Prime Packings of the Pentomino Y: Torsten Sillke, Bielefeld list completed: 1992-12-12 Update: 3x3xZ 1993-03 Update: no 3x3xZ15 1997-07-23, H. Postl Update: handed Y solved 1999-03-06, Y. Aksyonov x x x x x the Y pentomino A 'p' following a number indicates a prime box. Rectangles with the Y-pentomino: 10x 5p, 10, 14p, 15, 16p, 19, 20, 21, 23p, 24, 25, 26, 27p, ... {23..27} + 5n 15x 10, 14p, 15p, 16p, 17p, 19p, 20,21p,22p,23p,24,25,26,27,28,...{19..28}+10n 20x 5, 9p, 10, 11p, 13p, 14, 15, 16, 17p, ... {13..17} + 5n 25x 10, 14, 15, 16, 17p, 18p, 19, 20, 21, 22p, 23, ... {14..23} + 10n 30x 5, 9p, 10, 11p, 13p, 14, 15, 16, 17, ... {13..17} + 5n 35x 10, 11p, 13p, 14, 15, 16, 17, 18p, 19, 20, 21, 22, ... {13..22} + 10n 40x 5, 9, 10, 11, 13, 14, 15, 16, 17, ... {13..16} + 5n 45x 9p, 10, 11p, 13p, 14, 15, 16, ... 50x 5, 9, 10, 11, 12p, 13, ... {9..13} + 5n 55x 9p, 10, 11, 12p, 13, 14, 15, 16, ... 60x 5, 9, 10, 11, 12p, 13, 14, 15, 16, ... 65x 9, 10, 11, 12p, 13, 14, 15, 16, ... 70x 5, 9, 10, 11, 12p, 13, 14, 15, 16, ... 75x 9, 10, 11, 12p, 13, 14, 15, 16, ... 80x 5, 9, 10, 11, 12p, 13, 14, 15, 16, ... 85x 9, 10, 11, 12p, 13, 14, 15, 16, ... 90x 5, 9, 10, 11, 12p, 13, 14, 15, 16, ... 95x 9, 10, 11, 12p, 13, 14, 15, 16, ... 100x 5, 9, 10, 11, 12, 13, 14, 15, 16, ... /no new prime Strips (one side open) with the Y-pentomino: Nx 5, 6p, 8p, 9, 10, 11, 12, ... {8..12} + 5n Strips (two sides open) with the Y-pentomino: Zx 2p, 4, 5, ... {4, 5} + 2n Boxes with the Y-pentomino: 2x5x 6p, 8p, 10, 11p, 12, 13p, 14, 15p, ... {10..15} + 6n 3x5x 4p, 8, 9p, 10, 11p, ... {8..11} + 4n 4x5x 3p, 4p, 5p, ... {3..5} + 3n 5x5x 4p, 5p, 6p, 7p, ... {4..7} + 4n 6x5x 2p, 4, 5p, ... {4, 5} + 2n 7x5x 4, 5p, 6, 7p, ... {4..7} + 4n 8x5x 2p, 3, ... {2, 3} + 2n 9x5x 3p, 4, 5, ... {3..5} + 3n kx5x 2, 3, ... {2, 3} + 2n (k>=10) see 2x5xk, and 3x5xk 2x10x 4p, 5, 6, 7p, ... {4..7} + 4n 3x10x 4, 5, 6p, 7p, ... {4..7} + 4n 2x15x 4p, 5p, 6, 7p, ... {4..7} + 4n 3x15x 4, 5, 6p, 7p, ... {4..7} + 4n 3d-Strips (one side open) with the Y-pentomino: Nx2x 4, 5, 6, 7, ... {4..7} + 4n Nx3x 4, 5, 6, 7, ... {4..7} + 4n 3d-Strips (two sides open) with the Y-pentomino: Zx2x 1p, ... n Zx3x 2, 3p, ... {2, 3} + 2n Hyperboxes with the Y-pentomino: 2x2x5x 4p, 6, 8, 9p, 10, 11, ... {8..11} + 4n 2x3x5x 4, 6, 8, 9, 10, 11, ... {8..11} + 4n 3x3x5x 4, 8, 9, 10, 11, ... {8..11} + 4n Complete list of prime rectangles (# = 40): 5x10, 9x{20, 30, 45, 55}, 10x{14, 16, 23, 27}, 11x{20, 30, 35, 45}, 12x{50, 55, 60, 65, 70, 75, 80, 85, 90, 95}, 13x{20, 30, 35, 45}, 14x15, 15x{15, 16, 17, 19, 21, 22, 23}, 17x{20, 25}, 18x{25, 35}, 22x25. Complete list of prime boxes (# = 22): 2x4x{10, 15}, 2x5x{6, 8, 11, 13, 15}, 2x7x{10, 15}, 3x4x5, 3x5x{9, 11}, 3x6x{10, 15}, 3x7x{10, 15}, 4x4x5, 4x5x5, 5x5x{5, 6, 7}, 5x7x7. Complete list of prime hyperboxes (# = 2): 2x2x4x5, 2x2x5x9. Impossible hyperboxes with the Y-pentomino: 3xZ {4,7}xN both die out after 3 pieces 5xk if k <> 0 (modulo 10) 6xk for all k 8xk for all k 9x{10, 15, 25, 35} 11x{10, 15, 25} 12x{10, 15, 20, 25, 30, 35, 40, 45} 13x{10, 15, 25} 17x{10} 18x{10, 15} 22x{10} 2x5x{4, 9} 2x..x2x3x..x3xN 2x..x2x3x..x3x5x5 2x..x2x3x..x3x5x7 3x..x3x5x6 Thm: no 3xZ strip possible Proof: The I4 part of Y5 can not be at border in all cases. So there must be some I4 part in the middle. Names it 'x'. . . . . x 1 1 1 1 . . . . x x x x 1 . . . . . . . A . . . . Then piece 1 is forthed and square A can not be covered. Thm: no 2x..x2x3x..x3xN possible Proof: It will be shown that the 2x..x2x3x..x3 cannot be filled. Look at the one-dimensional projection only keeping the last axis. The resulting one-dimensional problem is k*N that is k k k k k . . . with k the multiplyer how often this point must be filled. As the I4 part of Y5 is to long for the short axis we have the two placements 1 2 1 1 and 1 1 2 1 in the one-dimensional problem. Now look at the first three points A B B . . . . . If a point A is filled three counts will go into B. As the ratio A:B = k:2k = 1:2 this is too much. Thm: