11-Jan-2022 14:12:30 polyominoes_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2. Test polyominoes(). cell_ij_fill_test(): cell_ij_fill() fills in unit cells indexed by (I,J) using matrix coordinate system. cell_ij_fill_test(): Normal end of execution i4_modp_test(): I4_MODP factors a number into a multiple and a remainder. Number Divisor Multiple Remainder 107 50 2 7 107 -50 -2 7 -107 50 -3 43 -107 -50 3 43 Repeat using MOD: 107 50 2 7 107 -50 -3 -43 -107 50 -3 43 -107 -50 2 -7 i4_modp_test(): Normal end of execution. I4_WRAP_TEST I4_WRAP forces an integer to lie within given limits. ILO = 4 IHI = 8 I I4_WRAP(I) -10 5 -9 6 -8 7 -7 8 -6 4 -5 5 -4 6 -3 7 -2 8 -1 4 0 5 1 6 2 7 3 8 4 4 5 5 6 6 7 7 8 8 9 4 10 5 11 6 12 7 13 8 14 4 15 5 16 6 17 7 18 8 19 4 20 5 I4_WRAP_TEST Normal end of execution. i4row_neighbors_test(): Return the immediate neighbors of a grid point. Grid point: 7 9 Neighbors: 6 9 7 8 7 10 8 9 Grid point: -8 9 Neighbors: -9 9 -8 8 -8 10 -7 9 Grid point: 3 -8 Neighbors: 2 -8 3 -9 3 -7 4 -8 Grid point: -5 1 Neighbors: -6 1 -5 0 -5 2 -4 1 Grid point: 10 10 Neighbors: 9 10 10 9 10 11 11 10 i4row_neighbors_test(): Normal end of execution. i4row_sorted_insert_test(): Insert rows into a sorted array of rows. Initial matrix A: 1 1 7 5 8 10 8 10 7 10 4 0 10 10 9 New row #1 8 8 4 Updated matrix A: 1 1 7 5 8 10 8 8 4 8 10 7 10 4 0 10 10 9 New row #2 7 1 7 Updated matrix A: 1 1 7 5 8 10 7 1 7 8 8 4 8 10 7 10 4 0 10 10 9 New row #3 0 3 0 Updated matrix A: 0 3 0 1 1 7 5 8 10 7 1 7 8 8 4 8 10 7 10 4 0 10 10 9 New row #4 1 9 7 Updated matrix A: 0 3 0 1 1 7 1 9 7 5 8 10 7 1 7 8 8 4 8 10 7 10 4 0 10 10 9 New row #5 3 10 0 Updated matrix A: 0 3 0 1 1 7 1 9 7 3 10 0 5 8 10 7 1 7 8 8 4 8 10 7 10 4 0 10 10 9 i4row_sorted_insert_test(): Normal end of execution. i4row_sorted_minus_test(): Remove from S any rows which also occur in sorted A. Sorted row matrix A: 1 1 2 1 1 3 1 1 3 1 3 2 2 1 1 2 1 2 2 2 1 2 2 3 2 3 1 2 3 2 3 2 1 3 2 3 3 2 3 3 3 2 3 3 2 Row matrix S: 1 1 2 3 2 1 3 2 2 2 1 3 2 2 3 1 1 2 1 3 1 2 1 1 3 2 3 3 1 1 Rows of S that do not occur in A: 3 2 2 2 1 3 1 3 1 3 1 1 i4row_sorted_minus_test(): Normal end of execution. i4row_sorted_search_test: Search for an occurrence of a row vector in an array of rows. Row matrix A: 1 1 1 1 1 2 1 2 2 1 2 2 1 2 3 2 1 1 2 2 3 2 3 1 3 1 1 3 1 2 3 2 1 3 2 1 3 2 3 3 3 1 3 3 2 Search object #1 3 2 1 This object occurs in row 12 of A Search object #2 2 1 1 This object occurs in row 6 of A Search object #3 3 3 2 This object occurs in row 15 of A Search object #4 1 1 2 This object occurs in row 2 of A Search object #5 3 1 1 This object occurs in row 9 of A i4row_sorted_search_test: Normal end of execution. i4row_take_random_test: Take a random row from an I4ROW Current matrix: 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 51 52 53 54 61 62 63 64 71 72 73 74 81 82 83 84 91 92 93 94 101 102 103 104 Choice #1 is row 2 21 22 23 24 Current matrix: 31 32 33 34 41 42 43 44 51 52 53 54 61 62 63 64 71 72 73 74 81 82 83 84 91 92 93 94 101 102 103 104 26 26 22 30 30 19 12 8 Choice #2 is row 7 91 92 93 94 Current matrix: 101 102 103 104 26 26 22 30 30 19 12 8 8 26 4 32 15 32 38 20 Choice #3 is row 3 30 19 12 8 Current matrix: 8 26 4 32 15 32 38 20 18 21 33 26 13 21 32 16 Choice #4 is row 4 13 21 32 16 Current matrix: 22 38 23 24 15 36 25 9 Choice #5 is row 1 22 38 23 24 Final matrix A: 15 36 25 9 19 34 10 10 10 8 7 18 i4row_take_random_test: Normal end of execution. i4vec_compare_test: i4vec_compare compares the order of two i4vec's. V1: 1 2 3 V2: 0 5 10 compare = 1 V1: 1 2 3 V2: 1 2 3 compare = 0 V1: 1 2 3 V2: 1 3 0 compare = -1 i4vec_compare_test: Normal end of execution. pentomino_display_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 pentomino_display() displays a picture of a pentomino. Graphics saved as "F.png" Graphics saved as "I.png" Graphics saved as "L.png" Graphics saved as "N.png" Graphics saved as "P.png" Graphics saved as "T.png" Graphics saved as "U.png" Graphics saved as "V.png" Graphics saved as "W.png" Graphics saved as "X.png" Graphics saved as "Y.png" Graphics saved as "Z.png" pentomino_display_test(): Normal end of execution. PENTOMINO_MATRIX_TEST MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 PENTOMINO_MATRIX returns a 0/1 matrix representing a pentomino. F pentomino (3,3): 0 1 1 1 1 0 0 1 0 I pentomino (1,5): 1 1 1 1 1 L pentomino (2,4): 0 0 0 1 1 1 1 1 N pentomino (2,4): 1 1 0 0 0 1 1 1 P pentomino (3,2): 1 1 1 1 1 0 T pentomino (3,3): 1 1 1 0 1 0 0 1 0 U pentomino (2,3): 1 0 1 1 1 1 V pentomino (3,3): 1 0 0 1 0 0 1 1 1 W pentomino (3,3): 1 0 0 1 1 0 0 1 1 X pentomino (3,3): 0 1 0 1 1 1 0 1 0 Y pentomino (2,4): 0 0 1 0 1 1 1 1 Z pentomino (3,3): 1 1 0 0 1 0 0 1 1 PENTOMINO_MATRIX_TEST Normal end of execution. PENTOMINO_NAME_TEST MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 PENTOMINO_NAME returns the "name" of a pentomino, given its index between 1 and 12. Index Name 0 "?" 1 "F" 2 "I" 3 "L" 4 "N" 5 "P" 6 "T" 7 "U" 8 "V" 9 "W" 10 "X" 11 "Y" 12 "Z" 13 "?" PENTOMINO_NAME_TEST Normal end of execution. PENTOMINO_PACK_TEST MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 PENTOMINO_PACK packs all 12 pentominoes into an MxNx12 array. Pentomino 1: "F": 0 1 1 0 0 1 1 0 0 0 0 1 0 0 0 Pentomino 2: "I": 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 Pentomino 3: "L": 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 Pentomino 4: "N": 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 Pentomino 5: "P": 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 Pentomino 6: "T": 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 Pentomino 7: "U": 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 Pentomino 8: "V": 1 0 0 0 0 1 0 0 0 0 1 1 1 0 0 Pentomino 9: "W": 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 Pentomino 10: "X": 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 Pentomino 11: "Y": 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 Pentomino 12: "Z": 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 PENTOMINO_PACK_TEST Normal end of execution. PENTOMINO_PRINT_TEST MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 PENTOMINO_PRINT prints a pentomino. F pentomino (3,3): 0 1 1 1 1 0 0 1 0 U pentomino (2,3): 1 0 1 1 1 1 PENTOMINO_PRINT_TEST Normal end of execution. polyomino_area_test: polyomino_area