Figures from the non-block tetracubes: Torsten Sillke, 1999-03-06 last Update: 2000-11-20 Pieces: 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 2-4 3-4 5-4 6-4 -6-4 7-4 <- name - number of cubes Even and Odd Pieces: The pieces can be split into two sets. The odd pieces have an odd number of white and black cubes if the cubes are colored alternatingly. 2 pieces (3) and (5) are odd. The even pieces have an even number of white and black cubes if the cubes are colored alternatingly. 4 pieces are even. As the number of odd pieces if even we can build only even figues. Blocks: 2x3x4 2 3 2 made from 2 times 2 1 2 2x2x6 impossible Towers: No maximal tower is possible. Piece (5) is always at the border (or the figur dissects.) t t t t heigth subsets 2 impossible as the complement of each piece is like it. 3 (2,6,-6) (3,5,6) (3,5,-6) 4 (2,3,5,6) (2,3,5,-6) (2,6,-6,7) (3,5,6,7) (3,5,-6,7) 5 (-6) (6) complements 6 no 1 2 2 Otherwise the 2x2xZ6 is possible as 2 times 2 2 2 1 is possible or 2 2 2 2 2 2 can be made from 2 times 2.2 2 2 2 2 2 2 2 1 2 2 2 t t t t heigth subsets 1 (2) 2 (2,6) (2,-6) 3 (2,6,-6) (3,5,6) (3,5,-6) 4 (2,3,5,6) (2,3,5,-6) (2,3,5,7) 5 (7) (-6) (6) complements 6 no t t t t heigth subsets 1 (3) 2 (2,6) (2,-6) 3 (2,3,6) (2,3,-6) 4 no x Only pieces 2,3,6,-6 are possible (use coloring x o x) t t t t heigth subsets 1 (7) 2 (6,-6) 3 (2,6,-6) (3,5,6) (3,5,-6) (6,-6,7) 4 (2,3,5,6) (2,3,5,-6) (2,6,-6,7) (3,5,6,7) (3,5,-6,7) 5 (-6) (6) complements 6 no It is impossible to build a 3-tower pair. (You only have to check the T, N combination.) Tromino enlagements: I3 gives 2x2x6 which is impossible. L3 is possible (19 solutions) a b b b 8 solutions are a b b b generated by this a a dissection by a a rotation of the 'b' part. 4 4 4 subsets: (2,3,5) (2,6,7) (2,-6,7) 8 8 8 impossible x - coloring the figure like o x gives an defect of 0. - But maximal 'x' positions of (5,6,-6) are border places. - Therefore the figure dissects into two ones of heigth 4. impossible: 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 1 2 2 1 color x o o x . x x . Not all 'x' can be filled. 2 1 1 2 the o o o o x . . x Note: pieces 2 and 3 fill at most one 'x' 2 1 1 2 figure o o o o x . . x otherwise a singe cube is isolated. 1 2 2 1 x o o x . x x . 2 1 2 1 checker the figure: 1 2 1 2 The ratio 16:8 is not fillable. 2 1 2 1 1 2 1 2 2 2 1 2 1 1 1 2 2 1 1 1 2 1 2 2 2 2 1 2 2 1 1 1 1 1 1 2 2 1 2 2 2 2 1 1 2 1 2 1 1 2 1 2 1 1 2 2 1 2 1 2 2 2 1 1 1 1 2 2 2 1 2 1 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 Figures with 4x4 base: 30 30 30 60 unique 30 80 80 40 70 80 40 40 70 50 50 40 . . 60 60 . . 80 60 70 70 . . 50 50 . . 