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# Family Games America’s Quantumino solver

This module allows to solve the Quantumino puzzle made by Family Games America (see also this video on Youtube). This puzzle was left at the dinner room of the Laboratoire de Combinatoire Informatique Mathematique in Montreal by Franco Saliola during winter 2011.

The solution uses the dancing links code which is in Sage and is based on the more general code available in the module `sage.combinat.tiling`. Dancing links were originally introduced by Donald Knuth in 2000 (arXiv:cs/0011047). In particular, Knuth used dancing links to solve tilings of a region by 2D pentaminos. Here we extend the method for 3D pentaminos.

This module defines two classes :

- `sage.games.quantumino.QuantuminoState` class, to represent a state of the Quantumino game, i.e. a solution or a partial solution.
- `sage.games.quantumino.QuantuminoSolver` class, to find, enumerate and count the number of solutions of the Quantumino game where one of the piece is put aside.

AUTHOR:

> - Sebastien Labbe, April 28th, 2011

DESCRIPTION (from [1]):

> ” Pentamino games have been taken to a whole different level; a 3-D level, with this colorful creation! Using the original pentamino arrangements of 5 connected squares which date from 1907, players are encouraged to “think inside the box” as they try to fit 16 of the 17 3-D pentamino pieces inside the playing perimeters. Remove a different piece each time you play for an entirely new challenge! Thousands of solutions to be found! Quantumino hands-on educational tool where players learn how shapes can be transformed or arranged into predefined shapes and spaces. Includes: 1 wooden frame, 17 wooden blocks, instruction booklet. Age: 8+ “

EXAMPLES:

Here are the 17 wooden blocks of the Quantumino puzzle numbered from 0 to 16 in the following 3d picture. They will show up in 3D in your default (=Jmol) viewer:

    sage: from sage.games.quantumino import show_pentaminos
    sage: show_pentaminos()
    Graphics3d Object

To solve the puzzle where the pentamino numbered 12 is put aside:

    sage: from sage.games.quantumino import QuantuminoSolver
    sage: s = QuantuminoSolver(12).solve().next()         # long time (10 s)
    sage: s                                               # long time (<1s)
    Quantumino state where the following pentamino is put aside :
    Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (2, 1, 1)], Color: blue
    sage: s.show3d()                                      # long time (<1s)
    Graphics3d Object

To remove the frame:

    sage: s.show3d().show(frame=False)                    # long time (<1s)

To solve the puzzle where the pentamino numbered 7 is put aside:

    sage: s = QuantuminoSolver(7).solve().next()          # long time (10 s)
    sage: s                                               # long time (<1s)
    Quantumino state where the following pentamino is put aside :
    Polyomino: [(0, 0, 0), (0, 1, 0), (0, 2, 0), (0, 2, 1), (1, 0, 0)], Color: orange
    sage: s.show3d()                                      # long time (<1s)
    Graphics3d Object

The solution is iterable. This may be used to explicitly list the positions of each pentamino:

    sage: for p in s: p                                   # long time (<1s)
    Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink
    Polyomino: [(0, 0, 1), (0, 1, 0), (0, 1, 1), (0, 2, 1), (1, 2, 1)], Color: deeppink
    Polyomino: [(0, 2, 0), (0, 3, 0), (0, 4, 0), (1, 4, 0), (1, 4, 1)], Color: green
    Polyomino: [(0, 3, 1), (1, 3, 1), (2, 2, 0), (2, 2, 1), (2, 3, 1)], Color: green
    Polyomino: [(1, 3, 0), (2, 3, 0), (2, 4, 0), (2, 4, 1), (3, 4, 0)], Color: red
    Polyomino: [(1, 0, 1), (2, 0, 1), (2, 1, 0), (2, 1, 1), (3, 1, 1)], Color: red
    Polyomino: [(2, 0, 0), (3, 0, 0), (3, 0, 1), (3, 1, 0), (4, 0, 0)], Color: gray
    Polyomino: [(3, 2, 0), (4, 0, 1), (4, 1, 0), (4, 1, 1), (4, 2, 0)], Color: purple
    Polyomino: [(3, 2, 1), (3, 3, 0), (3, 3, 1), (4, 2, 1), (4, 3, 1)], Color: yellow
    Polyomino: [(3, 4, 1), (3, 5, 1), (4, 3, 0), (4, 4, 0), (4, 4, 1)], Color: blue
    Polyomino: [(0, 4, 1), (0, 5, 0), (0, 5, 1), (0, 6, 1), (1, 5, 0)], Color: midnightblue
    Polyomino: [(0, 6, 0), (0, 7, 0), (0, 7, 1), (1, 7, 0), (2, 7, 0)], Color: darkblue
    Polyomino: [(1, 7, 1), (2, 6, 0), (2, 6, 1), (2, 7, 1), (3, 6, 0)], Color: blue
    Polyomino: [(1, 5, 1), (1, 6, 0), (1, 6, 1), (2, 5, 0), (2, 5, 1)], Color: yellow
    Polyomino: [(3, 6, 1), (3, 7, 0), (3, 7, 1), (4, 5, 1), (4, 6, 1)], Color: purple
    Polyomino: [(3, 5, 0), (4, 5, 0), (4, 6, 0), (4, 7, 0), (4, 7, 1)], Color: orange

