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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 #6

Next puzzle in this series Tiling rectangles with a Heptomino plus 2x2 square

The goal is to tile rectangles as small as possible with the given heptomino, in this case number 7 of the 108 heptominoes (see example below). We allow the addition of copies of a rectangle. For each rectangle $a\times b$, find the smallest area larger rectangle that copies of $a\times b$ plus at least one of the given heptomino will tile.

Example with the $1\times 1$ you can tile a $2\times 5$ as follows:

1x1_2x5

Now we don't need to consider $1\times 1$ further as we have found the smallest rectangle tilable with copies of the heptomino plus copies of $1\times 1$.

I found only 7 more. I considered component rectangles of width 1 through 11 and length to 31 but my search may not be complete.

List of known sizes:

  • Width 1: Lengths 1 to 5
  • Width 2: Lengths 2, 3, 5

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.

theonetruepath
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    There seem to be an awful lot of these. Are they just going to continue indefinitely? (Looking at the history, the earliest ones attracted rather more upvotes and favourite-ings than more recent ones have.) There are 108 different heptominoes and I'm not sure we really want 108 "tiling rectangle with heptomino plus rectangles" puzzles... – Gareth McCaughan Jun 24 '18 at 23:53
  • @GarethMcCaughan I think there are an awful lot of riddles. But this isn't my personal puzzle emporium so I let it slide. The real question is, are they that unpopular and/or without merit that they should simply not be here? I'm open to persuasion. – theonetruepath Jun 25 '18 at 00:48
  • I'm not suggesting that they be deleted, or anything like that. I just don't look forward with particular joy to the next hundred puzzles that are exactly like this one but with a different choice of heptomino. And yes, there are too many bad riddles -- but they're all posted by different people, often newcomers to PSE, rather than one person posting a dozen riddles that all work the same way :-). – Gareth McCaughan Jun 25 '18 at 02:47
  • Well I have some sympathy for your feeling here: I don't look forward to the work involved in posting 100 more very similar puzzles. Maybe instead I will 'condense' them in some way to concentrate the more interesting ones in far fewer puzzles. – theonetruepath Jun 25 '18 at 03:47

2 Answers2

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I found a solution for the $2\times5$. It obviously also works for $1\times5$.

$16\times16$:
enter image description here

Here is a better $1\times5$ solution.

$9\times16$:
enter image description here

Jaap Scherphuis
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$1 \times 2$:

$3 \times 5 = 15$
enter image description here

$2 \times 2$:

$4 \times 10 = 20$
enter image description here

$1 \times 3$:

$7 \times 10 = 70$
enter image description here

and a variation on that theme which works for $2 \times 3$:

$13 \times 10 = 130$
enter image description here

This is about as hard as I can solve without using even pen and paper to think them out (obviously I'm using Excel to make the pictures). FWIW I'm working on an iPad app to help with tiling by hand, it's a good exercise to polish my Swift. the app works now (more or less), and helped me finding a solution for $1 \times 4$:

$9 \times 10 = 90$
enter image description here

Glorfindel
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