no 2x..x2x3x..x3x5x5 possible Proof: The I4 part of Y5 can only fit into the 5*5 sections. Look at the 5*5 sections. There is only one possibility to fill the corners (ignoring reflections). w w w w x Only the border is shown. y . . . x The rest is "don't care". y . . . x Then no further piece of lenght 4 can be placed. y . . . x y z z z z Thm: no 3x..x3x5x6 possible Proof: The I4 part of Y5 can only fit into the 5*6 sections. Look at the 5*6 sections. There is only one possibility to fill the corners (ignoring reflections). w w w w A x Only the border is shown. y . . . . x The rest is "don't care". y . . . . x y . . . . x y A z z z z How can the empty places 'A' be covered? With an I4 part of the y-pentomino is impossible. w w w w A x This is impossible as 'y' y q . . . x will interfere with 'q', y q . . . x as the hole 'y' will by y q . . . x in the 5*6 section. y q z z z z Therefore all 'A' places are filled by horizontal border pices. Projection into the 5*6 plane gives with each position stands for 3*...*3 cubes an odd number. Note: Excluding the I4 case is not possible in if working with the projection to the 5*6 plane. If the number would be even in the last step then a solution is possible as the 2*5*6 box is possible. x A x x A x Cubes of the y-pentomino which can x . . . . x can fill 'A': x . . . . x 0 1 x . . . . x 0 1 0 0 0 0 1 0 x A x x A x The horizontal y-pentominoes can fill the 'A' positions if each fills 2 cubes into it. As each y-pentomino must deliver an even number of cubes for the 'A' positions we cannot get the required odd sums. Thm: no 2x..x2x3x..x3x5x7 possible Proof: The I4 part of Y5 can only fit into the 5*7 sections. Look at the 5*7 sections. There is only one possibility to fill the corners (ignoring reflections). w w w w C A x Only the border is shown. y B . . . . x The rest is "don't care". y . . . . . x y . . . . B x y A C z z z z How can the empty places 'A' be covered? The same reasoning as in the previous proof shows that the 'A' positions must be filled with horizontal y-pentominoes from the border (w, z type). Therefore the 'B' and 'C' positions are still empty. But filling a B dissects a single cube C in the same 5*7 section which is impossible to fill. Thm: the period of a 3x3xZ packing is a multiple of 15. Proof: As each peace contains five cube the period is a multiple of five. Color the cube (x,y,z) with the two colors according to x+y (mod 2) each piece covers each color with 1 (mod 3). Therefore the number of cubes in both colors must be equal (mod 3). This means the period is a multiple of three. Thm[7]: there is no 3x3xZ15 packing (period 15 packing) (H. Postl) Proof: Color the cube (x,y,z) with the two colors according to x+y (mod 2). That means the cross section looks like 0 1 0 / 1 0 1 / 0 1 0. Let's name 0 B(lack) and 1 W(hite). Then there are the Y5 can cover 4B+W (type a) or 4W+B (type b) cubes. In total we have 5*15 black and 4*15 white cubes. This gives the linear system: a + 4b = 4*15 (the white cubes) 4a + b = 5*15 (the black cubes) The unique solution is (a, b) = (16, 11). Each of the 5 black bars (length 15) can hold at most 3 I4 parts of the Y5. Therefore at most 15 pieces of type a can be filled in. But there are 16 of type a. Annotations: When did I get the results: New (August 1992) 10*23, 10*27, 15*17, 15*19, 15*21, 20*13, 20*17 (September 1992) 30*11, 30*13, 25*17, 35*11, 45*11, 45*9, 55*9, 35*13, 45*13 (October 1992) 15*23, 18*25, 18*35, 3*5*9, 3*5*11, 5*5*6, 5*5*7, 5*7*7, 2*7*10, 3*6*10, 2*7*15 (November 1992) 3*7*10, 3*7*15, 2*2*4*5, 2*2*5*9 impossibility proofs for the hyperboxes (found Dec. 1992, added Aug. 1998) Added: (March 1993) 3*3*Z The H-symmetry: 1 1 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 2 2 The 5xN strip: All 5xN strips dissect into 5x10 rectangles. There are two solutions of the 5x10 rectangle. Both are symmetric. The first one uses only left Ys. |1 1 1 1 2 4 4 4 4 A |A 1 3 2 2 2 2 4 A A |A 3 3 3 3 3 3 3 3 A |A A 4 2 2 2 2 3 1 A |A 4 4 4 4 2 1 1 1 1 |a a a a a H H H H A |A a a a a a H H A A |A H H H H H H H H A |A A H H a a a a a A |A H H H H a a a a a Special 5x15 rectangles without a 1x5 block (5 placements) 22 20 20 20 20 18 15 15 15 15 13 11 11 11 11 22 22 20 18 18 18 18 16 15 13 13 13 13 11 10 22 21 21 21 21 16 16 16 16 14 12 12 12 12 10 22 23 21 19 19 19 19 17 14 14 14 14 12 10 10 23 23 23 23 19 17 17 17 17 . . . . . 