To get all the solutions, use the iterator returned by the `solve` method. Note that finding the first solution is the most time consuming because it needs to create the complete data to describe the problem:

    sage: it = QuantuminoSolver(7).solve()
    sage: it.next()                                     # not tested (10s)
    Quantumino state where the following pentamino is put aside :
    Polyomino: [(0, 0, 0), (0, 1, 0), (0, 2, 0), (0, 2, 1), (1, 0, 0)], Color: orange
    sage: it.next()                                     # not tested (0.001s)
    Quantumino state where the following pentamino is put aside :
    Polyomino: [(0, 0, 0), (0, 1, 0), (0, 2, 0), (0, 2, 1), (1, 0, 0)], Color: orange
    sage: it.next()                                     # not tested (0.001s)
    Quantumino state where the following pentamino is put aside :
    Polyomino: [(0, 0, 0), (0, 1, 0), (0, 2, 0), (0, 2, 1), (1, 0, 0)], Color: orange

To get the solution inside other boxes:

    sage: s = QuantuminoSolver(7, box=(4,4,5)).solve().next()       # not tested (2s)
    sage: s.show3d()                                                # not tested (<1s)

    sage: s = QuantuminoSolver(7, box=(2,2,20)).solve().next()      # not tested (1s)
    sage: s.show3d()                                                # not tested (<1s)

If there are no solution, a StopIteration error is raised:

    sage: QuantuminoSolver(7, box=(3,3,3)).solve().next()
    Traceback (most recent call last):
    ...
    StopIteration

The implementation allows a lot of introspection. From the `TilingSolver` object, it is possible to retrieve the rows that are passed to the DLX solver and count them. It is also possible to get an instance of the DLX solver to play with it:

    sage: q = QuantuminoSolver(0)
    sage: T = q.tiling_solver()
    sage: T
    Tiling solver of 16 pieces into the box (5, 8, 2)
    Rotation allowed: True
    Reflection allowed: False
    Reusing pieces allowed: False
    sage: rows = T.rows()                            # not tested (10 s)
    sage: len(rows)                                  # not tested (but fast)
    5484
    sage: x = T.dlx_solver()                         # long time (10 s)
    sage: x                                          # long time (fast)
    <sage.combinat.matrices.dancing_links.dancing_linksWrapper object at ...>

REFERENCES:

- [1] Family Games America’s Quantumino
- [2] Quantumino - How to Play on Youtube
- [3] Knuth, Donald (2000). “Dancing links”. arXiv:cs/0011047.

class `sage.games.quantumino.``QuantuminoSolver`(aside, box=(5, 8, 2))

  Bases: `sage.structure.sage_object.SageObject`

  Return the Quantumino solver for the given box where one of the pentamino is put aside.

  INPUT:

  - `aside` - integer, from 0 to 16, the aside pentamino
  - `box` - tuple of size three (optional, default: `(5,8,2)`), size of the box

  EXAMPLES:

      sage: from sage.games.quantumino import QuantuminoSolver
      sage: QuantuminoSolver(9)
      Quantumino solver for the box (5, 8, 2)
      Aside pentamino number: 9
      sage: QuantuminoSolver(12, box=(5,4,4))
      Quantumino solver for the box (5, 4, 4)
      Aside pentamino number: 12

  `number_of_solutions`()

    Return the number of solutions.

    OUTPUT:

    > integer

    EXAMPLES:

        sage: from sage.games.quantumino import QuantuminoSolver
        sage: QuantuminoSolver(4, box=(3,2,2)).number_of_solutions()
        0

    This computation takes several days:

        sage: QuantuminoSolver(0).number_of_solutions()                # not tested
        ??? hundreds of millions ???

