Problem of the Month (August 2010)

This month we examine 4 related tiling problems:

1. Given two polyominoes, what is the smallest (by area) rectangle that they tile, using at least one copy of each?

2. Given a polyomino reptile P, how many copies of P can be used to tile P?

3. Which polykings (vertex-connected unions of unit squares) are rectifiable (can tile a rectangle)?

4. Which polykings are reptiles? What are their orders? Are there polykings that are reptiles but not rectifiable?

Solvers this month include Mike Reid, George Sicherman, Joe DeVincentis, Bryce Herdt, Anti Sõlg, Rodolfo Kurchan, Maurizio Morandi, Patrick Hamlyn, Livio Zucca, and George Sicherman.

1.

Rodolfo Kurchan let me know that this problem has been studied before, with many solutions here and here, with most solutions by Mike Reid.

Below are the smallest-known rectangles containing pairs of polyominoes:

Tetrominoes and Tetrominoes
none

Tetrominoes and Pentominoes

(GS)
?
(GS)

(GS)
nonenonenone
(GS)

(GS)

(GS)
none
nonenonenonenonenone

Pentominoes and Pentominoes
?
(GS)

(GS)
nonenonenonenonenone
nonenonenone
nonenonenone
(Mike Reid)
nonenone
(Mike Reid)
none
(Mike Reid)
nonenone
none

Tetrominoes and Hexominoes

(GS)

(GS)
none
(JD)
none
(MR)
none
(MR)
none
(MR)
none
(JD)
none
(MR)

(GS)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(MR)
none
(MR)
none
(MR)
none
(MR)
none
(MR)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)
none
(JD)
none
(JD)

(GS)

(MR)

(MR)

(GS)
none
(MR)
none
(MR)
none
(MR)
none
(MR)
none
(MR)

(GS)
?
(GS)

(GS)
none
(JD)
none
(JD)
none
(JD)
none
(JD)
none
(JD)
none
(JD)
none
(JD)

Pentominoes and Hexominoes

(GS)

(MR)

(GS)

(GS)

(GS)

(GS)
none
(GS)

(GS)

(GS)

(GS)

(GS)

(GS)

(GS)
none
(JD)
none
(JD)
none
(JD)
none
(GS)
none
(JD)

(MR)

(MR)
none
(GS)
none
(GS)

(MR)
none
(GS)
none
(GS)
none
(JD)
none
(GS)
none
(JD)

(GS)
?
(GS)

(MR)

(MR)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(GS)
none
(JD)
none
(GS)

(GS)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)

(MR)

(GS)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(GS)
none
(JD)
none
(JD)
none
(JD)

(MR)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)
none
(JD)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)

(GS)

(GS)
none
(JD)
none
(JD)
none
(JD)

(GS)

(GS)
none
(JD)
none
(JD)
none
(JD)
none
(GS)
none
(JD)
none
(GS)
none
(JD)
none
(GS)

(GS)
none
(GS)

(GS)
none
(GS)

(GS)
none
(MR)

(GS)
none
(GS)
none
(JD)
none
(JD)
none
(JD)
none
(GS)

(GS)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)
none
(JD)
none
(GS)
none
(JD)
none
(GS)

(GS)
none
(JD)
none
(GS)
none
(GS)

(MR)

(GS)

(MR)
none
(GS)

(GS)

(GS)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(GS)
none
(JD)
none
(GS)
none
(GS)
none
(GS)

(GS)

(GS)

(GS)

(GS)
none
(JD)

(GS)

(GS)
none
(JD)
none
(JD)
none
(JD)

(JD)

(GS)
none
(JD)

(GS)

(GS)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)

(MR)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)

(GS)

(GS)
none
(JD)
none
(JD)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)

(GS)

(GS)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)
none
(JD)
none
(JD)

(GS)
none
(JD)
none
(JD)
none
(JD)
?
none
(JD)
none
(GS)
none
(GS)

(MR)

(GS)
none
(JD)
none
(JD)
none
(JD)
?
none
(JD)
none
(GS)
none
(GS)

George Sicherman and Patrick Hamlyn hex-hex and hept-hept pictures. George has more pictures, which are color-coded and have areas, while Patrick's tilings are all minimal. George's hex-hex pictures are here and here, while Patrick's are here. George's hept-hept pictures are here, here, and here, while Patrick's are here.