60 60 80 30 2 solutions 60 50 80 30 50 50 70 30 50 40 70 70 60 80 80 30 . . . . . . . . 40 40 40 70 70 30 30 80 70 50 30 80 50 50 30 60 50 40 60 60 70 70 80 80 . . . . . . . . 40 40 40 60 60 40 40 40 9 solutions 60 60 50 50 70 70 30 80 30 30 30 80 . . 40 . 60 50 50 . . 70 80 80 . 70 . . 80 80 60 60 5 solutions 80 70 60 40 70 70 40 40 30 30 30 40 . 80 60 . . 70 50 . . 50 50 . . 50 30 . 30 30 30 60 5 solutions 30 80 60 60 80 80 70 70 40 40 40 70 . . 50 . . 50 50 60 80 50 70 . . 40 . . 70 30 80 80 4 solutions 70 30 30 30 40 40 40 60 50 50 60 60 70 70 80 . . . 80 . . 40 . . . 50 50 60 80 80 30 30 3 solutions 80 40 60 30 70 40 40 30 70 40 50 50 . 80 60 . . . 60 60 70 70 . . . 50 50 . 40 80 60 60 7 solutions 40 40 50 60 40 70 50 50 70 70 30 50 . 80 80 60 . . 80 . . 70 . . 30 30 30 . 70 30 30 30 16 solutions 70 80 50 30 60 80 50 50 40 40 40 50 70 70 . . 80 80 . . 60 60 . . 60 40 . . 60 40 40 40 10 solutions 60 60 40 30 80 80 70 30 50 50 30 30 . . . . 60 70 70 . 80 . 70 . 80 50 50 . 3 Times: The two odd pieces must be in the same figure. The only figure is 2 3 2 1 Twins: 2 2 3 subsets: (2,3,5) (2,6,-6) (2,6,7) (3,5,7) 2 2 1 2 1 3 subsets: (2,3,5) (2,6,-6) (2,-6,7) (3,5,7) 2 2 2 2 2 3 subsets: (2,3,6) (2,3,-6) (2,5,6) (2,5,-6) (3,6,7) (3,-6,7) 2 1 2 1 2 3 subsets: (2,3,6) (2,3,-6) (2,3,7) (2,5,6) (2,5,-6) (2,5,-6) 2 2 2 (3,6,-6) (3,6,7) (3,-6,7) (5,6,-6) (5,6,7) (5,-6,7) 2 2 3 subsets: (2,3,5) (2,6,-6) (2,6,7) (2,-6,7) (3,5,6) (6,-6,7) 1 2 2 2 2 3. subsets: (2,6,-6) (2,6,7) (2,-6,7) (3,5,6) (3,5,-6) (6,-6,7) 2 2 2 2 3 2 subsets: (2,6,-6) (2,6,7)-(3,5,-6) (2,-6,7)-(3,5,6) 2 1 2 sym 2 3.2 subsets: (2,6,7)-(3,5,-6) (2,-6,7)-(3,5,6) 2 2 2 sym 2 3 1 subsets: (2,6,-6) (2,-6,7) (3,5,6) (3,5,-6) (6,-6,7) 2 2 2 2 3 2 subsets: (2,3,6) (2,3,-6) (2,3,7) (2,5,6) (2,5,-6) 1 2 2 (3,6,-6) (3,6,7) (3,-6,7) (5,6,-6) 3 2 2 subsets: (2,3,6) (2,3,-6) (2,3,7) (2,5,-6) (3,6,7) (3,-6,7) 2 2 1 2 2 subsets: (2,6,7)-(3,5,-6) (2,-6,7)-(3,5,6) (6,-6,7) 2 2 sym 2 2 2 2 1 subsets: (2,6,-6)-(3,5,7) (2,6,7) (2,-6,7) 2 2 1 sym 1 1 1 2 1 2 subsets: (2,5,6)-(3,-6,7) (2,5,-6)-(3,6,7) 2.2 2 2. sym 2.2 2 2 subsets: (2,5,6)-(3,-6,7) (2,5,-6)-(3,6,7) 1 2 1 1 sym 1 1 2 1 subsets: (2,6,-6) (2,6,7) (3,5,7) 2 2 2 1 1 2 1 1 2 2 2 1 1 2 2 subsets: (2,3,6) (2,3,-6) (2,3,7)-(5,6,-6) (2,5,6) (2,5,-6) 2 2 2 1 sym 2 2 2 subsets: (2,6,7)-(3,5,-6) (2,-6,7)-(3,5,6) (6,-6,7) 2 2.2 1 sym (from mirror twins blocks) 2 1 2 subsets: (2,3,5) (2,6,-6) (2,6,7)-(3,5,-6) (2,-6,7)-(3,5,6) 2 2 2 1 sym 2 1 2 1 subsets: (2,3,6) (2,3,-6) (2,5,6) (2,5,7) (3,6,-6) (5,-6,7) 2 2 2 2.2 2 subsets: (2,5,6) (2,5,-6) (3,6,7) (3,-6,7) 1 2 2 2 1 2 2 1 1 1 2 2 1 1 2 2 1 1 2 2 1 2 2 2 2 2 1 1 2 2 2 2 2 2. 