# Tiling rectangles with a Heptomino plus 2x2 square

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:

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':

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.

• 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. – Darrel Hoffman Oct 17 '19 at 14:22
• 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. – theonetruepath Oct 18 '19 at 1:14

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

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

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

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

and the final three:

• You're over half way... all minimal I think, will check properly later. – theonetruepath Jun 27 '18 at 18:11
• 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'? – theonetruepath Jul 2 '18 at 2:57
• I can't see any missing ones... just a few (7?) non-minimal. – theonetruepath Jul 2 '18 at 3:13

All $14\times16$

All $14\times14$

All $8\times14$

Unlike the other solution sets, these three solutions are not related.

$10\times12$

$8\times10$

$5\times10$

All $16\times22$

• Big effort. 14x16: A,C minimal, B,D not. 14x14: E,F minimal, G not. 8x14: H,I,J not minimal – theonetruepath Jun 28 '18 at 2:49
• @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. – Jaap Scherphuis Jun 30 '18 at 5:24
• I found a better one for H (9x10 IIRC), will post it tomorrow. – Glorfindel Jun 30 '18 at 12:22
• 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. – theonetruepath Jul 2 '18 at 1:26