prime-boxes of polycubes Torsten Sillke, 1993-11-16 initial version 2000-12-20 last update My (incomplete) list of Polycubes, for which I have computed prime rectangles, boxes, hyperboxes. I use the hight notation, the Schroeppel notation has been appended two times in brackets. (The pentacube numbering is from E. Kuenzell.) News: 1997 Wolf showed that Z5 is boxable. News: July 1997 Helmut Postl showed there is no 3x3xZ15 tiling with Y5 (problem 6). News: Jan. 1998 Helmut Postl found packings for the pentacubes 33, 37 and 51. He completed the N5 packings. News: May. 1998 Michael Reid found all prime rectangles for the handed L5 pentomino. News: June 1998 All pentacubes are boxable with two exceptions: X and 35. The last cases have been determined by Helmut Postl. He completed the Q5 packings. News: July 1998 Some further list have been completed by Helmut Postl. He completed the 81, J4, glider packings. News: March 1999 Yura Aksyonov found an elegant proof that 5x10 is the only prime rectangle for the handed Y5. News: April 1999 Michael Reid completed the list of G6 rectangles by showing that one side must be a multiple of 4. News: April 1999 Michael Reid proofed that there is no odd rectangle for the handed P5. News: Dec. 2000 Michael Reid showed that the area of rectangles made of X and I3 is a multiple of 3. polyominoes with complete list a prime packings: I5 (trivial, as it is harmonious) P5 (everybody knows) L5 (3*3*Z, 3*3*3*30, 3*3*3*45 are new, 3*3*...*3*15 is impossible) Y5 U5 1 1 [ 1 1 ] 61: 2 1 [ 3 1 ] 1 71: 2 1 1 2 41: 1 1 1 1 1 31: 2 1 completed june 1994 list complete in 3-dim: V5 N5 (completed by Helmut Postl in Jan. 1998) 1 1 1 Q5: 1 1 (completed by Helmut Postl in June 1998) 1 (completed by Helmut Postl in July 1998) 81: 2 1 1 (3*5*6 (smallest)) 1 1 (completed by Helmut Postl in July 1998) glider: 1 1 1 (e.g. 6*10, 10*10, 13*25, 3*4*5) 1 1 (completed by Helmut Postl in July 1998) J4: 1 1 (14*36 smallest rectangle by Helmut Postl Dec. 1997) list almost complete: F5 the missing case in 3-dim is 5*7*7 list incomplete: T5 (3*10*10, 5*5*12 (smallest)) W5 (5*6*6 (smallest)) 1 1 U6 1 1 1 1 (2*4*6, 3*4*4) 1 1 6 1 1 1 1 (4*6) G6: 1 1 1 (2-dim complete) 1 1 1 21: 1 (4*5*6, 4*10*10, 4*8*20, 6*6*{15,20,25}) 1 1 2 (With reflections: 3*4*15, 3*5*6, 3*5*9, 4*4*10) 2 82: 1 1 1 (2*5*5 (smallest)) 1 1 1 1 large-U: 1 (e.g. 4*5*6, 3*4*10) 1 1 1 J5: 1 1 few packings are known: Z5: (6*6*25 and 6*10*10 found by Wolf 1997) (6*6*25 has been found by Helmut Postl too) 2 1 33: 1 1 (5*8*8 has been found by Helmut Postl in Jan. 1998) 1 51: 1 2 1 (7*8*20 has been found by Helmut Postl in Jan. 1998) 2 37: 1 2 (5*N*N is tilable, all n-dim packings found for n>=4) (4*9*20 has been found by Helmut Postl in Jan. 1998) not boxable in three dimension (even with reflections): 2 2' [ 3 2 ] (2' : only one cube at the second level) 35: 1 1 [ 1 1 ] (shown by Helmut Postl in June 1998) not boxable in any dimension: X5 (N*N*...*N is not tilable, as the corner can not be filled.) Open Problems: 1) The last tetracube tiling problem: (2nd. Update) --------------------------------------------------- ______ |\ \ Is it possible to tile a 3*2n*2m box | \_____\ with the tetracube shown left? | | |____ |\| | \ All other (hyper-) box tiling problems with | *_____|_____\ only alike tetracubes are solved. See: | |\ \ | A. L. Clarke, Packing Boxes with Congruent Polycubes, \| \_____\ | J. of Recreational Mathematics 10 (1977/78) 177-182 * | |___| \| | *_____| You can show 2 | nm. According to my computations n and m had to be greater then 9. I found no 3*2n*N for n = 1..9 (n=10 is halve done). The search tree dies out before reaching 4/3 n (3rd dimension). Mike Beeler confirmed my computations for n = 1..8. The 3*4*Z is tileble: build two times 2 1 1 2 2 1 1 2 2 1 1 2 . . 1 2 2 1 1 2 2 1 1 2 2 1 . . . . 1 2 2 1 1 2 2 1 1 2 2 1 . . 2 1 1 2 2 1 1 2 2 1 1 2 use the a a b b a a b b a a b b dissection a c c b a c c b a c c b b c c a b c c a b c c a b b a a b b a a b b a a The 3*N*N is tileble: as the 3*(4*Z bent) is tileble. . . . . 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 Try for yourself! 3 3 3 3 3 3 3 3 . . 3 3 3 3 3 3 3 3 . . 3 3 3 3 3 3 3 3 My Conjecture: - the 3*n*N box (one side open) is not tileble for all n>0. 