1 2 1 1 2 2 1 1 1 1 2 1 1 1 2 2 1 1 3 3 1 subsets: (2,5,6)-(3,-6,7) (2,5,-6)-(3,6,7) 3 1 sym 1 3 3 2. subsets: (2,3,6)-(5,-6,7) (2,3,-6)-(5,6,7) (3,6,7) (3,-6,7) 3 1 sym 2. 2 3 1 subsets: (2,3,6) (2,3,-6) (2,5,6)-(3,-6,7) (2,5,-6)-(3,6,7) 3 3' sym 1 2 3 1 subsets: (2,6,-6)-(3,5,7) 3 3. sym 1 3 2 1 2 2 2 2 2 1 2 2 3 2 2. subsets: (2,6,7)-(3,5,-6) (2,-6,7)-(3,5,6) (6,-6,7) 2 3 2 sym 2 2 3. subsets: (2,6,7)-(3,5,-6) (2,-6,7)-(3,5,6) 2 2 2 sym 2 2 2 2 2 2 2. 1 1 2 2 2 1 2 2 2 2 2 2 1 1 2 2 subsets: (2,3,6) (2,3,-6) (2,5,6)-(3,-6,7) (2,5,-6)-(3,6,7) (5,6,-6) 2 2 2 sym 1 2 1 2 subsets: (2,3,6)-(5,-6,7) (2,3,-6)-(5,6,7) (2,5,6) (2,5,-6) 2 3 2 sym 1 1 1 1 subsets: (2,5,6)-(3,-6,7) (2,5,-6)-(3,6,7) (5,6,7) (5,-6,7) 2 3 2 sym 1 2 1 1 subsets: (2,3,6)-(5,-6,7) (2,3,-6)-(5,6,7) (2,5,6) (2,5,-6) 1 3 1 sym 2 2 2 3^ subsets: (2,3,6)-(5,-6,7) (2,3,-6)-(5,6,7) (5,6,-6) 2 3 2 sym 1 2 1 2 subsets: (2,3,6)-(5,-6,7) (2,3,-6)-(5,6,7) 1 1 1 sym 2 3 2 2 subsets: (2,3,6) (2,3,-6) (3,6,-6) (5,-6,7) 2 2 1 2 2 2. 1 2 subsets: (2,6,7)-(3,5,-6) (2,-6,7)-(3,5,6) 1 2 2 2. sym 1 2 2 1 1 2 2 1 2 1 1 2 1 2 2 1 1 2 1 2 1 1 2 2 1 2 1 1 subsets: (2,3,6) (3,6,-6) (3,-6,7) (5,6,7) (5,-6,7) 2 1 2 2 2 1 2 2 subsets: (2,3,5) (2,6,-6) (2,6,7) (3,5,6) (3,5,-6) 1 1 2 2 1 1 1 2 subsets: (2,3,6) (2,3,-6) (2,5,6) (2,5,-6) (2,5,7) (5,6,7) (5,-6,7) 1 2 2 2 1 1 1 1 subsets: (2,6,-6) (2,6,7) (3,5,7) (6,-6,7) 1 2 2 2 2 1 2 1 1 subsets: (2,3,7) (2,5,6) (2,5,-6) (2,6,-6) (2,6,7) (2,-6,7) 2 1 1 (3,5,7) (3,6,-6) (3,6,7) (6,-6,7) 1 1 2 2 1 1 subsets: (2,3,7) (2,5,6) (2,5,-6) (3,6,-6) (3,6,7) 2 1 2 1 1 1 1 2 1 1 2 1 1 2 2. 1 1 2 1 1 2 1 2 2. 1 1 2 2 1 2 2 1 1 1 2 2 2 2 1 1 1 2. 1 2 2 2 2 1 1 2 2 2 2 2 1 1 2 2 2 2 2 2. 1 2 2 2 2 2 1 1 2 2 2 2 2 2. 1 2 2 2 2 2 1 1 2 2 2 2 2 2. 2 2 2 1 2 2 1 2 2 2 2 1 2 2. 2 2 2 1 1 2 2 2 1 subsets: (2,3,6)-(5,-6,7) (2,3,-6)-(5,6,7) 2 2 1 1 sym 2 1 2 1 subsets: (2,5,7)-(3,6,-6) 1 2 2 1 sym 2 1 2 1 subsets: (2,3,6)-(5,-6,7) (2,3,-6)-(5,6,7) 3.2 1 1 sym 2 1 1 1 1 2 2 2 2 1 1 1 2 2 1 2 2 1 2 2 1 1 1 2 2 1 2 2 1 1 2 2 1 1 2 2 2 1 2 1 1 1 2 2 2 1 1 2 1 1 2 subsets: (2,3,5) (2,6,-6) (2,-6,7) (6,-6,7) 2 2 1 1 2 1 1 1 2 subsets: (2,3,5) (2,6,7) (2,-6,7) (3,5,-6) (6,-6,7) 2 2 1 2 1 1 1 2 2 2 1 1 2 1 1 1 1 2 2 2 2 1 4 3. 2 2 2 4.3 2 2 2 3 4. 2 2 2 3.4 2 2 2 1 2 2 2 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 2 2 2 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 2 2 1 1 2 1 1 1 1 1 1 1 2 2 2 1 1 1 2 2 1 1 1 2 1 2 1 2 2 1 1 1 1 1 1 1 1 2 2 1 1 2 1 1 1 2 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 2 2 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 1 2 1 1 1 1 1 1 2 1 2 1 1 subsets: (2,3,5) (2,6,7)-(3,5,-6) (2,-6,7)-(3,5,6) 1 1 1 1 sym 2 1 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 2 1 2 1 1 1 1 1 1 1 1 2 1 2 1 1 1 1 1 1 2 1 1 2 1 3 1 3 3 2 1 subsets: (2,3,5) (2,6,7)-(3,5,-6) 3 4 3 1 sym 3. subsets: (2,6,7)-(3,5,-6) (2,-6,7)-(3,5,6) (6,-6,7) 1 3 3 3^ sym point (BW) 2 2 1 2 2 2 sym point 2.2 3 3 3 2 1 3 3 subsets: (2,3,-6) (2,3,7) (3,6,-6) (5,6,7) (5,-6,7) 3 2 1 1 1 1 2 . 