prime packings of the P-pentomino: Torsten Sillke, list complete x x open: handed P5 rectangles x x x Prime: 2x5, 7x15, 3x3x5 Impossible: 3xZ, 5xu with u odd Proof: impossibility of 3xZ, trivial Proof: impossibility of 5xu with u odd, 5*N dissects in 1 1 1 1 1 4 1 1 1 X X X 1 1 1 X X X 3 3 3 2 1 1 1 1 4 4 1 1 X X X X 1 1 X X X X 3 3 2 2 1 2 2 2 4 4 2 2 X X X 4 2 2 X X X Y Y Y 2 2 . . . 2 2 2 2 3 3 2 2 3 3 4 4 2 2 3 3 Y Y Y Y 1 1 2 2 2 3 3 3 2 3 3 3 4 4 2 3 3 3 Y Y Y 1 1 1 With X X X 1 1 1 But 1 2 2 gives a disscetion X X X X = 2 2 1 1 1 1 2 2 into 2x5 and 4x5 X X X 2 2 2 1 1 2 rectangles. Annotations: 7x(10+5n) rectangle with n>=0 a a a a a 1 1 2 2 2 3 g g b b a a a a a 1 1 2 2 3 3 g g b b c c d d e e 1 4 4 3 3 g g b b c c d d e e 4 4 4 5 5 g g b b c c d d e e 6 6 5 5 5 g g b b c c d d e e 6 6 7 7 f f f f f c c d d e e 6 7 7 7 f f f f f 3x3x5 box 1 1 2 p p 2 = 2x5 + 2x5 + 5xP5 p p p References: - David Klarner, Packing a Rectangle with Congruent N-ominoes, JoCT 7 (1969) 107-115, gives in figur ?? an example of a 7x15 rectangle with P5. - Solomon W. Golomb, Polyominoes, 1994, 2nd Ed., Chap. 8: Tiling Rectangles with Polyominoes figure 161 shows the 7x15 rectangle with P5 of Klarner Handed P5: ========== Michael Reid found that there is no odd rectangle with the handed P5 for rectangles of width <= 21.