George Sicherman also sent 1 tet-hept picture, 1 2 pent-hept pictures, 1 2 3 4 5 6 7 8 pent-oct pictures, and 1 2 3 4 5 hex-hept pictures.

George Sicherman also investigated tiling tilted rectangles with two different polyominoes:

Tetrominoes and Tetrominoes

Tetrominoes and Pentominoes

Tetrominoes and Hexominoes

Pentominoes and Pentominoes

Pentominoes and Hexominoes

Hexominoes and Hexominoes

George Sicherman also investigated tiling a rectangle with two different polyaboloes:

Small Polyaboloes and Small Polyaboloes

none none none
none

Small Polyaboloes and Tetraboloes

?

? ? none none

?

? none

none

? ? none

none

? ? none none

? none none

? none none none
? ? none none

? ? none none

? ? none none

Tetraboloes and Tetraboloes

?

none??none
????

?
none???

??

????
????
?
???
??
?

???
??

??

?
nonenone??

?
?

2.

Here are the best-known impossible values for n for reptile polyominoes of area 6 or less:

 2, 3, 5 4 6 8

 2, 3 4 5 6

 2, 3, 5, 8 4 6

 2, 3, 5, 6 4 8(AS) 9

 2-15, 17, 18, 21 16 19(JD) 20(JD) 22(JD) 24(JD) 25(JD) 29(MM)

 2, 3, 5, 8 4 6 11

 2-9, 11, 12 10 13(JD) 14(MM) 15(AS) 17(MM) 18

 2-11 12(JD) 13 14 15(AS) 16(MM) 17(AS)

 2-8, 10, 12 9 11(AS) 13(GS) 14(AS) 16(AS) 18

 2-17, 20-21, 25 18 19(GS) 22 23

 2-7, 9-13, 16, 17 8 19(JD) 23(MM)

 2-29, 33, 35, 39, 42

 30(GS) 31(AS) 32 34(MM) 36 37(MM) 38(MM) 40 41(MM) 45(JD) 47(JD)

 2-21, 23-25, 27, 29, 31, 35 22 26(JD) 30(MM) 33(JD)

 2-39, 41, 42, 45, 47, 48, 50, 51, 53, 54, 57

 40 43(JD) 44(JD) 46(JD) 49(JD) 56(MM) 59(MM) 60(JD) 62(JD) 63(JD) 71(MM) 81(MR)

2-62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82-124, 126, 127, 129, 130, 132, 133, 135,
136, 138, 139, 142, 145, 148, 151, 154, 160, 173, 175, 179, 185, 187, 191, 193, 197
 141(JD) 144(MM) 146(AS)

 147 162(AS) 163(AS)

 164(AS) 165(AS) 167(AS)

 63(LZ) 66(MR) 69(MR) 72(MR)

 75(MR) 78(MR) 81(MR) 169(MR)

 ? 888(JD)

2-16, 18, 21
 17(MR) 19(MR) 20(MR) 22(MR) 23(MR)

 24(MR) 25(MR) 26(MR) 27(MR) 28(MR)

 29(MR) 30(MR) 31(MR) 32(MR) 34(MR)

2-16, 18, 19, 21, 24
 17(MR) 20(MR) 22(MR) 23(MR) 25(MR)

 26(MR) 27(MR) 28(MR) 29(MR) 30(MR)

 31(MR) 32(MR) 34(MR) 35(MR) 37(MR) 40(MR)

3.

Here are the known rectifiable polykings (most by George Sicherman):

4-Kings

5-Kings

4.

Here are the smallest known orders for non-polyomino reptile polykings:

reptilingorder
4

(BH)
10
18
reptilingorder
4
10
10

(MM) (AS)
10

(MM) (AS)
10

(MM) (AS)
16

(MM) (AS)
16

(MM) (AS)
16

(AS)
216

(AS)
301
reptilingorder

(MM)
13

(MM)
18

(MM)
20

(MM)
26

(MM)
26

(AS)
28

(AS)
28

(AS)
28

(AS)
34

(AS)
34

(AS)
40

(MM)
88

(MM)
104

(MM)
136

(MM)
156

(MM)
400

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 2/15/13.