returns the area of a polyomino. Random polyomino of 7 squares on 1 by 2 grid 1 1 Area is 2 Random polyomino of 17 squares on 2 by 2 grid 1 1 1 1 Area is 4 Random polyomino of 10 squares on 2 by 2 grid 1 1 1 1 Area is 4 Random polyomino of 2 squares on 1 by 1 grid 1 Area is 1 Random polyomino of 3 squares on 1 by 1 grid 1 Area is 1 polyomino_area_test(): Normal end of execution. POLYOMINO_CONDENSE_TEST: MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 POLYOMINO_CONDENSE "cleans up" a matrix that is supposed to represent a polyomino: * nonzero entries are set to 1;: * initial and final zero rows and columns are deleted. The initial (3,3) polyomino P: 0 1 1 1 1 0 0 1 0 The condensed (3,3) polyomino Q: 0 1 1 1 1 0 0 1 0 The initial (3,3) polyomino P: 0 1 2 1 3 0 0 -9 0 The condensed (3,3) polyomino Q: 0 1 2 1 3 0 0 -9 0 The initial (3,4) polyomino P: 0 0 0 0 1 3 0 0 0 0 0 0 The condensed (1,2) polyomino Q: 1 3 The initial (2,4) polyomino P: 0 0 0 0 0 0 0 0 The condensed (0,0) polyomino Q: [ Null matrix ] POLYOMINO_CONDENSE_TEST: Normal end of execution. polyomino_display_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 polyomino_display() displays a polyomino. Graphics saved as "polyomino_display.png" polyomino_display_test(): Normal end of execution. polyomino_embed_number_test: MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 polyomino_embed_number reports the number of ways a fixed polyomino can be embedded in a region. The given region R: 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 The given polyomino P: 0 1 0 1 1 1 As a fixed polyomino, P can be embedded in R in 3 ways polyomino_embed_number_test: Normal end of execution. POLYOMINO_EMBED_LIST_TEST: MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 POLYOMINO_EMBED_LIST lists the offsets used to embed a fixed polyomino in a region. The given region R: 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 The given polyomino P: 0 1 0 1 1 1 As a fixed polyomino, P can be embedded in R in 3 ways Embedding number 1: 0 2 1 1 1 2 0 1 2 2 1 1 1 0 1 1 Embedding number 2: 0 1 1 2 1 1 0 2 1 1 2 2 1 0 1 1 Embedding number 3: 0 1 1 1 1 1 0 2 1 1 1 2 1 0 2 2 POLYOMINO_EMBED_LIST_TEST: Normal end of execution. POLYOMINO_ENUMERATE_CHIRAL_TEST: MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 POLYOMINO_ENUMERATE_CHIRAL returns counts of the number of chiral or one-sided polyominoes. Order Number 0 1 1 1 2 1 3 2 4 7 5 18 6 60 7 196 8 704 9 2500 10 9189 11 33896 12 126759 13 476270 14 1802312 15 6849777 16 26152418 17 100203194 18 385221143 19 1485200848 20 5741256764 21 22245940545 22 86383382827 23 336093325058 24 1309998125640 25 5114451441106 26 19998172734786 27 78306011677182 28 307022182222506 29 1205243866707468 30 4736694001644862 POLYOMINO_ENUMERATE_CHIRAL_TEST: Normal end of execution. polyomino_enumerate_fixed_test: MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 polyomino_enumerate_fixed returns counts of the number of fixed polyominoes. Order Number 0 1 1 1 2 2 3 6 4 19 5 63 6 216 7 760 8 2725 9 9910 10 36446 11 135268 12 505861 13 1903890 14 7204874 15 27394666 16 104592937 17 400795844 18 1540820542 19 5940738676 20 22964779660 21 88983512783 22 345532572678 23 1344372335524 24 5239988770268 25 20457802016011 26 79992676367108 27 313224032098244 28 1228088671826973 polyomino_enumerate_fixed_test: Normal end of execution. POLYOMINO_ENUMERATE_FREE_TEST: MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 POLYOMINO_ENUMERATE_FREE returns counts of the number of free polyominoes. Order Number 0 1 1 1 2 1 3 2 4 5 5 12 6 35 7 108 8 369 9 1285 10 4655 11 17073 12 63600 13 238591 14 901971 15 3426576 16 13079255 17 50107909 18 192622052 19 742624232 20 2870671950 21 11123060678 22 43191857688 23 168047007728 24 654999700403 25 2557227044764 26 9999088822075 27 39153010938487 28 153511100594603 POLYOMINO_ENUMERATE_FREE_TEST: Normal end of execution. POLYOMINO_INDEX_TEST MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 POLYOMINO_INDEX assigns an index to each nonzero entry of a polyomino. The polyomino P: 1 0 1 1 1 1 1 0 0 1 1 0 PIN: Index vector for P: 1 0 2 3 4 5 6 0 0 7 8 0 POLYOMINO_INDEX_TEST Normal end of execution. POLYOMINO_LP_WRITE_TEST: MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 POLYOMINO_LP_WRITE writes an LP file associated with a binary programming problem for tiling a region with copies of a single polyomino. POLYOMINO_LP_WRITE created the LP file "reid.lp" POLYOMINO_LP_WRITE_TEST: Normal end of execution. polyomino_monohedral_test: MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 polyomino_monohedral investigates solutions to the problem of tiling a given region R, using multiple copies of a single polyomino P. polyomino_monohedral_test01 Region R: 1 1 0 1 1 1 1 1 1 Polyomino P: 1 1 VERBOSE: The internal variable "verbose" is set to "true"; Print statements marked "VERBOSE" can be suppressed by setting "verbose" to "false". VERBOSE: polyomino_monohedral: Analyze the problem of tiling a region R using copies, possibly rotated or reflected, of a single polyomino P. VERBOSE: Input R has shape (3,3). VERBOSE: Input R is a binary matrix. VERBOSE: Condensed R has shape (3,3). VERBOSE: Input P has shape (1,2). VERBOSE: Input P is a binary matrix. VERBOSE: Condensed P has shape (1,2). VERBOSE: MAX_P = 2 <= MAX_R = 3. VERBOSE: MIN_P = 1 <= MIN_R = 3. VERBOSE: AREA_R = 8 ~= 0. VERBOSE: AREA_P = 2 ~= 0. VERBOSE: AREA_R = 8 is an exact multiple of AREA_P = 2 VERBOSE: 8x10 system matrix A and right hand side B: 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 Linear system saved in LP file "reid.lp" VERBOSE: RREF has determinant = -1 9x10 Row-Reduced Echelon Form system matrix A and right hand side B: 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1-1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1-1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1-1 0 0 0 0 0 0 0 0 0 0 0 1-1 1 1 0 0 0 0 0 0 0 0 0 0 0 VERBOSE: Seek binary solutions with exactly 4 nonzeros VERBOSE: System has 3 degrees of freedom. VERBOSE: Augmented Row-Reduced Echelon Form system matrix A and right hand side B: Columns associated with a free variable are headed with a "*" : : : : : : * : * * 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1-1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1-1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1-1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1-1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 VERBOSE: Tried 8 right hands sides, found 4 solutions. 