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Inspired by Polyomino T hexomino and rectangle packing into rectangle

See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1

Previous puzzle in this series Tiling rectangles with Heptomino plus rectangle #7

The goal is to tile rectangles as small as possible with each of the given heptominoes (see diagram in example below) plus 2x2 squares.

Example Using the snake heptomino 'e' plus 2x2 squares you can tile a $3\times 5$ as follows: 2_86_3x5

Use each of these heptominoes. Solutions may exist for other heptominoes, I only found solutions for these ones. Note that I omitted labels 'W', 'j', 'l': Heptomino2x2KnownTilersLabelled2

All of these could all be tiled by hand, of course the bigger ones will be challenging. I'm making this one a 'hand tiling only' puzzle. In other words, use a computer to do anything except look up or compute the arrangements.

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  • $\begingroup$ There are trivial solutions for 'j' and 'l' - put two 'j's together to make a 2x7 rectangle. If a 2x2 is required, stick it on the end to make a 2x9. Likewise, two 'l's make a 2x7 and you can add a 2x2 on the end. Or if the 2x2 is not required, one 'l' by itself makes a rectangle already. 'W' may be impossible. $\endgroup$
    – Darrel Hoffman
    Oct 17, 2019 at 14:22
  • $\begingroup$ j and I were omitted because they trivially make rectangles on their own. W is probably impossible, I haven't thought of an elegant way to stop my program bogging down making width-7 right-angles when tiling the quarter plane. $\endgroup$
    – theonetruepath
    Oct 18, 2019 at 1:14

2 Answers 2

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Interesting twist; indeed, the $2 \times 2$ are often hard to tile. Here are some of the solutions:

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Here are two improvements on Jaap's tiles B and D:

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and G (can't believe I've missed that one)

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Here's a better ($9 \times 10$) version of H:

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and the final three:

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  • $\begingroup$ You're over half way... all minimal I think, will check properly later. $\endgroup$
    – theonetruepath
    Jun 27, 2018 at 18:11
  • $\begingroup$ OK all have labels now. Your first three are T,a=8x8 minimal, Q not. Then Y,U, neither is minimal. Z=3x6 b,n=3x5 all minimal. o no, m=4x12 yes. S=8x8 yes. P no. d=8x8 yes.f,c,V,P (again)=5x6 yes. k no, h=4x10 yes. X=8x10 yes. B=6x13 yes. D=12x16 yes. G=8x8 yes. H=9x10 yes. i nice but not minimal. R=10x12 yes. g=12x14 yes. Is that all, given you repeated 'P'? $\endgroup$
    – theonetruepath
    Jul 2, 2018 at 2:57
  • $\begingroup$ I can't see any missing ones... just a few (7?) non-minimal. $\endgroup$
    – theonetruepath
    Jul 2, 2018 at 3:13
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All $14\times16$
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All $14\times14$
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All $8\times14$
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Unlike the other solution sets, these three solutions are not related.

$10\times12$
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$8\times10$
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$5\times10$
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All $16\times22$
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  • $\begingroup$ Big effort. 14x16: A,C minimal, B,D not. 14x14: E,F minimal, G not. 8x14: H,I,J not minimal $\endgroup$
    – theonetruepath
    Jun 28, 2018 at 2:49
  • $\begingroup$ @theonetruepath Glorfindel has found improvements for B,D,G, so I've been trying to find better solutions for H,I,J, but not getting very far. They all have $7\times16$ solutions, but that is the same area as the $8\times14$ I already found. $\endgroup$
    – Jaap Scherphuis
    Jun 30, 2018 at 5:24
  • $\begingroup$ I found a better one for H (9x10 IIRC), will post it tomorrow. $\endgroup$
    – Glorfindel
    Jun 30, 2018 at 12:22
  • $\begingroup$ I added labels which should agree with yours so far. Also I checked H,I,J. There is a smaller one for H but I and J are indeed the same as yours in area (7x16), apologies for that, hope you didn't waste too much time. $\endgroup$
    – theonetruepath
    Jul 2, 2018 at 1:26

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