7
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See my similar question about T hexomino (Polyomino T hexomino and rectangle packing into rectangle)

This is exactly same but with other polyomino - Z pentomino.

Let's pack some (one or more) Z pentominoes 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 one Z pentomino and four $1\times1$ rectangles into a $3\times3$ 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 Z pentominoes 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.

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:
$1\times1$, $1\times2$, $1\times3$, $1\times4$, $1\times5$, $1\times6$, $1\times8$,
$2\times2$, $2\times3$, $2\times4$, $2\times5$

1 week after posting, I'll put up my own answers for unfound combinations (from list above).

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Added 1x8.
Added 1x5 and 1x6. Replaced 1x3.

1 x 1 (Area = 9), 1 x 2 (Area = 9), 1 x 3 (Area = 42), 1 x 4 (Area = 56)

Pentamino1

1 x 5 (Area = 165), 1 x 6 (Area = 156)

Pentamino3

1 x 8 (Area = 432)

Pentamino4

2 x 2 (Area = 36), 2 x 3 (Area = 104)

Pentamino2

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A few additions... firstly for the 1x5 a different answer, smaller area but more rectangles than Len's so just for interest

1 x 5 (Area = 160)

enter image description here

3 x 4 (Area = 400)

enter image description here

This one's the same area as Len's but fewer rectangles

1 x 8 (Area = 432)

enter image description here

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1x4, area 36

.___________.
| | |_______|
| |___ | _|
| | _|_| | |
|_| | |___| |
|___|___ | |
|_______|_|_|

2x4, area 120

_________________________
| | | | |
| |_______|_______| |
| | _| |_ | |
|___| | |_______| | |___|
| |___|___ | ___|___| |
|___ | _|_|_|_ | ___|
| |_| | | |_| |
| |___|_______|___| |
| | | | |
|___|_______|_______|___|

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3
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As I have promised, posting my own answers for unfound combinations (only one from listed was not found).

2x5, area 680

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2
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I found smaller ones for 2x5:

20x19 = 380, 28 rectangles

enter image description here

and for 1x8:

26x16 = 416, 22 rectangles

enter image description here

And here is one for a new rectangle size, 2x6:

30x20 = 600, 40 rectangles
enter image description here

Wait. Haven't I seen that one before?

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  • $\begingroup$ Already known of 2x6 solution, got it the same way that you, from triomino and pentomino solution, was too lazy to draw an image an post it. $\endgroup$ – Somnium May 7 '18 at 22:01

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