  `solve`(partial=None)

    Return an iterator over the solutions where one of the pentamino is put aside.

    INPUT:

    - `partial` - string (optional, default: `None`), whether to include partial (incomplete) solutions. It can be one of the following:
      - `None` - include only complete solution
      - `'common'` - common part between two consecutive solutions
      - `'incremental'` - one piece change at a time

    OUTPUT:

    > iterator of QuantuminoState

    EXAMPLES:

    Get one solution:

        sage: from sage.games.quantumino import QuantuminoSolver
        sage: s = QuantuminoSolver(8).solve().next()         # long time (9s)
        sage: s                                              # long time (fast)
        Quantumino state where the following pentamino is put aside :
        Polyomino: [(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 0)], Color: yellow
        sage: s.show3d()                                     # long time (< 1s)
        Graphics3d Object

    The explicit solution:

        sage: for p in s: p                                  # long time (fast)
        Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink
        Polyomino: [(0, 0, 1), (0, 1, 0), (0, 1, 1), (0, 2, 1), (1, 2, 1)], Color: deeppink
        Polyomino: [(0, 2, 0), (0, 3, 0), (0, 4, 0), (1, 4, 0), (1, 4, 1)], Color: green
        Polyomino: [(0, 3, 1), (1, 3, 1), (2, 2, 0), (2, 2, 1), (2, 3, 1)], Color: green
        Polyomino: [(1, 3, 0), (2, 3, 0), (2, 4, 0), (2, 4, 1), (3, 4, 0)], Color: red
        Polyomino: [(1, 0, 1), (2, 0, 0), (2, 0, 1), (2, 1, 0), (3, 0, 1)], Color: midnightblue
        Polyomino: [(0, 4, 1), (0, 5, 0), (0, 5, 1), (0, 6, 0), (1, 5, 0)], Color: red
        Polyomino: [(2, 1, 1), (3, 0, 0), (3, 1, 0), (3, 1, 1), (4, 0, 0)], Color: blue
        Polyomino: [(3, 2, 0), (4, 0, 1), (4, 1, 0), (4, 1, 1), (4, 2, 0)], Color: purple
        Polyomino: [(3, 2, 1), (3, 3, 0), (4, 2, 1), (4, 3, 0), (4, 3, 1)], Color: yellow
        Polyomino: [(3, 3, 1), (3, 4, 1), (4, 4, 0), (4, 4, 1), (4, 5, 0)], Color: blue
        Polyomino: [(0, 6, 1), (0, 7, 0), (0, 7, 1), (1, 5, 1), (1, 6, 1)], Color: purple
        Polyomino: [(1, 6, 0), (1, 7, 0), (1, 7, 1), (2, 7, 0), (3, 7, 0)], Color: darkblue
        Polyomino: [(2, 5, 0), (2, 6, 0), (3, 6, 0), (4, 6, 0), (4, 6, 1)], Color: orange
        Polyomino: [(2, 5, 1), (3, 5, 0), (3, 5, 1), (3, 6, 1), (4, 5, 1)], Color: gray
        Polyomino: [(2, 6, 1), (2, 7, 1), (3, 7, 1), (4, 7, 0), (4, 7, 1)], Color: orange

    Enumerate the solutions:

        sage: it = QuantuminoSolver(0).solve()
        sage: it.next()                                          # not tested
        Quantumino state where the following pentamino is put aside :
        Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink
        sage: it.next()                                          # not tested
        Quantumino state where the following pentamino is put aside :
        Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink

    With the partial solutions included, one can see the evolution between consecutive solutions (an animation would be better):

        sage: it = QuantuminoSolver(0).solve(partial='common')
        sage: it.next().show3d()               # not tested (2s)
        sage: it.next().show3d()               # not tested (< 1s)
        sage: it.next().show3d()               # not tested (< 1s)

    Generalizations of the game inside different boxes:

        sage: QuantuminoSolver(7, (4,4,5)).solve().next()       # long time (2s)
        Quantumino state where the following pentamino is put aside :
        Polyomino: [(0, 0, 0), (0, 1, 0), (0, 2, 0), (0, 2, 1), (1, 0, 0)], Color: orange
        sage: QuantuminoSolver(7, (2,2,20)).solve().next()      # long time (1s)
        Quantumino state where the following pentamino is put aside :
        Polyomino: [(0, 0, 0), (0, 1, 0), (0, 2, 0), (0, 2, 1), (1, 0, 0)], Color: orange
        sage: QuantuminoSolver(3, (2,2,20)).solve().next()      # long time (1s)
        Quantumino state where the following pentamino is put aside :
        Polyomino: [(0, 0, 0), (0, 1, 0), (0, 2, 0), (1, 0, 0), (1, 0, 1)], Color: green