4 binary solutions were found. Binary solution vectors x: 1 1 0 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 1 Check residuals ||Ax-b||: All solutions had zero residual. Translate each correct solution into a tiling: Tiling based on solution 1 Numeric Labels 1 1 0 4 2 2 4 3 3 Tiling based on solution 1 "Colors" 1 1 0 1 1 1 1 1 1 Tiling based on solution 2 Numeric Labels 1 1 0 2 2 4 3 3 4 Tiling based on solution 2 "Colors" 1 1 0 1 1 1 1 1 1 Tiling based on solution 3 Numeric Labels 2 3 0 2 3 4 1 1 4 Tiling based on solution 3 "Colors" 1 1 0 1 1 1 1 1 1 Tiling based on solution 4 Numeric Labels 1 1 0 2 3 4 2 3 4 Tiling based on solution 4 "Colors" 1 1 0 1 1 1 1 1 1 polyomino_monohedral: Normal end of execution. polyomino_monohedral_test02 Region R: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Polyomino P: 1 1 0 1 1 1 VERBOSE: The internal variable "verbose" is set to "true"; Print statements marked "VERBOSE" can be suppressed by setting "verbose" to "false". VERBOSE: polyomino_monohedral: Analyze the problem of tiling a region R using copies, possibly rotated or reflected, of a single polyomino P. VERBOSE: Input R has shape (4,5). VERBOSE: Input R is a binary matrix. VERBOSE: Condensed R has shape (4,5). VERBOSE: Input P has shape (2,3). VERBOSE: Input P is a binary matrix. VERBOSE: Condensed P has shape (2,3). VERBOSE: MAX_P = 3 <= MAX_R = 5. VERBOSE: MIN_P = 2 <= MIN_R = 4. VERBOSE: AREA_R = 20 ~= 0. VERBOSE: AREA_P = 5 ~= 0. VERBOSE: AREA_R = 20 is an exact multiple of AREA_P = 5 VERBOSE: 20x68 system matrix A and right hand side B: 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1 1 0 1 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 0 1 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 Linear system saved in LP file "4x5.lp" VERBOSE: RREF has determinant = -5 21x68 Row-Reduced Echelon Form system matrix A and right hand side B: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0-1 0 1 0 0 1 1 0-1 0 0 0-1 1 0 1-1 0 1 0 0 0 1 1 0-1 1 0 0 0 1 2-1 0 1 1 0-1 2 1 0-1 2 0 1-1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0-1 1 0 0-1 0 1 0-1 0 0 1 0-1 1 0 0 0 1 0 1-1 0 0 0 0 1 1 0-1 1 0 1-1 2 0 1-1 1-1 1-1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0-1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0-1 0 0 0 0 0 1 0 0 1-1 0 1 0 0 0 1-1 0 1 1 0 0 0 1 0-1 1 0 0 0 0 1-1 1 0 1 0 0 0 1-1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1-1 0 0 0 0 1 0 1 0 0 1 1 0-1 2 0 1-1 1 0 1-1 1 1 0 0 1 1 0-1 2 0 1-1 2 0 1-1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1-1 1 0 0 0 0 1-1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0-1 0 0 0 0 0 0 1 0-1 0 0-1 0 0 1 0 0 0 1-1 0-1 1 0-1 0 0 0-1 0 0 1-1 0 0 0-1-1 1 0-1-1 0 1-2 0 0 1-2 0-1 1-1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0-1 0 0 0 0 0 0 0 0 0 1 0-1 0 1 0-1 0 0 1 0 0 0 0 1-1 0 0 0-1 0 0 1 0 0 0 0-1 0 1 0 0-1 0 0-1-1 1 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0-1 0 1 1 0 0 1 0 0-1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0-1 1 0 0 0 1 0 1-1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0-1 0 0-1 0 0 0 0 0 0 2-1-1 0 0 0-1 1 0 0 0 0-1 1 0-1-1 1 0-2-1 0 1-1 0 0 1-2 0 1-1 0-2 2-1-1-1 1 0-1-1 1 0-2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1-1 0 0-1 0 0 0 0 0 0 1 1-2 0 0 0-1 0 1 1 0 0-1 0 1-1 0 0 1-2 0-1 1 0 0 0 1-1-1 1 0-1-1 0 1-2-1 0 1-1 0 0 1-2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1 1 0 0 0 0 0-1 1 0-1 0 0 0 0-1 0 1-1 0 0 0 0 0 1-1 0 0 0 0 0-1 1 0-1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1 0 1 0 0 0 0-1 0 1-1 0 0 0 0 0-1 1 0 0 0 0 0-1 1 0-1 0 0 0 0-1 0 1-1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0-1-1 1 0 0 0 1 1-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1 0 1-1 1-1 1-1 0-1 1-1 1-1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0-1-1 0 1 0 0 1 1 0-1-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1 1 0 1 0 0 0 1-1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0-1-1 0 0-1 0 1 1 0 0 0-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 1-1 0 0 0-1 0-1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1-1-1 0 0 0-1 1 1 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1 1-1 1-1 1 0 1-1 1-1 1-1 0 1 0 VERBOSE: Seek binary solutions with exactly 4 nonzeros VERBOSE: System has 48 degrees of freedom. 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Binary solution vectors x: 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Check residuals ||Ax-b||: All solutions had zero residual. Translate each correct solution into a tiling: Tiling based on solution 1 Numeric Labels 1 1 3 3 3 1 1 1 3 3 2 2 4 4 4 2 2 2 4 4 Tiling based on solution 1 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Tiling based on solution 2 Numeric Labels 4 4 4 3 3 4 4 3 3 3 1 1 2 2 2 1 1 1 2 2 Tiling based on solution 2 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Tiling based on solution 3 Numeric Labels 1 1 2 2 2 1 1 1 2 2 4 4 4 3 3 4 4 3 3 3 Tiling based on solution 3 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Tiling based on solution 4 Numeric Labels 2 2 2 3 3 4 2 2 3 3 4 4 1 1 3 4 4 1 1 1 Tiling based on solution 4 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Tiling based on solution 5 Numeric Labels 3 3 3 1 1 3 3 1 1 1 4 4 4 2 2 4 4 2 2 2 Tiling based on solution 5 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Tiling based on solution 6 Numeric Labels 4 4 2 2 2 4 4 2 2 3 4 1 1 3 3 1 1 1 3 3 Tiling based on solution 6 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 polyomino_monohedral: Normal end of execution. polyomino_monohedral_test: Normal end of execution. polyomino_monohedral_example_reid_test(): polyomino_monohedral_example_reid() sets up the Reid polyomino tiling example. Region R: 1 1 0 1 1 1 1 1 1 Polyomino P: 1 1 System matrix A and right hand side B: 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 Wrote the LP file "reid.lp" RREF has determinant -1 Row-Reduced Echelon Form system matrix A and right hand side B: 1 0 0 0 0 0 1 0-1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0-1 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1-1 0 1 0 1 0 0 0 0 0 0 0 1 