10 22 22 22 22 18 16 16 16 16 14 12 12 12 12 10 This can be extended 23 22 20 18 18 18 18 16 14 14 14 14 12 10 10 to a 15x15 square. 23 20 20 20 20 17 17 17 17 15 13 13 13 13 10 23 23 21 19 19 19 19 17 15 15 15 15 13 11 10 23 21 21 21 21 19 . . . . . 11 11 11 11 22 22 22 22 18 17 17 17 17 14 12 12 12 12 10 23 22 20 18 18 18 18 17 14 14 14 14 12 10 10 23 20 20 20 20 19 16 16 16 16 13 13 13 13 10 23 23 21 19 19 19 19 16 15 15 15 15 13 11 10 23 21 21 21 21 . . . . . 15 11 11 11 11 22 22 22 22 18 17 17 17 17 14 12 12 12 12 10 23 22 20 18 18 18 18 17 14 14 14 14 12 10 10 23 20 20 20 20 . . . . . 13 13 13 13 10 23 23 21 19 19 19 19 16 15 15 15 15 13 11 10 23 21 21 21 21 19 16 16 16 16 15 11 11 11 11 22 22 22 22 19 16 16 16 16 14 12 12 12 12 10 23 22 19 19 19 19 17 16 14 14 14 14 12 10 10 23 20 20 20 20 17 17 17 17 . . . . . 10 23 23 21 20 18 18 18 18 15 13 13 13 13 11 10 23 21 21 21 21 18 15 15 15 15 13 11 11 11 11 Special 5x15 rectangles plus a 1x5 block (7 placements) 23 23 23 23 20 18 18 18 18 14 12 12 12 12 10 24 23 20 20 20 20 18 16 14 14 14 14 12 10 10 24 22 22 22 22 19 16 16 16 16 13 13 13 13 10 24 24 22 19 19 19 19 17 15 15 15 15 13 11 10 24 25 21 21 21 21 17 17 17 17 15 11 11 11 11 25 25 25 25 21 24 23 23 23 23 18 16 16 16 16 14 11 11 11 11 24 24 21 23 20 18 18 16 17 14 14 14 14 11 10 24 25 21 21 20 18 17 17 17 17 12 12 12 12 10 24 25 21 20 20 18 19 15 15 15 15 12 13 10 10 25 25 21 22 20 19 19 19 19 15 13 13 13 13 10 25 22 22 22 22 24 22 22 22 22 19 16 16 16 16 14 11 11 11 11 24 24 22 19 19 19 19 16 17 14 14 14 14 11 10 24 23 23 23 23 20 17 17 17 17 12 12 12 12 10 24 25 23 20 20 20 20 15 15 15 15 12 13 10 10 25 25 25 25 21 18 18 18 18 15 13 13 13 13 10 21 21 21 21 18 24 22 22 22 22 19 16 16 16 16 14 11 11 11 11 24 24 22 23 19 19 19 19 16 14 14 14 14 11 10 24 23 23 23 23 18 18 18 18 15 12 12 12 12 10 24 25 21 21 21 21 18 15 15 15 15 12 13 10 10 25 25 25 25 21 20 17 17 17 17 13 13 13 13 10 20 20 20 20 17 24 22 22 22 22 19 16 16 16 16 14 11 11 11 11 24 24 22 23 19 19 19 19 16 14 14 14 14 11 10 24 23 23 23 23 18 18 18 18 15 12 12 12 12 10 24 25 21 21 21 21 18 15 15 15 15 12 13 10 10 25 25 25 25 21 20 17 17 17 17 13 13 13 13 10 20 20 20 20 17 24 22 22 22 22 20 16 16 16 16 14 11 11 11 11 24 24 22 20 20 20 20 16 17 14 14 14 14 11 10 24 23 23 23 23 21 17 17 17 17 12 12 12 12 10 24 25 23 21 21 21 21 15 15 15 15 12 13 10 10 25 25 25 25 19 19 19 19 18 15 13 13 13 13 10 19 18 18 18 18 23 23 23 23 20 18 16 16 16 16 14 11 11 11 11 24 23 22 20 20 18 18 16 17 14 14 14 14 11 10 24 24 22 21 20 18 17 17 17 17 12 12 12 12 10 24 22 22 21 20 18 19 15 15 15 15 12 13 10 10 24 25 22 21 21 19 19 19 19 15 13 13 13 13 10 25 25 25 25 21 The 5xZ strip: All 5xZ strips dissect into 5x10 rectangles. There are two cases. First there is an I4 part arcross the strip. Then one easily sees that the only possible extension results in the 5xN case. In the second case all I4 parts are horizontal. Case checking shows this case has no solution. The 4:1 coloring do not discard this case as the Y5, L5 5xZ strip show 1 1 1 1 t t t t 2 2 2 2 1 1 1 1 t t t t 1 m m m m t t m m m m 2 1 m m m m t t b b b b m m t t t t m m b b b b m m t t t t b b m m m m 2 1 m m m m b b m m m m 2 b b b b 2 2 2 2 1 1 1 1 b b b b 2 2 2 2 The 6xN strip: There are only three differenz border types which are extendable. | a a a a a |A a a a a a |A A +6 b b b b b |A --> b b b b b |A a a a a a | a a a a a | c c c c c c c c b b b b b | c c c c c c b b b b b | +6 c c +6 a a a a a ----> c c c c --> a a a a a a a a a a b b b b b a a a a a b b b b b | a a a a a b b b b b | a a a a a b b b b b | +6 c c c c c c c c +6 