2 2 2 1 mirror Twins (only): 2 2 subsets: (2,3,6) (2,5,6) (2,5,-6) (5,6,7) 1 2 1 2 1 1 3 2 2. subsets: (2,3,5) (2,-6,7) (3,5,-6) 2.2 3 sym 2.2 1 2 2 1 2 2. sym (from mirror twins blocks) 1 2 2 1 1 2 1 2 2 1 2 1 1 1 1 2 1 1 1 2 1 1 2 1 2 1 1 1 2 2 1 1 1 2 1 2 2 2 1 1 2 1 1 1 2 2 1 2 2 1 1 1 2 2 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 1 mirror Twins blocks (excluding 6, -6): 2 1 1 1 1 1 1 2 2 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 2 2. Hexominoes of height 4: (19 of the 35 hexominoes are possible) 4 4 4 61 solutions 2 3 2 4 4 4 made from 2 times 2 1 2 4 5 solutions 3 4 4 2 2 4 4 4 made from 2 times 2 2 1 4 4 4 4 4 4 16 solutions 4 4 4 4 4 4 2 solutions 4 4 2 2 4 4 4 4 5 solutions made from the twins 3 2 2 1 4 4 4 4 2 solutions 4 4 4 4 4 4 4 4 2 solutions 4 4 4 4 4 4 5 solutions 4 4 4 4 4 2 solutions 4 4 4 4 4 4 unique 4 4 4 2 2 4 4 34 solutions 2 2 4 4 one solution is 2 2 two times. 4 4 4 4 6 solutions 4 4 4 4 4 8 solutions 2 2 3 4 4 4 one solution is 1 2 2 and its mirror immage. 4 2 solutions 4 4 4 4 4 4 4 4 2 solutions 4 4 4 4 4 solutions 4 4 4 4 4 4 4 solutions 4 4 4 4 4 4 2 solutions 4 4 4 4 4 4 4 14 solutions 4 4 4 4 impossible 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Octominoe single and double: 1 1 2 2 only possibility as N4 and L4 must be together. 1 1 1 2 2 2 (T4 has the wrong parity.) 1 1 1 2 2 2 Octominoes of height 3: (?? of the 53 symetric octominoes are possible) 3 3 3 3 61 solutions 3 3 3 3 3 3 3 3 18 solutions 3 3 3 3 3 3 3 3 2 solutions 3 3 3 3 3 3 129 solutions 3 3 3 3 3 3 3 3 3 9 solutions 3 3 3 3 3 with a twin solution 3 3 4 solutions 3 3 3 3 3 3 3 3 21 solutions 3 3 3 3 3 3 3 3 23 solutions 3 3 3 3 3 3 3 3 3 3 9 solutions 3 3 3 3 3 3 3 6 solutions 3 3 3 3 3 3 3 7 solutions 3 3 3 3 3 3 3 3 3 3 3 solutions 3 3 3 3 3 3 3 solutions 3 3 3 3 3 3 3 2 solutions 3 3 3 3 3 3 3 3 3 3 3 2 solutions 3 3 3 3 3 3 2 solutions 3 3 3 3 3 3 3 3 unique 3 3 3 3 3 3 3 3 unique 3 3 3 3 3 3 3 unique 3 3 3 3 3 3 3 3 3 3 impossible (odd figure) 3 3 3 3 3 3 3 3 impossible (odd figure) 3 3 3 3 3 3 3 impossible 3 3 3 3 3 3 3 3 impossible 3 3 3 3 3 3 3 impossible (odd figure) 3 3 3 3 3 3 3 3 3 3 3 impossible o o o x 3 3 3 3 x o o o 3 3 impossible (odd figure) 3 3 3 3 3 3 3 3 impossible (odd figure) 3 3 3 3 3 3 3 3 3 impossible 3 3 3 3 3 3 3 3 3 3 impossible o o x o o (crossing column) 3 o 3 o 3 o 3 impossible (column checkering) 3 3 3 3 3 