0-1 0 0 0 0 0 0 0 0 0 0 0 0 Found 4 binary solution vectors x: 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 Translate each correct solution into a tiling: Tiling based on solution 1 6 6 2 4 5 2 4 5 Tiling based on solution 2 6 6 2 8 8 2 10 10 Tiling based on solution 3 1 3 1 3 5 9 9 5 Tiling based on solution 4 6 6 7 7 5 9 9 5 polyomino_monohedral_example_reid_test(): Normal end of execution. POLYOMINO_MONOHEDRAL_MATRIX_TEST(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 POLYOMINO_MONOHEDRAL_MATRIX sets up the linear system A*x=b used to search for solutions to the problem of tiling a region R with copies of a polyomino P that may be reflected or rotated. The region R: 1 1 0 1 1 1 1 1 1 The polyomino P: 1 1 Linear system A | b 1 0 0 0 0 1 0 0 0 0 | 1 1 0 0 0 0 0 1 0 0 0 | 1 0 1 0 0 0 1 0 1 0 0 | 1 0 1 1 0 0 0 1 0 1 0 | 1 0 0 1 0 0 0 0 0 0 1 | 1 0 0 0 1 0 0 0 1 0 0 | 1 0 0 0 1 1 0 0 0 1 0 | 1 0 0 0 0 1 0 0 0 0 1 | 1 Wrote the LP file "matrix.lp" POLYOMINO_MONOHEDRAL_MATRIX_TEST: Normal end of execution. polyomino_monohedral_tiling_plot_test(): POLYOMINO_MONOHEDRAL_TILING_PLOT plots solutions to the problem of tiling a given region R, using multiple copies of a single polyomino P. POLYOMINO_MONOHEDRAL_TILING_PLOT_TEST01 Given 4 solutions for the Reid polyomino tiling problem, plot a representation of the tiling corresponding to each solution. Saved plot as "reid01.png" Saved plot as "reid02.png" Saved plot as "reid03.png" Saved plot as "reid04.png" POLYOMINO_MONOHEDRAL_TILING_PLOT_TEST02 Given 6 solutions to the 4x5 rectangle tiling problem, plot a representation of the tiling corresponding to each solution. Saved plot as "rectangle01.png" Saved plot as "rectangle02.png" Saved plot as "rectangle03.png" Saved plot as "rectangle04.png" Saved plot as "rectangle05.png" Saved plot as "rectangle06.png" POLYOMINO_MONOHEDRAL_TILING_PLOT_TEST: Normal end of execution. POLYOMINO_MONOHEDRAL_TILING_PRINT_TEST: MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 POLYOMINO_MONOHEDRAL_TILING_PRINT investigates solutions to the problem of tiling a given region R, using multiple copies of a single polyomino P. POLYOMINO_MONOHEDRAL_TILING_PRINT_TEST01 Given 4 solutions for the Reid polyomino tiling problem, print a representation of the tiling corresponding to each solution. Reid Example Solution #1 Numeric Labels 1 1 0 4 2 2 4 3 3 Reid Example Solution #1 "Colors" 1 1 0 1 1 1 1 1 1 Reid Example Solution #2 Numeric Labels 1 1 0 2 2 4 3 3 4 Reid Example Solution #2 "Colors" 1 1 0 1 1 1 1 1 1 Reid Example Solution #3 Numeric Labels 1 1 0 2 3 4 2 3 4 Reid Example Solution #3 "Colors" 1 1 0 1 1 1 1 1 1 Reid Example Solution #4 Numeric Labels 2 3 0 2 3 4 1 1 4 Reid Example Solution #4 "Colors" 1 1 0 1 1 1 1 1 1 POLYOMINO_MONOHEDRAL_TILING_PRINT_TEST02 Given 6 solutions to the 4x5 rectangle tiling problem, print a representation of the tiling corresponding to each solution. 4x5 Rectangle Example Solution #1 Numeric Labels 1 1 3 3 3 1 1 1 3 3 2 2 4 4 4 2 2 2 4 4 4x5 Rectangle Example Solution #1 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4x5 Rectangle Example Solution #2 Numeric Labels 4 4 4 3 3 4 4 3 3 3 1 1 2 2 2 1 1 1 2 2 4x5 Rectangle Example Solution #2 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4x5 Rectangle Example Solution #3 Numeric Labels 1 1 2 2 2 1 1 1 2 2 4 4 4 3 3 4 4 3 3 3 4x5 Rectangle Example Solution #3 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4x5 Rectangle Example Solution #4 Numeric Labels 2 2 2 3 3 4 2 2 3 3 4 4 1 1 3 4 4 1 1 1 4x5 Rectangle Example Solution #4 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4x5 Rectangle Example Solution #5 Numeric Labels 3 3 3 1 1 3 3 1 1 1 4 4 4 2 2 4 4 2 2 2 4x5 Rectangle Example Solution #5 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4x5 Rectangle Example Solution #6 Numeric Labels 4 4 2 2 2 4 4 2 2 3 4 1 1 3 3 1 1 1 3 3 4x5 Rectangle Example Solution #6 "Colors" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 POLYOMINO_MONOHEDRAL_TILING_PRINT_TEST: Normal end of execution. polyomino_monohedral_variants_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 POLYOMINO_MONOHEDRAL_VARIANTS determines the number of variants of a polyomino. Test polyomino F 0 1 1 1 1 0 0 1 0 F has 8 (M,N) variants (M,N) variant 1 0 1 1 1 1 0 0 1 0 (M,N) variant 2 1 0 0 1 1 1 0 1 0 (M,N) variant 3 0 1 0 0 1 1 1 1 0 (M,N) variant 4 0 1 0 1 1 1 0 0 1 (M,N) variant 5 1 1 0 0 1 1 0 1 0 (M,N) variant 6 0 1 0 1 1 1 1 0 0 (M,N) variant 7 0 1 0 1 1 0 0 1 1 (M,N) variant 8 0 0 1 1 1 1 0 1 0 F has 0 (N,M) variants Test polyomino I 1 1 1 1 1 I has 1 (M,N) variants (M,N) variant 1 1 1 1 1 1 I has 1 (N,M) variants (N,M) variant 1 1 1 1 1 1 Test polyomino L 0 0 0 1 1 1 1 1 L has 4 (M,N) variants (M,N) variant 1 0 0 0 1 1 1 1 1 (M,N) variant 2 1 1 1 1 1 0 0 0 (M,N) variant 3 1 0 0 0 1 1 1 1 (M,N) variant 4 1 1 1 1 0 0 0 1 L has 4 (N,M) variants (N,M) variant 1 1 1 0 1 0 1 0 1 (N,M) variant 2 1 0 1 0 1 0 1 1 (N,M) variant 3 0 1 0 1 0 1 1 1 (N,M) variant 4 1 1 1 0 1 0 1 0 Test polyomino N 1 1 0 0 0 1 1 1 N has 4 (M,N) variants (M,N) variant 1 1 1 0 0 0 1 1 1 (M,N) variant 2 1 1 1 0 0 0 1 1 (M,N) variant 3 0 0 1 1 1 1 1 0 (M,N) variant 4 0 1 1 1 1 1 0 0 N has 4 (N,M) variants (N,M) variant 1 0 1 0 1 1 1 1 0 (N,M) variant 2 0 1 1 1 1 0 1 0 (N,M) variant 3 1 0 1 1 0 1 0 1 (N,M) variant 4 1 0 1 0 1 1 0 1 Test polyomino P 1 1 1 1 1 0 P has 4 (M,N) variants (M,N) variant 1 1 1 1 1 1 0 (M,N) variant 2 0 1 1 1 1 1 (M,N) variant 3 1 1 1 1 0 1 (M,N) variant 4 1 0 1 1 1 1 P has 4 (N,M) variants (N,M) variant 1 1 1 0 1 1 1 (N,M) variant 2 1 1 1 0 1 1 (N,M) variant 3 1 1 1 1 1 0 (N,M) variant 4 0 1 1 1 1 1 Test polyomino T 1 1 1 0 1 0 0 1 0 T has 4 (M,N) variants (M,N) variant 1 1 1 1 0 1 0 0 1 0 (M,N) variant 2 1 0 0 1 1 1 1 0 0 (M,N) variant 3 0 1 0 0 1 0 1 1 1 (M,N) variant 4 0 0 1 1 1 1 0 0 1 T has 0 (N,M) variants Test polyomino U 1 0 1 1 1 1 U has 2 (M,N) variants (M,N) variant 1 1 0 1 1 1 1 (M,N) variant 2 1 1 1 1 0 1 U has 2 (N,M) variants (N,M) variant 1 1 1 0 1 1 1 (N,M) variant 2 1 1 1 0 1 1 Test polyomino V 1 0 0 1 0 0 1 1 1 V has 4 (M,N) variants (M,N) variant 1 1 0 0 1 0 0 1 1 1 (M,N) variant 2 0 0 1 0 0 1 1 1 1 (M,N) variant 3 1 1 1 0 0 1 0 0 1 (M,N) variant 4 1 1 1 1 0 0 1 0 0 V has 0 (N,M) variants Test polyomino W 1 0 0 1 1 0 0 1 1 W has 4 (M,N) variants (M,N) variant 1 1 0 0 1 1 0 0 1 1 (M,N) variant 2 0 0 1 0 1 1 1 1 0 (M,N) variant 3 1 1 0 0 1 1 0 0 1 (M,N) variant 4 0 1 1 1 1 0 1 0 0 W has 0 (N,M) variants Test polyomino X 0 1 0 1 1 1 0 1 0 X has 1 (M,N) variants (M,N) variant 1 0 1 0 1 1 1 0 1 0 X has 0 (N,M) variants Test polyomino Y 0 0 1 0 1 1 1 1 Y has 4 (M,N) variants (M,N) variant 1 0 0 1 0 1 1 1 1 (M,N) variant 2 1 1 1 1 0 1 0 0 (M,N) variant 3 0 1 0 0 1 1 1 1 (M,N) variant 4 1 1 1 1 0 0 1 0 Y has 4 (N,M) variants (N,M) variant 1 0 1 1 1 0 1 0 1 (N,M) variant 2 1 0 1 0 1 1 1 0 (N,M) variant 3 0 1 0 1 1 1 0 1 (N,M) variant 4 1 0 1 1 1 0 1 0 Test polyomino Z 1 1 0 0 1 0 0 1 1 Z has 4 (M,N) variants (M,N) variant 1 1 1 0 0 1 0 0 1 1 (M,N) variant 2 0 0 1 1 1 1 1 0 0 (M,N) variant 3 0 1 1 0 1 0 1 1 0 (M,N) variant 4 1 0 0 1 1 1 0 0 1 Z has 0 (N,M) variants polyomino_monohedral_variants_test(): Normal end of execution. polyomino_multihedral_test(): polyomino_multihedral() sets up and solves the linear system associated with a multi-polyomino tiling problem. polyomino_multihedral_test01(): polyomino_multihedral must solve a multihedral polyomino tiling problem for a 2x4 rectangle. Region R: 1 1 1 1 1 1 1 1 Polyomino N: 1 Polyomino O: 1 1 1 Polyomino P: 0 0 1 1 1 1 VERBOSE: The internal variable "verbose" is set to "true"; Print statements marked "VERBOSE" can be suppressed by setting "verbose" to "false". VERBOSE: polyomino_multihedral: Analyze the problem of tiling a region R using copies, possibly rotated or reflected, of several polyominoes. VERBOSE: Input R_SHAPE has shape (2,4). VERBOSE: Input R_SHAPE is a binary matrix. VERBOSE: Condensed R_SHAPE has shape (2,4). VERBOSE: Input P(1) has shape (2,4). VERBOSE: Input P(1) is a binary matrix. VERBOSE: Condensed P(1) has shape (1,1). VERBOSE: Input P(2) has shape (2,4). VERBOSE: Input P(2) is a binary matrix. VERBOSE: Condensed P(2) has shape (1,3). VERBOSE: Input P(3) has shape (2,4). VERBOSE: Input P(3) is a binary matrix. VERBOSE: Condensed P(3) has shape (2,3). 11x20 system matrix A and right hand side B: 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 VERBOSE: RREF has determinant = 4 12x20 Row-Reduced Echelon Form system matrix A and right hand side B: 1 0 0 0 0 0 0 0 0-1-1-1 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0-1-1 0 0 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0-1-1 0-1 0 0-1-1 0 0 -1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0-1 0-1 0-1-1-1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0-1 0 0 0-1-1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0-1-1 0 0 0-1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 VERBOSE: Seek binary solutions with exactly 3 nonzeros VERBOSE: System has 10 degrees of freedom. VERBOSE: Augmented Row-Reduced Echelon Form system matrix A and right hand side B: Columns associated with a free variable are headed with a "*" : : : : : : : : : * * * : * * * * * * * 1 0 0 0 0 0 0 0 0-1-1-1 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0-1-1 0 0 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0-1-1 0-1 0 0-1-1 0 0 -1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0-1 0-1 0-1-1-1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0-1 0 0 0-1-1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0-1-1 0 0 0-1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 VERBOSE: Tried 176 right hands sides, found 4 solutions. 4 binary solutions were found. Binary solution vectors x: 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 Check Loo residuals ||Ax-b||: All solutions had zero residual. Check Loo residuals ||Ax-b||: All solutions had zero residual. Translate each correct solution into a tiling: ans = 11 20 ans = 20 1 Tiling based on solution 1 Numeric Labels 2 2 2 3 1 3 3 3 Tiling based on solution 1 "Colors" 2 2 2 3 1 3 3 3 ans = 11 20 ans = 20 1 Tiling based on solution 2 Numeric Labels 3 3 3 1 3 2 2 2 Tiling based on solution 2 "Colors" 3 3 3 1 3 2 2 2 ans = 11 20 ans = 20 1 Tiling based on solution 3 Numeric Labels 3 2 2 2 3 3 3 1 Tiling based on solution 3 "Colors" 3 2 2 2 3 3 3 1 ans = 11 20 ans = 20 1 Tiling based on solution 4 Numeric Labels 1 3 3 3 2 2 2 3 Tiling based on solution 4 "Colors" 1 3 3 3 2 2 2 3 polyomino_multihedral_test02(): polyomino_multihedral() must solve a multihedral polyomino tiling problem for a subset of a 4x4 rectangle. Region R: 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 Polyomino N: 0 0 1 1 1 1 Polyomino O: 1 1 1 Polyomino P: 0 1 1 1 VERBOSE: The internal variable "verbose" is set to "true"; Print statements marked "VERBOSE" can be suppressed by setting "verbose" to "false". VERBOSE: polyomino_multihedral: Analyze the problem of tiling a region R using copies, possibly rotated or reflected, of several polyominoes. VERBOSE: Input R_SHAPE has shape (4,4). VERBOSE: Input R_SHAPE is a binary matrix. VERBOSE: Condensed R_SHAPE has shape (4,4). VERBOSE: Input P(1) has shape (4,4). VERBOSE: Input P(1) is a binary matrix. VERBOSE: Condensed P(1) has shape (2,3). VERBOSE: Input P(2) has shape (4,4). VERBOSE: Input P(2) is a binary matrix. VERBOSE: Condensed P(2) has shape (1,3). VERBOSE: Input P(3) has shape (4,4). VERBOSE: Input P(3) is a binary matrix. VERBOSE: Condensed P(3) has shape (2,2). 13x30 system matrix A and right hand side B: 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 VERBOSE: RREF has determinant = -6 14x30 Row-Reduced Echelon Form system matrix A and right hand side B: 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0-1 1 1-1 1 1-1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0-1 1 1 0 0 1 2 1 1 3 1 1 2 1 1 0 2 2 0 0 1 0 0 0 0 0 0 0 1 0 0-1-1-1 0 0-1 0-1-1 1 1-1-1-1 0-1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0-1-1-1-1 0 0-1 1-1-1-1 1-1-1 0 0-1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0-1 0 0 0 1 1 1 1 0-1-1-1-2-1-1-1-2-1 0-1-1 0 0 0 0 0 0 0 1 0 0 1 0 0 0-1-1-1 0 0 1 1 1 2 1 1 1 2 2 0 1 1 0 0 0 0 0 0 0 0 1 0 1-1 0 0 1-1-1 0 0 0 0 1 1-2 1-1 1 1 0 1-1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1-1 0 0-1-1-1-1-2-1-1-2-1-1 0-1 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0-1 0-1 0-1 0-1 0 0-1 0-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 VERBOSE: Seek binary solutions with exactly 3 nonzeros VERBOSE: System has 18 degrees of freedom. VERBOSE: Augmented Row-Reduced Echelon Form system matrix A and right hand side B: Columns associated with a free variable are headed with a "*" : : : : : : : : : * * : : * * * * : * * * * * * * * * * * * 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0-1 1 1-1 1 1-1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0-1 1 1 0 0 1 2 1 1 3 1 1 2 1 1 0 2 2 0 0 1 0 0 0 0 0 0 0 1 0 0-1-1-1 0 0-1 0-1-1 1 1-1-1-1 0-1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0-1-1-1-1 0 0-1 1-1-1-1 1-1-1 0 0-1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0-1 0 0 0 1 1 1 1 0-1-1-1-2-1-1-1-2-1 0-1-1 0 0 0 0 0 0 0 1 0 0 1 0 0 0-1-1-1 0 0 1 1 1 2 1 1 1 2 2 0 1 1 0 0 0 0 0 0 0 0 1 0 1-1 0 0 1-1-1 0 0 0 0 1 1-2 1-1 1 1 0 1-1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1-1 0 0-1-1-1-1-2-1-1-2-1-1 0-1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0-1 0-1 0-1 0-1 0 0-1 0-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 VERBOSE: Tried 988 right hands sides, found 2 solutions. 