    If the volume of the box is not 80, there is no solution:

        sage: QuantuminoSolver(7, box=(3,3,9)).solve().next()
        Traceback (most recent call last):
        ...
        StopIteration

    If the box is too small, there is no solution:

        sage: QuantuminoSolver(4, box=(40,2,1)).solve().next()
        Traceback (most recent call last):
        ...
        StopIteration

  `tiling_solver`()

    Return the Tiling solver of the Quantumino Game where one of the pentamino is put aside.

    EXAMPLES:

        sage: from sage.games.quantumino import QuantuminoSolver
        sage: QuantuminoSolver(0).tiling_solver()
        Tiling solver of 16 pieces into the box (5, 8, 2)
        Rotation allowed: True
        Reflection allowed: False
        Reusing pieces allowed: False
        sage: QuantuminoSolver(14).tiling_solver()
        Tiling solver of 16 pieces into the box (5, 8, 2)
        Rotation allowed: True
        Reflection allowed: False
        Reusing pieces allowed: False
        sage: QuantuminoSolver(14, box=(5,4,4)).tiling_solver()
        Tiling solver of 16 pieces into the box (5, 4, 4)
        Rotation allowed: True
        Reflection allowed: False
        Reusing pieces allowed: False

class `sage.games.quantumino.``QuantuminoState`(pentos, aside)

  Bases: `sage.structure.sage_object.SageObject`

  A state of the Quantumino puzzle.

  Used to represent an solution or a partial solution of the Quantumino puzzle.

  INPUT:

  - `pentos` - list of 16 3d pentamino representing the (partial) solution
  - `aside` - 3d polyomino, the unused 3D pentamino

  EXAMPLES:

      sage: from sage.games.quantumino import pentaminos, QuantuminoState
      sage: p = pentaminos[0]
      sage: q = pentaminos[5]
      sage: r = pentaminos[11]
      sage: S = QuantuminoState([p,q], r)
      sage: S
      Quantumino state where the following pentamino is put aside :
      Polyomino: [(0, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 2, 0)], Color: darkblue

      sage: from sage.games.quantumino import QuantuminoSolver
      sage: QuantuminoSolver(3).solve().next()      # not tested (1.5s)
      Quantumino state where the following pentamino is put aside :
      Polyomino: [(0, 0, 0), (0, 1, 0), (0, 2, 0), (1, 0, 0), (1, 0, 1)], Color: green

  `list`()

    Return the list of 3d polyomino making the solution.

    EXAMPLES:

        sage: from sage.games.quantumino import pentaminos, QuantuminoState
        sage: p = pentaminos[0]
        sage: q = pentaminos[5]
        sage: r = pentaminos[11]
        sage: S = QuantuminoState([p,q], r)
        sage: L = S.list()
        sage: L[0]
        Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink
        sage: L[1]
        Polyomino: [(0, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (2, 0, 1)], Color: red

  `show3d`(size=0.85)

    Return the solution as a 3D Graphic object.

    OUTPUT:

    > 3D Graphic Object

    EXAMPLES:

        sage: from sage.games.quantumino import QuantuminoSolver
        sage: s = QuantuminoSolver(0).solve().next()    # not tested (1.5s)
        sage: G = s.show3d()                            # not tested (<1s)
        sage: type(G)                                   # not tested
        <class 'sage.plot.plot3d.base.Graphics3dGroup'>

    To remove the frame:

        sage: G.show(frame=False) # not tested

    To see the solution with Tachyon viewer:

        sage: G.show(viewer='tachyon', frame=False) # not tested

`sage.games.quantumino.``show_pentaminos`(box=(5, 8, 2))

  Show the 17 3-D pentaminos included in the game and the \(5 \times 8 \times 2\) box where 16 of them must fit.

  INPUT:

  - `box` - tuple of size three (optional, default: `(5,8,2)`), size of the box

  OUTPUT:

  > 3D Graphic object

  EXAMPLES:

      sage: from sage.games.quantumino import show_pentaminos
      sage: show_pentaminos()    # not tested (1s)

  To remove the frame do:

      sage: show_pentaminos().show(frame=False)  # not tested (1s)

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