/polyominoes-1234-skewered-7x3.txt>`__ .. image:: images/ominoes/polyominoes-1234-skewered-7x3.png * Astroid (design from Kadon / Kate Jones): `18 solutions <../solutions/ominoes/polyominoes-1234-astroid.txt>`__ .. image:: images/ominoes/polyominoes-1234-astroid.png * 7x5 with cross-shaped hole (design by Dan Klarskov): `19 solutions <../solutions/ominoes/polyominoes-1234-7x5-cross-hole.txt>`__ .. image:: images/ominoes/polyominoes-1234-7x5-cross-hole.png * 7x4 plus one (design by Dan Klarskov): `1,522 solutions <../solutions/ominoes/polyominoes-1234-7x4-plus-one.txt>`__ .. image:: images/ominoes/polyominoes-1234-7x4-plus-one.png This is just a 7x4 rectangle using the polyominoes of order 2 through 4, plus the monomino separately. I thought it was cute. One-Sided Polyominoes of Order 1 Through 4 ========================================== These puzzles use the 1 monomino, 1 domino, 2 trominoes, and 7 tetrominoes, for a total of 37 squares. * Octagon: `solutions incomplete <../solutions/ominoes/one-sided-polyominoes-1234-octagon.txt>`__ .. image:: images/ominoes/one-sided-polyominoes-1234-octagon.png One-Sided Polyominoes of Order 2 Through 4 ========================================== These puzzles use the 1 domino, 2 trominoes, and 7 tetrominoes, for a total of 36 squares. * Square: `7,252 solutions <../solutions/ominoes/one-sided-polyominoes-234-square.txt>`__ .. image:: images/ominoes/one-sided-polyominoes-234-square.png * Octagon: `1,023 solutions <../solutions/ominoes/one-sided-polyominoes-234-octagon.txt>`__ .. image:: images/ominoes/one-sided-polyominoes-234-octagon.png Polyominoes of Order 4 & 5 (Tetrominoes & Pentominoes) ====================================================== These puzzles use the 5 tetrominoes and 12 pentominoes, for a total of 80 squares. Rectangles ---------- * 8x10: `solutions incomplete <../solutions/ominoes/polyominoes-45-8x10.txt>`__ .. image:: images/ominoes/polyominoes-45-8x10.png * 5x16: `solutions incomplete <../solutions/ominoes/polyominoes-45-5x16.txt>`__ .. image:: images/ominoes/polyominoes-45-5x16.png * 4x20: `solutions incomplete <../solutions/ominoes/polyominoes-45-4x20.txt>`__ .. image:: images/ominoes/polyominoes-45-4x20.png * 9x9 Square (with a hole in the middle): `solutions incomplete <../solutions/ominoes/polyominoes-45-square.txt>`__ .. image:: images/ominoes/polyominoes-45-square.png Miscellaneous ------------- * Diamond: `7,302 solutions <../solutions/ominoes/polyominoes-45-diamond.txt>`__ .. image:: images/ominoes/polyominoes-45-diamond.png * Aztec Diamond: `11,162 solutions <../solutions/ominoes/polyominoes-45-aztec-diamond.txt>`__ .. image:: images/ominoes/polyominoes-45-aztec-diamond.png Polyominoes of Order 2 Through 5 ================================ These puzzles use the 1 domino, 2 trominoes, 5 tetrominoes, and 12 pentominoes, for a total of 88 squares. * X (`designed for G4G10 `_): `178,355,676 solutions <../solutions/ominoes/polyominoes-2345-x-1.txt>`__ .. image:: images/ominoes/polyominoes-2345-x-1.png Polyominoes of Order 1 Through 5 ================================ These puzzles use the 1 monomino, 1 domino, 2 trominoes, 5 tetrominoes, and 12 pentominoes, for a total of 89 squares. * .. _poly-5 diamond: Diamond (a.k.a. `Kadon's "Poly-5" `_): `solutions incomplete <../solutions/ominoes/polyominoes-12345-diamond.txt>`__ .. image:: images/ominoes/polyominoes-12345-diamond.png The configuration below has the monomono, domino, & trominoes restricted to a central 3×3 square, resulting in `4,579 unique solutions for the outer ring of pentominoes & tetrominoes`__ alone. Combined with `6 unique independent solutions for the inner square`__, and 8 relative orientations (the inner square can rotate to 4 different positions, and 4 more flipped), the grand total is 219,792 unique solutions (4,579 × 6 × 8). __ ../solutions/ominoes/polyominoes-12345-diamond-2.txt __ polyominoes-123-square_ .. image:: images/ominoes/polyominoes-12345-diamond-2.png Also see the `poly-6 star`_ below. * Crosses: .. list-table:: :class: borderless * - .. figure:: images/ominoes/polyominoes-12345-cross-1.png `solutions incomplete <../solutions/ominoes/polyominoes-12345-cross-1.txt>`__ (design by Kadon) - .. figure:: images/ominoes/polyominoes-12345-cross-2.png `solutions incomplete <../solutions/ominoes/polyominoes-12345-cross-2.txt>`__ Polyominoes of Order 1 Through 6 ================================ These puzzles use the 1 monomino, 1 domino, 2 trominoes, 5 tetrominoes, 12 pentominoes, and 35 hexominoes, for a total of 299 squares. * 23x13 rectangle: `solutions incomplete <../solutions/ominoes/polyominoes-123456-23x13.txt>`__ .. image:: images/ominoes/polyominoes-123456-23x13.png * .. _poly-6 star: Star (design by `Jack Wetterer and Chris Patterson, with symmetry refinements by Darian Jenkins `__, extending `Kadon's "Poly-5"`_ a.k.a. `poly-5 diamond`_ above): `solutions incomplete <../solutions/ominoes/polyominoes-123456-star.txt>`__ .. image:: images/ominoes/polyominoes-123456-star.png As with the `poly-5 diamond`_ above, the various orders of polyominoes occupy different areas. The coloured version below shows the monomino (black), domino (gray), and trominoes (purple) fixed in a central 3×3 square, with the tetrominoes (blue) and pentominoes (green) occupying a middle ring around the square. The hexominoes (red) occupy the outer ring of the configuration. .. image:: images/ominoes/polyominoes-123456-star-coloured.png