2 binary solutions were found. Binary solution vectors x: 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 Check Loo residuals ||Ax-b||: Solution vector 1 has a nonzero Loo residual of 2 Solution vector 2 has a nonzero Loo residual of 2 Check Loo residuals ||Ax-b||: All solutions had zero residual. Translate each correct solution into a tiling: ans = 13 30 ans = 30 1 Tiling based on solution 1 Numeric Labels 3 0 0 0 3 0 0 0 4 7 3 2 4 4 1 2 Tiling based on solution 1 "Colors" 2 0 0 0 2 0 0 0 3 5 2 1 3 3 1 1 ans = 13 30 ans = 30 1 Tiling based on solution 2 Numeric Labels 3 0 0 0 3 0 0 0 4 7 3 2 1 4 4 2 Tiling based on solution 2 "Colors" 2 0 0 0 2 0 0 0 3 5 2 1 1 3 3 1 polyomino_multihedral_test(): Normal end of execution. polyomino_multihedral_example_2x4_test(): Set up and solve the 2x4 rectangle polyomino tiling example. Region R: 1 1 1 1 1 1 1 1 Polyomino N: 1 Polyomino O: 1 1 1 Polyomino P: 0 0 1 1 1 1 System matrix A and right hand side B: 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4 8 Wrote the LP file "2x4.lp" RREF has determinant 4 Row-Reduced Echelon Form system matrix A and right hand side B: 1 0 0 0 0 0 0 0 0-1-1-1 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0-1-1 0 0 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0-1-1 0-1 0 0-1-1 0 0 -1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0-1 0-1 0-1-1-1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0-1 0 0 0-1-1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0-1-1 0 0 0-1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 binary solution vectors x: 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 Translate each correct solution into a tiling: Tiling based on solution 1 9 9 9 14 5 14 14 14 Tiling based on solution 2 15 15 15 4 15 12 12 12 Tiling based on solution 3 1 20 20 20 11 11 11 20 Tiling based on solution 4 17 10 10 10 17 17 17 8 polyomino_multihedral_example_2x4_test(): Normal end of execution. polyomino_multihedral_example_4x5_test(): Define the data for an example in which the region is a 4x5 rectangle with a 1x2 hole, tiled with a 2x2 square, 2 copies of the P pentomino, and 1 copy of the L pentomino. VERBOSE: The internal variable "verbose" is set to "true"; Print statements marked "VERBOSE" can be suppressed by setting "verbose" to "false". VERBOSE: polyomino_multihedral_example_4x5_test(): Analyze the problem of tiling a region R using copies, possibly rotated or reflected, of several polyominoes. VERBOSE: Input R_SHAPE has shape (4,5). VERBOSE: Input R_SHAPE is a binary matrix. VERBOSE: Condensed R_SHAPE has shape (4,5). VERBOSE: Input P(1) has shape (4,5). VERBOSE: Input P(1) is a binary matrix. VERBOSE: Condensed P(1) has shape (2,3). VERBOSE: Input P(2) has shape (4,5). VERBOSE: Input P(2) is a binary matrix. VERBOSE: Condensed P(2) has shape (2,2). VERBOSE: Input P(3) has shape (4,5). VERBOSE: Input P(3) is a binary matrix. VERBOSE: Condensed P(3) has shape (2,3). 21x62 system matrix A and right hand side B: 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 0 1 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 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VERBOSE: Augmented Row-Reduced Echelon Form system matrix A and right hand side B: Columns associated with a free variable are headed with a "*" : : : : : : * : : : : : : : : : * * * * * : * * : * * * * * : * * : * * * : * * * * * * * * * * * * * * * * * * * * * * * * 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0-1-1 1-1 1 0-1-2 0-1 0-1-1-2 0-2-2 0-3-1-1 0 0-2-1-1-3-1-1-1 0-3-1-1-2-4-3-1-1-2-1-1-1-3-2-2 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0 0 0-1 0 0 1 0 0 0 1 1 0-1 0 0 0 0-1-1 0-1 0 0 0 0 0 1-1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0-1 0-1 0 0-1 0 0 0-1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0-1 0 0-1-1-1 0-1-1-1 0 0-1-1-1-1 0 0-1-1 0-1 0-1-1-1 0-1 0 0-1 0 0 0 0 0 1 0-1 0 0 0 0 0 0 0 0 0-1-1 0-2 1 0-1-1 0-1 1 0 0 0 0 0-1 0 0 1 0 0 1 0 1 0-1 2 0 0 1-2 0 0 0-1-1 0 1 0 1 0 1-1-1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1-1 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 VERBOSE: Tried 124314 right hands sides, found 4 solutions. 4 binary solutions were found. Binary solution vectors x: 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Check Loo residuals ||Ax-b||: All solutions had zero residual. Translate each correct solution into a tiling: ans = 21 62 ans = 62 1 Tiling based on solution 1 Numeric Labels 2 2 2 3 3 1 2 2 3 3 1 1 0 0 4 1 1 4 4 4 Tiling based on solution 1 "Colors" 1 1 1 2 2 1 1 1 2 2 1 1 0 0 3 1 1 3 3 3 ans = 21 62 ans = 62 1 Tiling based on solution 2 Numeric Labels 1 1 1 2 2 1 1 2 2 2 3 3 0 0 4 3 3 4 4 4 Tiling based on solution 2 "Colors" 1 1 1 1 1 1 1 1 1 1 2 2 0 0 3 2 2 3 3 3 ans = 21 62 ans = 62 1 Tiling based on solution 3 Numeric Labels 3 3 1 1 1 3 3 1 1 4 2 2 0 0 4 2 2 2 4 4 Tiling based on solution 3 "Colors" 2 2 1 1 1 2 2 1 1 3 1 1 0 0 3 1 1 1 3 3 ans = 21 62 ans = 62 1 Tiling based on solution 4 Numeric Labels 2 2 1 1 1 2 2 2 1 1 3 3 0 0 4 3 3 4 4 4 Tiling based on solution 4 "Colors" 1 1 1 1 1 1 1 1 1 1 2 2 0 0 3 2 2 3 3 3 polyomino_multihedral_example_4x5_test(): Normal end of execution. polyomino_multihedral_example_octomino_test(): polyomino_multihedral_example_octomino() defines a polyomino tiling using octominoes. polyomino_multihedral_example_octomino_matrix: Set up the linear system for the 