We call a polyomino P distinguishable if there are integers n, k, a, b, so that P is the only nomino where a maximum of k copies of P can be packed in an a×b box. If P is dintinguishable, we call the smallest such k the distinguishing mark of P.
Can you show that not all polyominoes are distinguishable? What is the smallest one? What are the distinguishing marks of some larger polyominoes? What about polyaboloes? What about polykings?
We call a polyomino P superdistinguishable if there are integers k, a, b, so that P is the only polyomino where a maximum of k copies of P can be packed in an a×b box. Which polyominoes are superdistinguishable?
We call a polyomino P 2distinguishable if there are integers n, k_{1}, a_{1}, b_{1}, k_{2}, a_{2}, b_{2}, so that P is the only nomino where a maximum of k_{1} copies of P can be packed in an a_{1}×b_{1} box, and a maximum of k_{2} copies of P can be packed in an a_{2}×b_{2} box. Which polyominoes are 2distinguishable?
2.
For a given positive integer n, what is the smallest polyomino jigsaw puzzle in a square tray that has exactly n solutions, including rotations and reflections? (Here "smallest" means smallest tray, and of those, smallest number of pieces.) What are the solutions for rectangular trays?
Solutions were received from George Sicherman, Berend van der Zwaag, Andrew Bayly, David Wilson, Bryce Herdt, and Gavin Theobald.
Berend van der Zwaag and George Sicherman showed that not all polyominoes are distinguishable.
Here are the known distinguishing marks of distinguishable polyominoes:





Here are the the known superdistinguishable polyominoes:



Here are some distinguishable polyaboloes:



Here are some distinguishable polykings:



If there is a k×k npuzzle, then putting 2piece rings around it can create a (k+2)×(k+2) 2npuzzle, 4npuzzle, or 8npuzzle.
If there is a solution for n, David Wilson pointed out there is a solution for 3n by putting a copy of the npuzzle inside a modified 3puzzle due to George Sicherman.
There is a n×n 2npuzzle comprised of (n1) n×1 rectangles and n 1×1 squares.
There is a (4n+1)puzzle comprised of a 2×2 square inside a totally symmetric field of 1×1 squares.
Here are the bestknown solutions for a square tray:
1  2  3 (DW)  4  5  6  7 (GS)  8  9  10 (GS) 
11 (GS)  12  13  14 (GS)  15 (DW)  16  17  18  19 (DW)  20 
21 (GS)  22  23 (GS)  24 (GS)  25  26  27  28 (GS)  29 (GS)  30 (GS) 
31 (DW)  32  33  34 (GS)  35 (DW)  36  37  38 (GS)  39 (GS)  40 
41 (GT)  42 (GS)  43 (GS)  44  45 (BH)  46  47 (GS)  48  49  50 (GS) 
51 (GS)  52  53  54 (GS)  55  56  57  58 (GS)  ? 59  60 
61  62 (GS)  63 (GT)  64 (GS)  65 (GS)  66 (GS)  67 (GS)  68  69  70 
23×23 71 (DW)  72 (GS)  73 (GS)  74 (GS)  75  76 (GS)  77 (GS)  78 (GS)  79  80 
81 (GS)  82 (GS)  83 (GS)  84 (GS)  85 (DW)  86 (GS)  26×26 87 (DW)  88 (GS)  89 (GS)  90 
91 (GT)  92 (GS)  93 (GS)  94 (GS)  95  96 (GS)  97  98 (GS)  99 (DW)  100 (GS) 
And here are the bestknown solutions for a rectangular tray:
1  2  3  4  5  6  7  8  9 (GS)  10 
11  12  13  14 (GS)  15  16  17 (GS)  18  19 (GS)  20 (GS) 
21  22  23 (GS)  24 (GS)  25 (GS)  26  27 (GS)  28 (GS)  29  30 
31 (GS)  32 (GS)  33 (GS)  34 (GS)  35  36  37  38 (GS)  39 (GS)  40 
41 (GS)  42  43 (GS)  44  45  46 (GS)  47 (GS)  48  49 (BH)  50 (GS) 
51 (GS)  52  53 (GS)  54 (GS)  55  56  57 (GS)  58 (GS)  59 (GS)  60 
61 (GS)  62 (GS)  63 (GS)  64 (GS)  65 (GS)  66 (GS)  67 (GS)  68 (GS)  69 (GS)  70 
71 (GS)  72 (GS)  73 (GS)  74 (GS)  75 (GS)  76 (GS)  77 (GS)  78 (GS)  79 (GS)  80 
81 (GS)  82 (GS)  83 (GS)  84 (GS)  85 (GS)  86 (GS)  87 (GS)  88 (GS)  89  90 (GS) 
91  92 (GS)  93 (GS)  94  95 (GS)  96 (GS)  97 (GS)  98 (GS)  99 (GS)  100 (GS) 
If you can extend any of these results, please email me. Click here to go back to Math Magic. Last updated 8/20/11.