a a a a a ----> c c c c c c --> a a a a a c c b b b b b c c c c b b b b b Note: 6x(2+5k) without two corners is always possible. . a b b b b a b b b b a a a h b h a a h b h a a a a h h h a a h h h a a a a h h h a a h h h a a a a h b h a a h b h a a a b b b b a b b b b a . The 8xN strip: There are only few differenz border types which are extendable. The state diagram looks like (Start) --> (**) <------------ (*) | --> (A) --> | --> (B) --> 2 | ----> (C) ----> (D) There are two possibilities for (Start)-->(C). The states (A), (B), (C), and (D) can be repeated. (Start) |A |A A |A C |A C |B C C |B C |B B |B (Start)-->(**) (*)-->(**) + 1 1 1 1 a a a a a a a a a a + + 1 2 a a a a a a a a a a + + 2 2 2 2 3 b + + h h h h 3 b b + + + h h 3 3 b b + + h h h h 3 b b + + a a a a a b b + a a a a a b (**)-->(A) (A)-->(*) (A)-->(A) + + + + + + + a a a a a + + + + + a a a a a + 1 1 1 1 b b b b b + 1 + 1 2 b b b b b + 1 + 2 2 a a a a a + 1 1 + + 2 a a a a a + 1 2 + + 2 b b b b b 2 2 2 2 + + + b b b b b (**)-->(B) (B)-->(*) (B)-->(B) + + + + + + + a a a a a + + + + + a a a a a + 1 1 1 1 b b b b b + 1 + 1 2 b b b b b + 1 1 + + 2 a a a a a + 1 + 2 2 a a a a a + 1 2 + + 2 b b b b b 2 2 2 2 + + + b b b b b (B)-->(D) (D)-->(D) + + + a a a a a a a a a a + + a a a a a a a a a a c c c c c c c c b b b b b + c c c c c c b b b b b + + c c a a a a a + c c c c a a a a a + + b b b b b + + + b b b b b (Start)-->(C) (C)-->(C) + 1 1 1 1 a a a a a + + 1 2 a a a a a + + 2 2 2 2 b b b b b + + h h h h 3 b b b b b + + + h h 3 3 3 3 a a a a a + + h h h h H H H H a a a a a + + a a a a a H H b b b b b + a a a a a H H H H b b b b b (Start)-->(C) + 1 1 1 1 + + 1 2 + + 2 2 2 2 + + a a a a a + + + a a a a a 3 + + 4 4 4 4 3 3 3 3 + + 5 4 a a a a a + 5 5 5 5 a a a a a The 9x(20+10k) rectangles: 46 43 43 43 43 y y y y y . . . . . . . . . . 46 46 41 43 40 y y y y y y . . . . . . . . . 46 47 41 41 40 y y y y y . . . . . . . . . . 46 47 41 40 40 y y y y . . . . . . . . . . . 47 47 41 42 40 y y y y y x x x x x . . . . . 48 47 42 42 42 42 38 32 32 32 32 x x x x . . . . . 48 44 44 44 44 38 38 38 38 32 x x x x x . . . . . 48 48 44 45 39 39 39 39 35 x x x x x x . . . . . 48 45 45 45 45 39 35 35 35 35 x x x x x . . . . . H H H H A a a a a a H H A A B a a a a a H H H H A B H H H H a a a a a A B B H H a a a a a B H H H H b b b b b c c c c c b b b b b c c c c c c c c c c b b b b b c c c c c b b b b b The 9x(45+10k) rectangles: 88 88 88 88 82 79 79 79 79 B2|C1 C1 C1 C1 C1 C2 C2 C2 C2 C2 C3 C3 C3 C3 C3 41 41 41 41 37|33 33 33 33 28 23 23 23 23 20 14 14 14 14 10 89 88 84 82 82 82 82 79 B2 B2|70 C1 C1 C1 C1 C1 C2 C2 C2 C2 C2 C3 C3 C3 C3 C3 42 41 37 37|34 33 28 28 28 28 23 20 20 20 20 15 14 10 10 89 89 84 83 83 83 83 B1 B2 B2|70 67 67 67 67 C4 C4 C4 C4 C4 C5 C5 C5 C5 C5 42 42 42 42 37|34 31 31 31 31 27 21 21 21 21 15 15 15 15 10 89 86 84 84 80 83 B1 B1 B2 B2|70 70 67 65 C4 C4 C4 C4 C4 C5 C5 C5 C5 C5 C6 C6 C6 C6 C6 37|34 34 31 27 27 27 27 24 21 18 18 18 18 11 10 89 86 84 85 80 80 B1 B1 B2 B2|70 71 65 65 65 65 62 56 56 56 56|50 50 50 50 C6 C6 C6 C6 C6|34 32 32 32 32 24 24 24 24 19 16 18 13 11 11 90 86 86 85 80 81 B1 B1 B2 74 71 71 71 71 62 62 62 62 58 56 53|51 50 47 43 43 43 43 38 35 35 35 35 32 29 25 25 25 25 19 16 13 13 11 12 90 86 85 85 80 81 B1 B1 74 74 74 74 66 66 66 66 58 58 58 58 53|51 47 47 47 47 43 38 38 38 38 35 29 29 29 29 26 25 19 19 16 16 13 11 12 90 90 87 85 81 81 B1 B8 75 75 75 75 68 66 63 63 63 63 59 53 53|51 51 48 44 44 44 44 39 36 36 36 36 30 26 26 26 26 22 19 16 17 13 12 12 90 87 87 87 87 81 78 78 78 78 75 68 68 68 68 63 59 59 59 59 53|51 48 48 48 48 44 39 39 39 39 36 30 30 30 30 22 22 22 22 17 17 17 17 12 The structure of this rectangle is a a a a a a a a a b c c c c c c c c c c c c c c c c c c c c e e e e e e e e e e f f f f f a a a a a a a a b b c c c c c c c c c c c c c c c c c c c c e e e e e e e e e e e f f f f a a a a a a a b b b c c c c c c c c c c c c c c c c c c c c e e e e e e e e e e f f f f f a a a a a a b b b b c c c c c c c c c c c c c c c c c c c c e e e e e e e e