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Let's pack some (one or more) T hexominoes together with some (one or more) small $a\times b$ rectangles into some bigger $m\times n$ rectangle without holes and overlapping pieces.

For example, I can pack two T hexominoes and eight $1\times1$ rectangles into a $5\times4$ rectangle:

Task

Find as many as you can different integer pairs $\{a,b\}$ so that one or more rectangles of size $a\times b$ can be packed together with T hexominoes into a $m\times n$ rectangle without holes and overlapping pieces.

Provide images of solutions if you claim that some combination is possible. Please, put your images in spoiler tags so that other users can try find themselves.

If some answers will have same number of pairs $\{a,b\}$, then answer with smaller total area of outer rectangles on example images is preferable. If area also equal, then earlier posted answer is preferable.

EDIT: Please, add to your answers total area of outer rectangles. If one tiling implicitly includes several $a\times b$ rectangles, then multiply area by number of combinations it includes.

Notes

I can guarantee that there exist solutions following integer pairs:
$1\times1$, $1\times2$, $1\times3$, $1\times4$, $1\times5$, $1\times6$, $1\times8$, $1\times9$,
$2\times2$, $2\times3$, $2\times4$, $2\times5$, $2\times6$

I'm wondering why a $1\times7$ rectangle solution is hard to find (maybe it's impossible?) when there exist solutions for with $1\times8$ and $1\times9$ rectangles.

1 week after posting, I'll put up my own answers for unfound combinations.

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    $\begingroup$ Please clarify your question; I can't understand what you're asking. Do you want to pack $m$ T-hexominoes and $n$ $a\times b$ rectangles into a $c\times d$ rectangle? What are the combinations you're looking for of $m$, $n$, $a$, $b$, $c$, $d$? Are some of these fixed? $\endgroup$
    – Rand al'Thor
    Commented Nov 19, 2014 at 21:56
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    $\begingroup$ Thanks for editing! But when you say 'a 1x7 rectangle solution', does this mean $a$ and $b$ are 1 and 7? What are the other parameters? Or are you just trying to pack some T-hexominoes and some 1x7 rectangles into a big rectangle of some size? $\endgroup$
    – Rand al'Thor
    Commented Nov 19, 2014 at 22:05
  • $\begingroup$ @randal'thor Yes, I'm "just trying to pack some T-hexominoes and some 1x7 rectangles into a big rectangle of some size". 1x7 is one that I'm suspicious about. $\endgroup$
    – Somnium
    Commented Nov 19, 2014 at 22:09
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    $\begingroup$ Under notes, you say $1 \times 7$ is impossible, then again that it is possible. Is the second supposed to be $1 \times 8$? In the list you have found, is the first $1 \times 2$ supposed to be $1 \times 1$? $\endgroup$
    – Ross Millikan
    Commented Nov 19, 2014 at 22:41
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    $\begingroup$ The more solutions I see, the more I like this question. A pity I can not up-vote it more than I have already! I'm eager to the see the link to your homepage once it is up! $\endgroup$
    – BmyGuest
    Commented Dec 14, 2014 at 17:48

5 Answers 5

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2x2 - area 108 - optimal

2x2

2x3 - area 72 - optimal

2x3 2x3 2x3

2x4 - area 108 - optimal

2x4

3x4 - area 84 - optimal

3x4

1x5 - area 54 - optimal

1x5

2x5 - area 304 - optimal

2x5

3x5 - area 576 - optimal

3x5

2x6 - area 240 - optimal

2x6

1x7 - area 1034

1x7

1x8 - area 432 - optimal

1x8

3x8 - area 5880

3x8

1x9 - area 585 - optimal

1x9

Note that by subdividing the yellow rectangles:
2x3 indirectly solves 1x1, 1x2, 1x3
2x4 indirectly solves 1x4 and 2x2
2x5 indirectly solves 1x5
2x6 indirectly solves 1x6

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  • $\begingroup$ Yes, it was a nice surprise. $\endgroup$
    – Florian F
    Commented Nov 22, 2014 at 22:10
  • $\begingroup$ Your 2x2 solution can be extended also for 2x4 - it'll be smaller then. $\endgroup$
    – Somnium
    Commented Nov 24, 2014 at 10:28
  • $\begingroup$ I know. I already did the picture and edited the answer. I must have forgotten to save the edits. I'll do that tonight. $\endgroup$
    – Florian F
    Commented Nov 24, 2014 at 11:12
  • $\begingroup$ No more rectangles found? I have some unfinished in my answer. Maybe they will help you. Seems like you have skill in finding tilings. $\endgroup$
    – Somnium
    Commented Nov 26, 2014 at 18:38
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    $\begingroup$ No more. I filled a number of pages with T patterns but managed only to improve on existing answers. $\endgroup$
    – Florian F
    Commented Nov 26, 2014 at 22:50
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Let's start with the easy ones.

