polyomino_parity

polyomino_parity, an Octave code which uses parity considerations to determine whether a given set of polyominoes can tile a specified region.

Licensing:

The computer code and data files made available on this web page are distributed under the MIT license

Languages:

polyomino_parity is available in a MATLAB version and an Octave versionand a Python version.

Related Data and Programs:

boundary_word, an Octave code which works with a polyomino that is described by a sequence of symbols that indicate how to trace out its boundary.

diophantine_nd, an Octave code which is given a Diophantine equation in N variables, and returns all strictly positive solutions, or all nonnegative solutions.

pariomino, an Octave code which considers pariominoes, which are polyominoes with a checkerboard parity.

polyomino_parity_test

polyominoes, an Octave code which defines, solves, and plots a variety of polyomino tiling problems, which are solved by a direct algebraic approach involving the reduced row echelon form (RREF) of a specific matrix, instead of the more typical brute-force or backtracking methods.

Reference:

1. Marcus Garvie, John Burkardt,
   A new mathematical model for tiling finite regions of the plane with polyominoes,
   Contributions to Discrete Mathematics,
   Volume 15, Number 2, July 2020.
2. Solomon Golomb,
   Polyominoes: Puzzles, Patterns, Problems, and Packings,
   Princeton University Press, 1996,
   ISBN: 9780691024448

Source Code:

• addmultisteps.m, seeks combinations of N1 elements of (-Q1,+Q1), N2 elements of (-Q2,Q2), ..., and NK elements of (-QK,+QK), that match a given sum.
• diophantine_nd_nonnegative.m, returns all nonnegative solutions of a Diophantine equation in N variables.
• diophantine_nd_positive.m, returns all strictly positive solutions of a Diophantine equation in N variables.
• pv_search.m, finds trivial and serious parity violations in proposed tilings of a region by a set of polyominoes.
• pv_search_post.m, prints the results from a call to pv_search().
• sumallsteps.m, finds all possible sums of A plus N signed copies of Q.
• timestamp.m, prints the YMDHMS date as a timestamp.