e f f f f f f a a a a a a b b b b c d c c c c d d d d d e e e e c c c c c e e e e e e e e e e f f f f f a a a a a a b b b d d d d d d d d d d d d e e e e e e e e e e e e e e e e e e e f f f f f a a a a a a b b d d d d d d d d d d d d d e e e e e e e e e e e e e e e e e e e f f f f f a a a a a a b d d d d d d d d d d d d d d e e e e e e e e e e e e e e e e e e e f f f f f a a a a a a d d d d d d d d d d d d d d d e e e e e e e e e e e e e e e e e e e f f f f f This can be extended by a 5x10 rectangle plus four 2x5 parallelograms. The 11x(20+10k) and 11x(35+10k) rectangles: 95 92 92 92 92 87 81 81 81 81 one half of the 11x20 95 95 92 93 87 87 87 87 81 95 93 93 93 93 83 83 83 83 95 96 89 89 89 89 83 84 96 96 96 96 89 84 84 84 84 97 97 97 97 90 85 85 85 85 79 98 97 90 90 90 90 85 79 79 79 79 98 98 94 88 88 88 88 82 78 78 78 78 98 94 94 94 94 88 82 82 82 82 78 98 99 91 91 91 91 86 80 80 80 80 99 99 99 99 91 86 86 86 86 80 a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a 1 a a a a a a a a a a b b b b b 1 a a a a a a a a a a b b b b b 1 1 a a a a a a a a a a a a a a a 1 b b b b b c c c c c a a a a a b b b b b c c c c c h h h h c c c c c b b b b b h h c c c c c b b b b b h h h h b b b b b c c c c c a a a a a b b b b b c c c c c a a a a a 50 48 48 48 48 40 34 34 34 34 28 24 24 24 24 18 12 12 12 12 50 50 48 45 40 40 40 40 34 28 28 28 28 24 18 18 18 18 12 10 50 51 45 45 45 45 35 35 35 35 26 26 26 26 19 15 15 15 15 10 50 51 46 43 43 43 43 35 32 32 32 32 26 23 19 19 16 15 10 10 51 51 46 46 43 41 36 36 36 36 32 29 23 23 19 16 16 16 16 10 53 51 46 47 41 41 41 41 36 29 29 29 29 23 19 20 13 13 13 13 53 49 46 47 42 42 42 42 33 30 30 30 30 23 20 20 20 20 13 11 53 53 49 47 47 44 39 42 33 33 33 33 30 27 22 22 22 22 17 11 11 53 49 49 47 44 39 37 37 37 37 27 27 27 27 22 17 17 17 17 11 49 52 44 44 39 39 38 37 31 31 31 31 25 21 21 21 21 14 11 52 52 52 52 44 39 38 38 38 38 31 25 25 25 25 21 14 14 14 14 The 12x(50+5k) rectangles: 94 94 94 94 88 82 82 82 82 76|68 68 68 68 62 58 58 58 58 51 47 47 47 47 95 94 88 88 88 88 82 76 76 76 76|69 68 62 62 62 62 58 51 51 51 51 47 42 95 92 92 92 92 84 80 80 80 80|69 69 69 69 61 61 61 61 53 50 50 50 50 42 95 95 92 89 84 84 84 84 80|77 70 70 70 70 66 61 57 53 53 53 53 50 42 42 95 96 89 89 89 89 85|77 77 77 77 70 66 66 66 66 57 54 54 54 54 48 43 42 41 96 96 96 96 85 85 85 85|75 75 75 75 67 67 67 67 57 57 55 54 48 48 43 41 41 97 97 97 97 86 86 86 86|78 75 71 71 71 71 67 63 57 55 55 55 55 48 43 43 41 98 97 90 90 90 90 86|78 78 78 78 74 71 63 63 63 63 56 56 56 56 48 43 44 41 98 98 93 90 87 87 87 87 81|74 74 74 74 64 64 64 64 59 56 52 49 44 44 44 44 40 98 93 93 93 93 87 81 81 81 81|72 72 72 72 64 59 59 59 59 52 49 45 45 45 45 40 98 99 91 91 91 91 83 79 79 79 79|73 72 65 65 65 65 60 52 52 49 49 46 45 40 40 99 99 99 99 91 83 83 83 83 79|73 73 73 73 65 60 60 60 60 52 49 46 46 46 46 40 94 94 94 94 88 82 82 82 82 77 70 70 70 70 62 58 58 58 58 51 46 46 46 46 95 94 88 88 88 88 82 77 77 77 77 68 70 66 62 62 59 58 51 51 51 51 46 95 92 92 92 92 87 79 79 79 79 74 68 66 66 62 59 59 59 59 53 47 47 47 47 95 95 92 87 87 87 87 80 79 74 74 68 68 66 62 63 56 53 53 53 53 47 43 95 96 89 89 89 89 80 80 80 80 74 68 69 66 63 63 56 54 54 54 54 43 43 43 43 96 96 96 96 89 85 81 81 81 81 74 69 69 69 69 63 56 56 55 54 48 48 48 48 97 97 97 97 85 85 85 85 81 78 71 71 71 71 67 63 56 55 55 55 55 48 42 42 42 42 98 97 90 90 90 90 86 78 78 78 78 71 67 67 67 67 57 57 57 57 49 49 49 49 42 98 98 93 90 86 86 86 86 75 75 75 75 72 64 64 64 64 60 57 52 50 49 44 40 40 40 40 98 93 93 93 93 83 83 83 83 75 72 72 72 72 64 60 60 60 60 52 50 44 44 44 44 40 98 99 91 91 91 91 84 83 76 76 76 76 73 65 65 65 65 61 52 52 50 50 45 41 41 41 41 99 99 99 99 91 84 84 84 84 76 73 73 73 73 65 61 61 61 61 52 50 45 45 45 45 41 The 13x20 (unique), 13x30, 13x35, 13x45 rectangles: 32 32 32 32 23 18 18 18 18 2 2 2 2 2 2 2 2 2 2 2 33 32 23 23 23 23 14 18 12 2 2 2 2 2 2 2 2 2 2 2 33 27 27 27 27 19 14 12 12 2 2 2 2 2 2 2 2 2 2 2 33 33 28 27 19 19 14 14 12 2 2 2 2 2 2 2 2 2 2 2 33 28 28 28 28 19 14 15 12 2 2 2 2 2 2 2 2 2 2 2 34 29 29 29 29 19 15 15 15 15 2 2 2 2 2 2 2 2 2 2 34 34 29 24 20 20 20 20 13 10 2 2 2 2 2 2 2 2 2 2 34 30 24 24 24 24 20 13 13 10 2 2 2 2 2 2 2 2 2 2 34 30 30 25 21 21 21 21 13 10 10 2 2 2 2 2 2 2 2 2 35 30 25 25 25 25 21 16 13 10 11 2 2 2 2 2 2 2 2 2 35 30 26 26 26 26 16 16 16 16 11 2 2 2 2 2 2 2 2 2 35 35 31 26 22 22 22 22 17 11 11 2 2 2 2 2 2 2 2 2 35 31 31 31 31 22 17 17 17 17 11 2 2 2 2 2 2 2 2 2 58 58 58 58 47 43 43 43 43 36 31 31 31 31 24 18 18 18 18 . . 