1x1

enter image description here

1x2

enter image description here

1x3

enter image description here

1x4

enter image description here

1x6

enter image description here

These ones took me a while.

1x5

enter image description here

2x3

enter image description here

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  • $\begingroup$ Just curious, what interface/software are you guys using to make the graphs? $\endgroup$
    – Leo
    Commented Nov 21, 2014 at 19:08
  • $\begingroup$ @Leo, I wrote a small image rendering application using Python and the Pillow library. I don't know what everyone else is using. $\endgroup$
    – Kevin
    Commented Nov 21, 2014 at 20:12
  • $\begingroup$ @Leo I wrote VB.NET application for creating polyomino tilings. In my opinion, fastest method without any special programs is to use PowerPoint - create figures from squares, group them and then it's easy to copy/move/rotate them. $\endgroup$
    – Somnium
    Commented Nov 21, 2014 at 22:20
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    $\begingroup$ Your 1x5 could be reduced to a 9 by 6 rectangle. $\endgroup$
    – Florian F
    Commented Nov 22, 2014 at 14:35
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Rev 4 - added 1 x 9 incorrect but still trying, added 1 x 5 using Florian's clue
Rev 3 - added 1 x 8 solution
Rev 2 - added 2 x 6 solution
Rev 1 - added 2 x 2 solution
Thanks for the clarification, Somnium. I will put some more answers here.

1 x 5 solution using Florian's clue (area = 54)

1x5 solution

1 x 8 solution (area = 560)

1x8 solution

1 x 9 incorrect still trying

1x9 incorrect

2 x 2 solution (area = 144)

2x2 solution

2 x 3 alternate (area = 72)

2x3 alternate

2 x 6 solution (area = 240)

2x6 solution

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  • $\begingroup$ BTW, your 2x3 is smallest possible. $\endgroup$
    – Somnium
    Commented Nov 20, 2014 at 22:56
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2x6 - area 240 Another solution

3x5 - area 1014

Following maybe will help someone.

2x9 unsuccessful try

1x7 unsuccessful try

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  • $\begingroup$ Interesting sparse solution for 3x5. $\endgroup$
    – Florian F
    Commented Dec 14, 2014 at 20:39
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I found one for a new size, $4 \times 5$:

36x39 = 1404
enter image description here

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  • $\begingroup$ That was unexpected! It's found by program or by hand? $\endgroup$
    – Somnium
    Commented Jun 9, 2018 at 8:54
  • $\begingroup$ By program. I'm getting better at hand-solving, but this is out of my league. $\endgroup$
    – Glorfindel
    Commented Jun 9, 2018 at 9:12
  • $\begingroup$ I usually try at first hand-solving, to see if there's some pattern, if there's something that can prevent solution existence, then do computer search. Do you have solutions of other size (may be bigger)? If I want to post your solution on my webpage (not so soon), do you want attribution? $\endgroup$
    – Somnium
    Commented Jun 9, 2018 at 9:48
  • $\begingroup$ It's not so much that I want attribution; because this content is posted on Stack Exchange, under the CC BY-SA 3.0 license you have to provide attribution; see the Terms of Service. (The flip side of this is that you don't even need to ask me for permission to publish the content elsewhere, though I appreciate it!) I don't have any more solutions at the moment, by the way. $\endgroup$
    – Glorfindel
    Commented Jun 9, 2018 at 15:12
  • $\begingroup$ @Somnium and as for something preventing solution existence, my program is far better than me at that as well. It takes half a second to realize that there's no solution for the F-pentomino + the 2x2 rectangle (well, at least not one with dimensions below 63x128). $\endgroup$
    – Glorfindel
    Commented Jun 9, 2018 at 15:28

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