3 3 3 impossible 3 3 3 3 3 3 3 3 impossible 3 3 3 3 3 3 3 3 3 impossible 3 3 3 3 3 3 3 3 3 impossible (odd figure) 3 3 3 3 3 3 impossible 3 3 3 3 3 3 3 Building Blocks Hexomino Pairs: As the hexominoes of height 2 are even figures the tetrominoes 3 and 5 must be in the same set. 2, 3, 5 | 6, -6, 7 -------------+--------------------- | 2 2 2 2 2 | 2 2 2 2 2 2 2 2 2 | 2 2 2 2 3, 5, 6 | 2, -6, 7 -------------+--------------------- | 2 2 2 2 | 2 2 2 2 2 2 2 | 2 2 2 2 2 2 2 | 2 2 | 2 2 2 2 2 | 2 2 2 2 2 2 | 2 2 2 2 2 | 2 | 2 2 2 2 2 | 2 2 2 2 2 2 | 2 2 2 2 2 2 In total we get 19 different pairs as one pair is generated two times. Building Blocks Tetromino plus Octomino: The tetrominoes L, N, and T can be build with height 2. Only two subsets (2,6) and (6,-6) build tetrominoes. Case: 2, 6 tetromino 2 2 2 2 2 2 2 2 combined with 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Case: 6,-6 tetromino 2 2 2 2 combined with 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Composable figures: a a b b a b b b a a a b a a a b a a a b a a a a a a a a a a a a a a a a a a b b a a b b b b b b a a a a a a a a b b b b a a a a a a b a a b b b b b b a b a a a a a a a a a b b a a . b b a a a a a a a a b b a a a a b b a a a b b a a a a a b b b b b b b a a a b b b a a a a a a a a a a a a a a a b b a a b b a a b b a a a a b b a a b b b b a a a a a a b b b b b b a a a a b a a a a a a a b b a a a b b b b b b b a a b b a a a b b b a b a a a a a b a b b b b b a a a a a b a b b b b b a a b a a . b b a a b b b a a a a a b . a a b b b a a a a a b a a b b a a b a a a a a a a a b b b b a a a b a a a b b a a b a a a b a a b a a a b b 12-omines of hight two: . 30 30 . 2 solutions 70 30 80 80 70 30 60 80 . 60 60 . . 50 50 . 70 70 50 50 40 40 40 80 . 40 60 . . 30 30 . 80 30 40 60 80 30 40 40 . 70 40 . . 50 50 . 50 50 60 60 80 80 70 60 . 70 70 . 2 2 2 2 2 2 2 2 2 made from the twins 2 1 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 made from the mirror twins 2 2 3 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 . 2 2 made from 2 2 2 two times or 2 2 two times 2 2 2 2 2. 1 2 2 2 2 2 2 2 2 2 . 2 2 made from 2 2 2 two times 2 2 2 2 2 2. 1 2 2 2 2 2 2 2 2 1 1 2 1 2 2 2 2 made from the mirror twins 2 2 2 1 2 2 2 2 2 2 2.2 2 2 2 2 2 2 2 made from the twins 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 made from the twins 2 2 2. 2 2 unique 2 2 2 2 2 2 2 2 2 2 The Cube with 3 holes: (2,0,0,1)=(v,e,f,c) 3 3 3. 