59 58 53 47 47 47 47 43 36 36 32 31 24 24 24 24 14 18 12 . . 59 53 53 53 53 44 44 44 44 36 32 25 25 25 25 19 14 14 12 . . 59 59 54 48 48 48 48 44 38 36 32 32 25 26 19 19 14 12 12 . . 59 54 54 54 54 48 38 38 38 38 32 26 26 26 26 19 14 15 12 . . 60 55 55 55 55 49 39 39 39 39 33 27 27 27 27 19 15 15 15 15 . 60 60 55 49 49 49 49 40 39 33 33 33 33 27 20 20 20 20 13 10 . 60 56 50 50 50 50 40 40 40 40 34 28 28 28 28 20 21 13 13 10 . 60 56 56 50 51 45 45 45 45 34 34 34 34 28 21 21 21 21 13 10 10 61 56 51 51 51 51 45 41 37 37 37 37 29 23 23 23 23 16 13 10 11 61 56 52 52 52 52 41 41 41 41 37 29 29 29 29 23 16 16 16 16 11 61 61 57 52 46 46 46 46 42 35 35 35 35 30 22 22 22 22 17 11 11 61 57 57 57 57 46 42 42 42 42 35 30 30 30 30 22 17 17 17 17 11 47 47 47 47 38 33 33 33 33 24 19 19 19 19 12 . . 48 47 45 38 38 34 33 31 24 24 24 24 19 12 12 . . 48 45 45 39 38 34 31 31 31 31 20 20 20 20 12 . . 48 48 45 39 38 34 34 27 25 25 25 25 20 18 12 . . 48 46 45 39 39 34 35 27 27 26 25 18 18 18 18 . . . 46 46 39 41 35 35 27 26 26 26 26 15 15 15 15 . . 46 41 41 41 41 35 27 28 22 22 22 22 16 15 11 . . 46 42 42 42 42 35 28 28 28 28 22 16 16 11 11 . . . 43 42 36 36 36 36 32 23 23 23 23 16 13 11 10 . . 43 43 40 36 32 32 32 32 23 21 17 16 13 11 10 . . 43 40 40 40 40 29 29 29 29 21 17 13 13 10 10 . . 43 44 37 37 37 37 30 29 21 21 17 17 13 14 10 . . 44 44 44 44 37 30 30 30 30 21 17 14 14 14 14 The 14x(10+5k) rectangles: 19 19 19 19 11 A A A A A 2 2 2 2 2 20 19 11 11 11 11 A A A A A 2 2 2 2 20 H H H H B B B B B 2 2 2 2 2 20 20 H H B B B B B 2 2 2 2 2 2 20 H H H H 8 8 8 8 7 2 2 2 2 2 21 16 16 16 16 5 8 6 7 7 2 2 2 2 2 21 21 H 16 H 5 5 6 6 7 2 2 2 2 2 21 22 H H H 5 4 6 2 7 2 2 2 2 2 21 22 H H H 5 4 6 2 1 2 2 2 2 2 22 22 H 13 H 4 4 2 2 1 2 2 2 2 2 23 22 13 13 13 13 4 3 2 1 1 2 2 2 2 23 H H H H 3 3 3 3 1 2 2 2 2 2 23 23 H H A A A A A 2 2 2 2 2 2 23 H H H H A A A A A 2 2 2 2 2 The 15x15 square has 16 solutions (without H-symmetry) . . . . . . . . . . 18 18 18 18 10 The special 15x15 . . . . . . . . . . 19 18 15 10 10 solution which . . . . . . . . . . 19 19 15 11 10 contains a . . . . . . . . . . 19 15 15 11 10 5x10 rectangle. . . . . . . . . . . 19 22 15 11 11 41 41 41 41 33 33 33 33 27 22 22 22 22 11 12 42 41 36 36 36 36 33 27 27 23 23 23 23 12 12 42 42 39 36 34 34 34 34 27 25 20 23 16 13 12 42 39 39 39 39 34 31 28 27 25 20 20 16 13 12 42 43 37 37 37 37 31 28 25 25 20 16 16 13 13 43 43 38 37 35 31 31 28 28 25 20 24 16 13 14 44 43 38 38 35 32 31 28 29 24 24 24 24 14 14 44 43 38 35 35 32 29 29 29 29 21 21 21 21 14 44 44 38 40 35 32 32 30 26 26 26 26 21 17 14 44 40 40 40 40 32 30 30 30 30 26 17 17 17 17 22 22 22 22 18 16 16 16 16 14 12 12 12 12 10 23 22 20 18 18 18 18 16 14 14 14 14 12 10 10 23 20 20 20 20 17 17 17 17 15 13 13 13 13 10 23 23 21 19 19 19 19 17 15 15 15 15 13 11 10 23 21 21 21 21 19 . . . . . 11 11 11 11 . . . . . . X X X X X . . . . . . . . . X X X X X . . . . . 35 33 33 33 33 . . . . . 