polyomino_monohedral_test 15-May-2018 09:43:17 POLYOMINO_MONOHEDRAL_TEST: Test POLYOMINO_MONOHEDRAL, which investigates solutions to the problem of tiling a given region R, using multiple copies of a single polyomino P. POLYOMINO_MONOHEDRAL_TEST01 POLYOMINO_MONOHEDRAL_DEMO: 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 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 Linear system saved in LP file "reid.lp" VERBOSE: 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. VERBOSE: POLYOMINO_MONOHEDRAL: Normal end of execution. 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 1 1 8 3 3 8 5 5 Tiling based on solution 2 1 1 2 2 10 4 4 10 Tiling based on solution 3 6 7 6 7 10 4 4 10 Tiling based on solution 4 1 1 8 9 10 8 9 10 POLYOMINO_MONOHEDRAL_TEST02 POLYOMINO_MONOHEDRAL_DEMO: 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 "reid.lp" VERBOSE: 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. 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 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 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VERBOSE: POLYOMINO_MONOHEDRAL: Normal end of execution. 6 binary solutions were found. 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 1 1 12 12 12 1 1 1 12 12 7 7 18 18 18 7 7 7 18 18 Tiling based on solution 2 28 28 28 21 21 28 28 21 21 21 7 7 18 18 18 7 7 7 18 18 Tiling based on solution 3 1 1 12 12 12 1 1 1 12 12 34 34 34 27 27 34 34 27 27 27 Tiling based on solution 4 10 10 10 56 56 65 10 10 56 56 65 65 9 9 56 65 65 9 9 9 Tiling based on solution 5 28 28 28 21 21 28 28 21 21 21 34 34 34 27 27 34 34 27 27 27 Tiling based on solution 6 45 45 30 30 30 45 45 30 30 44 45 25 25 44 44 25 25 25 44 44 POLYOMINO_MONOHEDRAL_TEST: Normal end of execution. 15-May-2018 09:43:21 exit