2) Rectangles with the One-sided Y5, P5, L5. -------------------------------------------- There are three pentominoes, which tile rectangles and are handed. These are: L, P, and Y. Show: the only prime rectangles for the one-sided pentominoes are: L: 2*5 -> see update (solved) P: 2*5 -> see update (solved) Y: 5*10 -> see update (solved) The handed L4 is long known (D. A. Klarner, AMM 70:7 (Sep. 1963) 760-61 E1543). If you allow for the bent tromino only the halve turn, so you have o o o o o and o only, then the prime rectangles are 2*3 and 3*2. There is no odd rectangle possible. This follows from Conway's proof of tiling a triangle with o o o = T2. Shearing T2 gives the two orientations of the tromino. This has been noticed by Noam Elkies. UPDATE: Michael Reid found (until May 1998) all prime rectangle of the handed L5: 2x5, 13x55, 15x39, 17x35, 19x25. He showed that there is no odd rectangle for the handed P5 of width <= 21. UPDATE: Yura Aksyonov (1999-03-06) He found a elegant proof that 5x10 is the only prime. UPDATE: Michael Reid (1999-04-14) He found a group proof that 2x5 is the only prime. 3) T4 rectangles with one hole ------------------------------ Is there a rectangle n*m with n*m = 1 (mod 4) which can be tiled with T4 leaving one hole. My Conjecture: No D. W. Walkup, Covering a Rectangle with T-tetrominoes, AMM 72:9 (1965) 986-88 4) n-bone conjecture -------------------- The n-bone are n spheres in a row glued together. So the 5-bone looks like: ooooo John H. Conway solved the 3-bone conjecture: It is impossible to tile with 3-bones a triangular arangement of spheres. See: W. P. Thurston, Conway's Tiling Groups, AMM 97 (1990) 757-773 You see easily that with 2-bones there is a tiling of the triangle or tetrahedron iff the number of spheres is odd. On the right is a 2 example tiling of 2 1 a triangle with 2-bones 0 0 1 My extended conjecture is: (the n-bone conjecture) There is no tiling of the d-simplex with n-bones, where d>=2 and n>=3. The simplex had to be finite of course. 5) N-pentasphere tetrahedron ---------------------------- Is there a easy proof, that 24 o o o o o do not tile an tetrahedron with 8 spheres on the edge. (last year I made a list of possible packings of the thetrahedron with polyspheres [written in german]. This tile needed the longest time.) My program takes 100h on a HP850 to show this. Is it possible to tile an tetrahedron with 9 spheres on the edge? 6) Y5 packings with minimal period ---------------------------------- A periodic tiling of 3x3xZ with the Y5 has a length which is a multiple of 15. I constructed a tiling with period 30. Find one with period 15. UPDATE: Helmut Postl solved this (letter from july 1997). He showed that period 15 is impossible by coloring arguments. He find constructions for all periods of length 15*n and n>1. It is possible to have a symmetric glide reflections of period 15. -- problem solved -- 7) F5 box 5x7x7 --------------- The last open problem for the F-Pentomino box packings is: Can the 5x7x7 box be packed? 8) Pu prime rectangles (with u = 2n-1) [Helmut Postl] -------------------------------------- The P(2n-1) is a 2xn rectangle without one corner. Helmut Postl thinks that the only prime rectangle is the trivial one 2x(2n-1) for n >= 5. For the P3, P5, P7 other prime rectangles have been found. UPDATE: odd prime rectangles are possible for general L-pieces. Michael Reid (1999-01) -- problem solved -- References: -) S. W. Golomb, Tiling with Polyominoes, JoCT 1 (1966) 280-296 -) D. A. Klarner, Packing a Rectangle with Congruent N-ominoes, JoCT 7 (1969) 107-115; Zbl 174.41 -) F. G"obel, D. A. Klarner, Packing Boxes with Congruent Figures, Kon. Ned. Akad. Wet. Amst. A (=Indagationes Math.) 72 (1969) 465-472 -) M. Gardner, Mathematical Magic Show, New York (1977), Chap. 13: Polyominoes and Rectification -) A. L. Clarke, Packing Boxes with Congruent Polycubes, J. of Recreational Mathematics 10 (1977/78) 177-182 -) S. W. Golomb, Polyominoes which tile rectangles JoCT A 51 (1989) 117-124; Zbl 723.05041 -) K. A. Dahlke, A Heptomino of order 76, JoCT A 51 (1989) 127-8 & 52 (1989) 321; Zbl 715.05014 -) K. A. Dahlke, A Y-hexomino has order 92, JoCT A 51 (1989) 125-6; Zbl 715.05013 -) I. Stewart, Another Fine Math You've Got Me Into, Freeman, 1992, Chap. 2: Tile and Error -) S. W. Golomb, Polyominoes, 1994, 2nd Ed., Chap. 8: Tiling Rectangles with Polyominoes -- Dr. Torsten Sillke mailto:sillke@mathematik.uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/PENTA/qu-prime