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?

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:

	none

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

			?	(GS)		(GS)
	none		none	none			none	none
	none		none	none
	none		none		none	(Mike Reid)
	none			none	(Mike Reid)
			none	(Mike Reid)
	none		none
	none

					(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)

	(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:

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

		none		none	none
				none

		?
		?		?	none	none
		?		?		none
					none
		?		?		none
					none
		?		?	none	none
		?			none	none
		?			none	none	none
		?		?	none	none
		?		?	none	none
		?		?	none	none

				?
	none	?	?	none
	?	?	?	?
	?
	none	?	?	?
	?			?
	?	?	?	?
	?	?	?	?
	?
	?	?	?
	?	?
	?

	?	?			?
	?	?			?	?
					?
	none	none	?	?
	?
	?

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)

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

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

reptiling order 4 (BH) 10 18

reptiling order 4 10 10 (MM) (AS) 10 (MM) (AS) 10 (MM) (AS) 16 (MM) (AS) 16 (MM) (AS) 16 (AS) 216 (AS) 301

reptiling order (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.