18 13 13 13 13 35 35 33 31 26 26 26 26 22 18 18 18 18 13 10 35 36 31 31 31 31 26 22 22 19 19 19 19 10 10 35 36 32 27 27 27 27 23 22 20 15 19 14 11 10 36 36 32 32 27 28 23 23 22 20 15 15 14 11 10 37 36 32 28 28 28 28 23 20 20 15 14 14 11 11 37 37 32 29 29 29 29 23 24 20 15 16 14 11 12 37 34 34 34 34 29 24 24 24 24 16 16 16 16 12 37 38 34 30 30 30 30 25 21 21 21 21 17 12 12 38 38 38 38 30 25 25 25 25 21 17 17 17 17 12 Jenifer Hasegrove first found a 15x15 square [3] also shown in [8]. Note that in his solution in the left upper corner the part x x x x x x x x x x x x x x x x x x x x can has C4 symmetry and can be rotated as its tiling has C2 symmetry. This part intersects a H-subblock. The 2x5x6 Box: (unique without H-symmetry) 1 x x x x x x x x x x 3 1 h h h h 2 4 h h h h 3 1 1 h h 2 2 4 4 h h 3 3 1 h h h h 2 4 h h h h 3 y y y y y 2 4 y y y y y The 2x5x8 Box: (unique without H-symmetry) h . h g g g g 1 one half h h h . g g 1 1 h h h g g g g 1 h 2 h 3(3)3 3 1 2 2 2 2 4 4(4)4 The 3x4x5 Box: (unique without H-symmetry) 1 1 1 1 2 3 g 1 one half h h h h 2 3 g g 3 h h 2 2 3 g h h h h 2 3 g The 4x4x5 Box: (unique without H-symmetry) 1 1 1 1 2 1 4 3 . . . . one half h h h h 2 4 4 3 . . . . 2 h h 3 2 5 4 3 5 5 5 5 h h h h 2 6 4 3 6 6 6 6 The 5x5x5 Box: 0 0 0 0 12 29 0 X4 17 14 29 X1 X4 14 14 29 25 X4 15 14 29 21 X4 16 14 0 0 0 0 12 26 0 17 17 H1 X4 X4 X4 X4 H1 29 25 X1 15 H1 28 21 21 16 H1 0 0 0 0 12 26 0 X4 17 12 26 X1 X4 15 H1 25 25 X4 15 H1 28 21 X4 16 16 0 0 0 0 12 26 0 24 17 H1 24 24 24 24 H1 28 25 23 15 H1 28 21 19 16 H1 0 0 0 0 13 26 0 22 13 13 22 22 22 22 13 23 23 23 23 13 28 19 19 19 19 . . . . . unique solution for crown configuration I . . 21 . . . 24 Y 15 . . . 17 . . . . . . . H1 26 16 15 11 H1 24 21 15 12 H1 24 Y 15 12 H1 24 17 15 12 29 24 17 23 12 26 26 16 14 11 H1 21 21 14 11 H1 Y Y 14 14 29 22 17 14 12 29 23 23 23 23 H1 26 16 16 11 H1 22 21 13 10 H1 22 Y 13 10 H1 22 17 13 10 29 22 20 13 10 25 26 16 18 11 25 18 18 18 18 25 25 19 13 10 25 19 19 19 19 29 20 20 20 20 . . . . . unique solution for crown configuration II . 26 . 15 . . . Y . . . 28 . 17 . . . . . . 25 25 25 25 13 26 26 25 15 H1 27 Y Y Y Y 28 28 17 17 H1 29 29 29 29 14 24 24 24 24 13 27 26 19 15 H1 27 19 19 15 H1 27 28 19 17 H1 27 29 19 14 14 22 24 16 13 13 22 26 16 15 H1 22 22 16 16 H1 22 28 16 17 H1 23 23 23 23 14 20 20 20 20 13 H2 20 H2 12 H1 H2 H2 H2 12 12 H2 H2 H2 12 H1 H2 23 H2 12 14 The 2x2x4x5 Hyperbox: The 2x2x5x9 Hyperbox: (half) 0 0 0 0 b c b b A 1 A 3 4 B B B B b 0 b b b c b b A A A 3 4 4 B B X A: 3 solutions a a a a b c b b A A A 3 4 B B B B B: 2 solutions a a a a b 0 b b A 2 A 3 4 6 6 6 6 a a a a 0 0 0 0 2 2 2 2 5 7 7 7 7 All solutions for 2x5x9 box 0 0 0 0 c c c c 1 1 1 1 9 0 0 0 0 with 2 defects d 0 d d c c c c x A 9 9 9 9 a 0 b at the X positions. d a d d c c c c x A A 3 8 a a b b Two of this box d a d d d 0 d d X A 8 8 8 8 a 6 b give a 2x2x5x9 box. d a d d 0 0 0 0 x A 5 5 5 5 a 7 b The 3x3xZ strip: A symmetric 3x3xZ solution which is reflection-periodic of order 15. . . . . 57 50 50 50 50 48 42 42 42 42 35 . . . . . . 57 57 57 57 53 48 48 48 48 42 39 39 39 39 33 . . 59 59 59 59 53 53 53 53 47 44 44 44 44 39 37 33 33 33 33 . . 55 55 55 55 51 50 46 43 43 43 43 35 35 35 35 . . . . 56 56 56 56 52 46 46 46 46 43 40 36 36 36 36 . . . . 59 54 54 54 54 47 47 47 47 44 41 37 37 37 37 . . . . . 55 51 51 51 51 45 45 45 45 38 38 38 38 . . . . . 58 56 52 52 52 52 49 45 40 40 40 40 38 36 34 . . . 58 58 58 58 54 49 49 49 49 41 41 41 41 34 34 34 34 . This gives a 3x3xZ30 solution. H. Postl [7] found a 3x3xZ45 solution too. Therefore there are 3x3xZ(15k) solutions for all k>=2 and all others are impossible by the impossibility theorems given above. Figures of height 5: 5 5 5 5 5 1 1 1 1 1 5 5 5 5 5 unique as 5 times 1 1 1 1 1 5 5 5 5 5 5 5 5 5 5 5 5 compare with the 2x5x6 box 20 17 17 17 17 12 20 18 18 18 18 11 20 20 18 19 11 11 20 19 19 19 19 11 21 21 21 21 15 11 . 16 17 12 12 12 12 . 16 13 13 13 13 10 . 16 16 13 14 10 10 . 16 14 14 14 14 10 . 21 15 15 15 15 10 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 compare with the 2x5x8 box 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 68 65 65 65 65 61 56 54 53 68 68 63 61 61 61 61 54 53 68 66 64 64 64 64 58 54 53 68 67 67 67 67 59 57 54 53 69 69 69 69 59 59 59 59 55 . 66 63 65 60 56 56 56 56 52 . 66 63 60 60 60 60 54 53 52 . 66 63 64 58 58 58 58 52 52 . 66 63 67 62 57 57 57 57 52 . 69 62 62 62 62 55 55 55 55 all longer figures are constructable from these 3 pieces. 