10 solutions 2.2 2 3 3'3 2 1 1 2 3 3 made from the mirror twins 1 1 1 3 3 2 21 solutions 3 3'3 2 3 3 2 3 2 28 solutions 3 3'3 3 3 3 (0,3,0,0)=(v,e,f,c) 3 3 3' 30 solutions 2 2 1 1 1 1 3 3 3 2 1 1 1 2 2 3'3 3' made from 1 1 1 and 1 2 1 3 2 3 57 solutions 2 3 3 3 2 3 3 2 3 38 solutions 3.3 3 3 2 3 3'2 3 85 solutions 3 3 3 3 2 3 3 3.3 65 solutions 2 3 3 3 2 3 3 3 3' 4 solutions 3.3 3 3 2 3 (0,2,0,1)=(v,e,f,c) 3 3 3' 6 solutions 1 3 3'3 3 1 3'3 3 made from the twins 3 3 1 3 3 3' 15 solutions 3 3'3 3 3 3' 3 2 3 24 solutions 3 3'2 3 3 3 3 2 3 32 solutions 3 3'3 3 3 3' (2,1,0,0)=(v,e,f,c) 3 3 129 solutions 3 3 3 3 3 3 (0,0,2,1)=(v,e,f,c) 3 3 3 9 solutions 3 3 3 3 3 with a twin solution 3 3'3 19 solutions 3 3'3' 3 3 3 (1,0,1,1)=(v,e,f,c) 3 3 2 26 solutions 3 1 3 3 3 3 3 3 3. 22 solutions 3 1 3 3 3 3 modified squares: A 5-square without corners plus 3 cubes in the second layer. Ignoring the second layer (so using 3 L-Trominoes) we have 23 possibilities to form the truncated 5-square. Replacing the L-Trominoes by 5, 6, and -6 generates the possible figues. The symmetric ones are 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 Other figures: 2 3 2 28 solutions 3 4 3 2 3 2 4 3 6 solutions 3 4 3 3 4 3.4 4 4 4 3.4 4 3. made from the twins 4 3' 1 2 2 1 3 3 3 3 1 2 2 1 1 1 unique 1 4 4 1 1 4 4 1 1 1 1 2 1 2 1 unique 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 2 1 2 1 made from the twins 1 1 1 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 2 1 made from the twins 1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 1 1 2 1 1 2 1 1 1 1 2 1 1 2 1 1 1 1 1 2 2 2 1 2 1 1 1 1 1 2 1 1 1 2 2 2 2 1 made from the twins 2 2 2 1 1 1 1 2 1 1 1 2 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 2 1 2 2 1 1 2 2 1 2 1 1 1 1 1 1 1 1 1 1 2 2 1 1 2 2 1 2 1 1 2 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 19 solutions 1 1 2 3 3 2 2 2 1 2 3 3 2 made from the twins 2 2 1 2 3 2 2 3 2 2 4 2 3 3 2 2 2 2 and 1 2 1 build 3 4 3 2 4 2 3 3 2 2 3 2 2 3 2 2 4 2 3 3 2 3 2 2 4 solutions 3 3 2 4 2 2 3 2 3 2 2 2 and 1 2 1 build 3 3 2 4 2 2 2 3 3 3 2 2 3 3 2 4 2 2 3 2 3 4 4 3 3 2 2 2 2 2 2 impossible x We have 8 'x'. 2 2 2 make the o o o Only 7 can be 2 2 . 2 2 coloration x o . o x fitted. 2 2 2 o o o 2 x 2 2 2 solutions 1 2 2 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 3 solutions 2 2 2 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 unique 2 2 1 1 1 2 2 1 1 1 2 2 1 1 1 1 1 1 impossible 1 2 2 1 1 1 2 2 1 1 1 2 2 1 1 1 Symmetric partitions: 1 1 1 2 1 3 2 1 1 1 2 2 2 1 1 1 1 1 1 1 2 1 2 1 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 2 2 2 2 1 1 1 2 2 1 1 1 1 1 1 2 1 this gives a flexible figure of height 3. 1 1 2 1 1 1 1 e.g. it forms the 2x3x4 block. 1 1 1 1 1 2 1 Nonplanar Tetracubes only: 2 1 2 2 2 1 2 2 2 2 2 1 2 1 References: - Torsten Sillke; Blockfree Tetracubes, CFF 50 Part 6 (Oct. 1999) 20-21 - some figures of this file -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/