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 (vertexconnected 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 most solutions by Mike Reid.
Below are the smallestknown 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 sent hexhex and hepthept pictures. George has more pictures, which are colorcoded and have areas, while Patrick's tilings are all minimal. George's hexhex pictures are here and here, while Patrick's are here. George's hepthept pictures are here, here, and here, while Patrick's are here.
George Sicherman also sent 1 tethept picture, 1 2 penthept pictures, 1 2 3 4 5 6 7 8 pentoct pictures, and 1 2 3 4 5 hexhept 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  
 
 

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?  ?  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 bestknown 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 
215, 17, 18, 21  16  19 (JD)  20 (JD)  22 (JD)  24 (JD)  25 (JD)  29 (MM) 
2, 3, 5, 8  4  6  11 
29, 11, 12  10  13 (JD)  14 (MM)  15 (AS)  17 (MM)  18 
211  12 (JD)  13  14  15 (AS)  16 (MM)  17 (AS) 
28, 10, 12  9  11 (AS)  13 (GS)  14 (AS)  16 (AS)  18 
217, 2021, 25  18  19 (GS)  22  23 
27, 913, 16, 17  8  19 (JD)  23 (MM) 
229, 33, 35, 39 
30 (GS)  31 (AS)  32  34 (MM)  36  37 (MM)  38 (MM)  40  41 (MM)  41  45 (JD)  47 (JD) 
221, 2325, 27, 29, 31, 35  22  26 (JD)  30 (MM)  33 (JD) 
239, 41, 42, 45, 47, 48, 50, 51, 53, 54, 57 
40  43 (JD)  44 (JD)  46 (JD)  49 (JD)  52 (AB)  56 (MM)  58 (AB)  59 (MM)  60 (JD) 
61 (AB)  62 (JD)  63 (JD)  64 (AB)  67 (AB)  68 (AB)  69 (AB)  70 (AB)  71 (MM)  81 (MR) 
Andrew Bayly proved that this next polyomino can be tiled with 121 or more smaller copies!
63 (LZ)  66 (MR)  69 (MR)  72 (MR) 
75 (MR)  78 (MR)  81 (MR) 
?  888 (JD) 
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) 
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 nonpolyomino reptile polykings:
order 4  order 10 (BH)  order 18 
order 4  order 10  order 10  order 10 (MM) (AS)  order 104 (GS) 
order 10 (MM) (AS)  order 16 (MM) (AS)  order 16 (MM) (AS)  order 16 (MM) (AS)  
order 139 (AB)  order 432 (GS) 
order 13 (MM)  order 17 (GS)  order 18 (MM)  order 26 (MM)  order 26 (MM) 
order 28 (AS)  order 28 (AS)  order 28 (AS)  order 34 (AS)  order 34 (AS) 
order 40 (AS)  order 104 (MM)  order 136 (MM)  order 53 (GS)  
order 76 (GS)  
order 124 (GS)  order 280 (GS)  
order 104 (AB)  order 1156 (GS)  
order 280 (GS) 
George Sicherman also investigated polymings:
order 4  order 4  order 10  order 4  order 9  order 9  order 9  order 20  order 38 (BH) 
order 4  order 4  order 15  order 18  order 26 (BH) 
order 44  order 10  order 44 (GS) 
order 57 (GS) 
If you can extend any of these results, please email me. Click here to go back to Math Magic. Last updated 9/16/20.