8x8 octomino example. Region R: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Polyomino O1: 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O2: 1 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O3: 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O4: 1 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O5: 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O6: 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O7: 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O8: 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 polyomino_multihedral_example_octomino_matrix: Normal end of execution. polyomino_multihedral_example_octomino_tiling_print Given 8 solutions for the 8x8 multihedral octomino tiling problem, print a representation of the tiling corresponding to each solution. Region R: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Polyomino O1: 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O2: 1 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O3: 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O4: 1 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O5: 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O6: 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O7: 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Polyomino O8: 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ans = 72 1736 ans = 1736 1 8x8 Octomino Tiling Numeric Labels 1 1 1 3 3 3 3 4 1 1 1 3 4 3 4 4 2 2 1 3 4 4 4 4 2 2 1 3 5 5 5 5 2 7 7 7 7 7 6 5 2 2 8 8 7 6 6 5 8 2 8 7 7 6 6 5 8 8 8 8 6 6 6 5 8x8 Octomino Tiling "Colors" 1 1 1 3 3 3 3 4 1 1 1 3 4 3 4 4 2 2 1 3 4 4 4 4 2 2 1 3 5 5 5 5 2 7 7 7 7 7 6 5 2 2 8 8 7 6 6 5 8 2 8 7 7 6 6 5 8 8 8 8 6 6 6 5 POLYOMINO_MULTIHEDRAL_EXAMPLE_OCTOMINO_TILING_PRINT Normal end of execution. polyomino_multihedral_example_octomino_tiling_plot(): Given 8 solutions for the 8x8 multihedral octomino tiling problem, plot a representation of the tiling corresponding to each solution. Region R: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Saved plot as "octomino01.png" Saved plot as "octomino02.png" Saved plot as "octomino03.png" Saved plot as "octomino04.png" Saved plot as "octomino05.png" Saved plot as "octomino06.png" Saved plot as "octomino07.png" Saved plot as "octomino08.png" POLYOMINO_MULTIHEDRAL_EXAMPLE_OCTOMINO_TILING_PLOT Normal end of execution. polyomino_multihedral_example_octomino_test(): Normal end of execution. polyomino_multihedral_example_pent18x30_test(): Plot a tiling of the 18x30 rectangle that uses 9 copies of each of the 12 pentominoes. polyomino_multihedral_example_pent18x30_define(): Define the data for the example in which an 18x30 rectangle is tiled by 9 copies of the 12 pentominoes. Saved plot as "pent18x30.png" polyomino_multihedral_example_pent18x30_test(): Normal end of execution. polyomino_multihedral_example_pentomino_test(): polyomino_multihedral_example_pentomino() sets up the matrix for a tiling problem involving pentominoes. polyomino_multihedral_example_pentomino_matrix(): Set up the linear system for the 8x8 pentomino example. Region R: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 The 12 Pentominoes: 8x8 array of 12 polyominoes: Polyomino #1 0 1 1 1 1 0 0 1 0 Polyomino #2 1 1 1 1 1 Polyomino #3 0 0 0 1 1 1 1 1 Polyomino #4 1 1 0 0 0 1 1 1 Polyomino #5 1 1 1 1 1 0 Polyomino #6 1 1 1 0 1 0 0 1 0 Polyomino #7 1 0 1 1 1 1 Polyomino #8 1 0 0 1 0 0 1 1 1 Polyomino #9 1 0 0 1 1 0 0 1 1 Polyomino #10 0 1 0 1 1 1 0 1 0 Polyomino #11 0 0 1 0 1 1 1 1 Polyomino #12 1 1 0 0 1 0 0 1 1 POLYOMINO_MULTIHEDRAL_EXAMPLE_PENTOMINO_MATRIX: Normal end of execution. polyomino_multihedral_example_pentomino_test(): Normal end of execution. polyomino_multihedral_matrix_test(): polyomino_multihedral_matrix() sets up the linear system associated with a multi-polyomino tiling problem. Region R: 1 1 1 1 1 1 1 1 Polyomino N: 1 Polyomino O: 1 1 1 Polyomino P: 0 0 1 1 1 1 System matrix A and right hand side B: 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 Linear system saved in LP file:"2x4.lp" polyomino_multihedral_matrix_test(): Normal end of execution. polyomino_multihedral_tiling_plot_test(): polyomino_multihedral_tiling_plot() plots tilings of a given region R, using copies of a set of polyominoes P. polyomino_multihedral_tiling_plot_test01() Given 4 solutions for the 2x4 multihedral polyomino tiling problem, plot corresponding tilings. Saved plot as "twobyfour01.png" Saved plot as "twobyfour02.png" Saved plot as "twobyfour03.png" Saved plot as "twobyfour04.png" polyomino_multihedral_tiling_plot_test02() Given 4 solutions for the 2x4 multihedral polyomino tiling problem, plot corresponding tilings. Region R: 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 Polyomino N: 0 0 1 1 1 1 Polyomino O: 1 1 1 Polyomino P: 0 1 1 1 Saved plot as "fourbyfour01.png" polyomino_multihedral_tiling_plot_test(): Normal end of execution. polyomino_multihedral_tiling_print_test(): polyomino_multihedral_tiling_print() investigates solutions to the problem of tiling a given region R, using copies of a set of polyominoes P. polyomino_multihedral_tiling_print_test01(): Given 4 solutions for the 2x4 multihedral polyomino tiling problem, print a representation of the tiling corresponding to each solution. Region R: 1 1 1 1 1 1 1 1 Polyomino N: 1 Polyomino O: 1 1 1 Polyomino P: 0 0 1 1 1 1 ans = 11 20 ans = 20 1 2x4 Multihedral Tiling #1 Numeric Labels 2 2 2 3 1 3 3 3 2x4 Multihedral Tiling #1 "Colors" 2 2 2 3 1 3 3 3 ans = 11 20 ans = 20 1 2x4 Multihedral Tiling #2 Numeric Labels 3 3 3 1 3 2 2 2 2x4 Multihedral Tiling #2 "Colors" 3 3 3 1 3 2 2 2 ans = 11 20 ans = 20 1 2x4 Multihedral Tiling #3 Numeric Labels 3 2 2 2 3 3 3 1 2x4 Multihedral Tiling #3 "Colors" 3 2 2 2 3 3 3 1 ans = 11 20 ans = 20 1 2x4 Multihedral Tiling #4 Numeric Labels 1 3 3 3 2 2 2 3 2x4 Multihedral Tiling #4 "Colors" 1 3 3 3 2 2 2 3 polyomino_multihedral_tiling_print_test02(): Given 4 solutions for the 2x4 multihedral polyomino tiling problem, print a representation of the tiling corresponding to each solution. Region R: 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 Polyomino N: 0 0 1 1 1 1 Polyomino O: 1 1 1 Polyomino P: 0 1 1 1 ans = 13 30 ans = 30 1 4x4 Multihedral Tiling #1 Numeric Labels 1 0 0 0 1 0 0 0 1 1 3 3 2 2 2 3 4x4 Multihedral Tiling #1 "Colors" 1 0 0 0 1 0 0 0 1 