5 5 5 5 5 5 5 5 5 5 5 5 5 5 unique 68 66 65 59 59 59 59 56 68 68 65 62 60 57 56 56 68 65 65 60 60 60 60 56 68 69 65 61 61 61 61 56 69 69 69 69 58 58 58 58 66 66 66 66 59 57 67 62 62 62 62 57 67 63 63 63 63 57 67 67 63 64 61 57 67 64 64 64 64 58 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 unique 68 68 68 68 59 56 56 56 56 52 69 63 63 63 63 57 57 57 57 52 69 64 64 64 64 58 54 57 52 52 69 69 65 61 61 61 61 53 55 52 69 65 65 65 65 61 55 55 55 55 66 68 62 59 59 59 59 56 66 62 62 63 60 58 54 53 66 66 62 64 60 58 54 53 66 67 62 60 60 58 54 53 67 67 67 67 60 58 54 53 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5 5 5 5 5 5|5 5 5 5 all longer figures are constructable from these pieces. 5 5 5 5 5 5 5 5|5 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 68 66 61 61 61 61 55 51 51 51 51 68 64 64 64 64 58 52 52 52 52 49 68 68 64 65 59 58 58 52 53 49 49 68 65 65 65 65 58 53 53 53 53 49 69 69 69 69 60 58 54 54 54 54 49 66 66 66 66 61 55 55 55 55 51 67 62 62 62 62 56 56 56 56 50 67 67 62 59 59 59 59 56 50 50 67 63 63 63 63 57 57 57 57 50 67 69 63 60 60 60 60 57 54 50 5 5 5 5 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5|5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 68 68 68 68 58 58 58 58 53 47 47 47 47 69 68 60 60 60 60 58 53 53 53 53 47 45 69 65 65 65 65 57 55 54 54 54 54 45 45 69 69 64 65 59 59 59 59 50 50 50 50 45 69 67 61 61 61 61 59 56 51 51 51 51 45 66 62 62 62 62 57 55 52 48 48 48 48 66 66 63 62 60 57 55 52 52 49 48 46 66 63 63 63 63 57 55 52 54 49 46 46 66 64 64 64 64 57 55 52 49 49 50 46 67 67 67 67 61 56 56 56 56 49 51 46 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 66 62 62 62 62 58 53 57 52 47 49 45 43 66 66 63 62 58 58 53 53 52 47 50 45 43 66 63 63 63 63 58 53 52 52 47 47 45 43 66 64 64 64 64 58 53 56 52 47 48 45 43 67 67 67 67 61 56 56 56 56 48 48 48 48 68 68 68 68 59 57 57 57 57 49 49 49 49 43 69 68 65 59 59 59 59 54 50 50 50 50 43 43 69 65 65 65 65 60 54 54 54 54 51 45 44 43 69 69 64 60 60 60 60 55 51 51 51 51 46 43 69 67 61 61 61 61 55 55 55 55 46 46 46 46 all longer figures are constructable from these pieces. 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 68 68 68 68 61 54 54 54 54 69 68 61 61 61 61 54 47 45 69 66 62 56 56 56 56 47 45 69 69 67 57 57 57 57 47 45 69 67 67 67 67 57 53 47 45 . 65 65 65 65 50 50 50 50 . 66 62 65 55 51 51 51 51 . 66 62 55 55 56 52 47 45 . 66 62 60 55 52 52 52 52 . 66 62 64 55 53 53 53 53 . 63 58 58 58 58 48 50 46 . 63 63 59 58 48 48 51 46 . 63 59 59 59 59 48 46 46 . 63 60 60 60 60 48 49 46 . 64 64 64 64 49 49 49 49 References: 0) Solomon W. Golomb, Tiling with Polyominoes, JoCT 1 (1966) 280-296 Y-primes: 5x10 0) David A. Klarner, Packing a Rectangle with Congruent N-ominoes, JoCT 7 (1969) 107-115; Zbl 174.41 Y-primes: 5x10 1) Chris J. Bouwkamp, David A. Klarner, Packing a Box with Y Pentacubes, JoRM 3:1 (1970) 2-26 Y-primes: 5x10, 10x16, 15x16, 15x22, 25x22 2) Frits G"obel, D. A. Klarner, Packing Boxes with Congruent Figures, Kon. Ned. Akad. Wet. Amst. A (=Indagationes Math.) 72 (1969) 465-472 3) Jenifer Hasegrove, Packing a Square with Y-pentominoes, JoRM 7 (1974) 229 Y-primes: 15x15, one part of the given solution can be rotated. 4) J. Bitner, Tiling 5n*12 Rectangles with Y-pentominoes, JoRM 7 (1974) 276-278 Y-primes: 5n*12, n=10..19 5) Karl Scherer, JoRM 12 (1979/80) 201 Y-primes: 10x14, 15x14, 20x9, 30x9, 20x11 6) Chris J. Bouwkamp, the cube-y problem, Cubism for fun 25 ( = CFF 25 silver aniversary ) (dec 1990 - jan 1991) part 3, pp. 30-43. the 1264 distinct solutions to the 5 x 5 x 5 cube. 7) Helmut Postl, letter from 23. July 1997 (He proofs there is no periodic solution of 3x3xZ with period 15. There are periodic solution of 3x3xZ with period 15*k and k>=2.) 8) Solomon W. Golomb, Polyominoes, 1994, 2nd Ed., Chap. 8: Tiling Rectangles with Polyominoes figure 162 shows Hasegrove's 15-square 9) Yuri Aksyonov Handed Y-pentomino problem is solved, email from 1999-03-06 10) Chris J. Bouwkamp, Tiling Squares with Pentominoes Y, Cubism for fun 49, (June 1999) 5-8 (some rotational symmetric square packings) -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/