1 3 3 2 2 2 3 polyomino_multihedral_tiling_print_test03 is FAILING Skip it for now! polyomino_multihedral_tiling_print_test(): Normal end of execution. polyomino_multihedral_variants_test(): polyomino_multihedral_variants() determines variants of an array of polyominoes. polyomino_multihedral_variants_test01(): polyomino_multihedral_variants() determines variants of an array of polyominoes. Region in which polyominoes must fit: 3x5 array of 1 polyominoes: Polyomino #1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Array of polyominoes to be analyzed: 3x5 array of 3 polyominoes: Polyomino #1 1 1 1 1 Polyomino #2 1 1 0 0 1 0 0 1 1 Polyomino #3 1 0 0 1 1 1 The polyominoes have 13 distinct variants Variant 1 of polyomino 1 1 1 1 1 Variant 2 of polyomino 2 1 1 0 0 1 0 0 1 1 Variant 3 of polyomino 2 0 0 1 1 1 1 1 0 0 Variant 4 of polyomino 2 0 1 1 0 1 0 1 1 0 Variant 5 of polyomino 2 1 0 0 1 1 1 0 0 1 Variant 6 of polyomino 3 1 0 0 1 1 1 Variant 7 of polyomino 3 0 1 0 1 1 1 Variant 8 of polyomino 3 1 1 1 0 0 1 Variant 9 of polyomino 3 1 1 1 0 1 0 Variant 10 of polyomino 3 0 0 1 1 1 1 Variant 11 of polyomino 3 1 1 0 1 0 1 Variant 12 of polyomino 3 1 1 1 1 0 0 Variant 13 of polyomino 3 1 0 1 0 1 1 polyomino_multihedral_variants_test02 polyomino_multihedral_variants() determines variants of an array of polyominoes. Region in which polyominoes must fit: 2x4 array of 1 polyominoes: Polyomino #1 1 1 1 1 1 1 1 1 Array of polyominoes to be analyzed: 2x4 array of 3 polyominoes: Polyomino #1 1 Polyomino #2 1 1 1 Polyomino #3 0 0 1 1 1 1 The polyominoes have 6 distinct variants Variant 1 of polyomino 1 1 Variant 2 of polyomino 2 1 1 1 Variant 3 of polyomino 3 0 0 1 1 1 1 Variant 4 of polyomino 3 1 1 1 1 0 0 Variant 5 of polyomino 3 1 0 0 1 1 1 Variant 6 of polyomino 3 1 1 1 0 0 1 polyomino_multihedral_variants_test(): Normal end of execution. polyomino_parity_test(): polyomino_parity() returns the parity of a polyomino. Random polyomino of 2 squares on 2 by 1 grid 1 1 parity is 0 Random polyomino of 17 squares on 6 by 9 grid 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 parity is 3 Random polyomino of 4 squares on 2 by 3 grid 0 1 1 1 1 0 parity is 0 Random polyomino of 12 squares on 7 by 4 grid 1 1 0 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 0 parity is 0 Random polyomino of 11 squares on 6 by 4 grid 0 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 parity is 1 polyomino_parity_test(): Normal end of execution. polyomino_periodic_matrix_test(): polyomino_periodic_matrix() sets up the matrix for a periodic polyomino tiling problem. polyomino_periodic_matrix_test(): Normal end of execution. polyomino_periodic_variants_test(): polyomino_periodic_variants() generates variants of polyominoes that can periodicly tile a region R. Region ito be tiled: 3x3 array of 1 polyominoes: Polyomino #1 1 1 1 1 1 1 1 1 1 Array of polyominoes to be analyzed: 3x3 array of 1 polyominoes: Polyomino #1 0 0 1 1 1 1 The polyominoes have 8 distinct variants Variant 1 of polyomino 1 0 0 1 1 1 1 Variant 2 of polyomino 1 1 1 0 1 0 1 Variant 3 of polyomino 1 1 1 1 1 0 0 Variant 4 of polyomino 1 1 0 1 0 1 1 Variant 5 of polyomino 1 1 0 0 1 1 1 Variant 6 of polyomino 1 0 1 0 1 1 1 Variant 7 of polyomino 1 1 1 1 0 0 1 Variant 8 of polyomino 1 1 1 1 0 1 0 polyomino_periodic_variants_test(): Normal end of execution. polyomino_periodicity_apply_test(): polyomino_periodicity_apply() applies periodicity to a polyomino P with respect to a region R. Region R to be tiled: 1 1 1 1 1 1 1 1 1 1 1 1 Polyomino P: 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 Periodicized polyomino T: 0 0 1 0 0 0 0 0 1 1 1 1 polyomino_periodicity_apply_test(): Normal end of execution. polyomino_random_test(): polyomino_random() generates a random polyomino. Random polyomino of 14 squares on 5 by 10 grid 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 Graphics saved as "polyomino_random_1.png" Random polyomino of 11 squares on 4 by 6 grid 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 1 1 Graphics saved as "polyomino_random_2.png" Random polyomino of 6 squares on 4 by 3 grid 0 1 1 1 1 0 1 0 0 1 0 0 Graphics saved as "polyomino_random_3.png" Random polyomino of 10 squares on 4 by 5 grid 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 Graphics saved as "polyomino_random_4.png" Random polyomino of 7 squares on 2 by 5 grid 0 0 1 1 1 1 1 1 1 0 Graphics saved as "polyomino_random_5.png" polyomino_random_test(): Normal end of execution. polyomino_reflect_test: MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 polyomino_reflect reflecs a polyomino about the vertical axis. The given polyomino P: 0 1 1 1 1 0 0 1 0 The reflected polyomino Q 1 1 0 0 1 1 0 1 0 polyomino_reflect_test: Normal end of execution. polyomino_transform_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 polyomino_transform() can transform a polyomino. Generate all 8 combinations of rotation and reflection applied to a polyomino represented by a binary matrix. The given polyomino P: 0 1 1 1 1 0 0 1 0 P after 0 reflections and 0 rotations: 0 1 1 1 1 0 0 1 0 P after 0 reflections and 1 rotations: 1 0 0 1 1 1 0 1 0 P after 0 reflections and 2 rotations: 0 1 0 0 1 1 1 1 0 P after 0 reflections and 3 rotations: 0 1 0 1 1 1 0 0 1 P after 1 reflections and 0 rotations: 1 1 0 0 1 1 0 1 0 P after 1 reflections and 1 rotations: 0 1 0 1 1 1 1 0 0 P after 1 reflections and 2 rotations: 0 1 0 1 1 0 0 1 1 P after 1 reflections and 3 rotations: 0 0 1 1 1 1 0 1 0 polyomino_transform_test(): Normal end of execution. POLYOMINOES_PRINT_TEST MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 POLYOMINOES_PRINT prints an array of polyominoes. Array of polyominoes to be analyzed: 3x5 array of 3 polyominoes: Polyomino #1 1 1 1 1 Polyomino #2 1 1 0 0 1 0 0 1 1 Polyomino #3 1 0 0 1 1 1 POLYOMINOES_PRINT_TEST Normal end of execution. rectangle_3x20_plot(): An illustration of polyomino tiling. In this example, a region of 3x20 squares is to be covered by the 12 pentominoes. Graphics saved as "rectangle_3x20.png" reid_plot(): A simple illustration of polyomino tiling.n In this example, a region of 8 squares is to be covered by 4 tiles, each of which is a pair of squares. Graphics saved as "reid.png" reid_plot(): Normal end of execution. polyominoes_test(): Normal end of execution. 11-Jan-2022 14:13:19