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Inspired by Polyomino Z pentomino and rectangle packing into rectangle

Also in this series: Tiling rectangles with F pentomino plus rectangles

Tiling rectangles with N pentomino plus rectangles

Tiling rectangles with T pentomino plus rectangles

Tiling rectangles with U pentomino plus rectangles

Tiling rectangles with W pentomino plus rectangles

Tiling rectangles with X pentomino plus rectangles

The goal is to tile rectangles as small as possible with the V pentomino. Of course this is impossible, so 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 V-pentomino will tile. Examples shown, with the $1\times 1$, $1\times 2$ or $2\times 2$, you can tile a $3\times 3$ as follows:

V plus 1x1, 1x2, 1x3

Now we don't need to consider $1\times 1$, $1\times 2$, or $2\times 2$ any longer as we have found the smallest rectangle tilable with copies of V plus copies of each of those three.

There are at least 20 more solutions. I tagged it 'computer-puzzle' but you can certainly work some of these out by hand. The larger ones might be a bit challenging.

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2 Answers 2

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Here are (most of) the remaining ones. An easy one for 1x5:

enter image description here

and a more difficult one for 1x6:

enter image description here

1x7 takes a lot more:

24x11 = 264

enter image description here

1x8:

16x11 = 176

enter image description here

1x9:

22x12 = 264
enter image description here

1x10:

13x30 = 390

enter image description here

1x12:

14x42 = 588

enter image description here

2x3:

4x8 = 32

enter image description here

2x5:

5x6 = 30

enter image description here

2x6:

8x12 = 96

enter image description here

2x7:

16x19 = 304

enter image description here

2x8:

17x18 = 306

enter image description here

2x9:

21x24 = 504

enter image description here

3x4:

7x18 = 126
enter image description here

3x5:

11x35 = 385

enter image description here

3x7:

13x48 = 624

enter image description here

3x8:

33x38 = 1254

enter image description here

4x5:

21x40 = 840

enter image description here

4x6:

26x36 = 936
enter image description here

5x6:

38x60 = 2280

enter image description here

I assume the number of solutions here is infinite (probably in both directions), I'll post more when I have them.

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  • $\begingroup$ The 1x5 into 5x6 is minimal, your 1x6 into 6x10 is nice, but there's a smaller one $\endgroup$ Commented Apr 21, 2018 at 0:19
  • $\begingroup$ @theonetruepath ah, of course. I think I found it. $\endgroup$ Commented Apr 21, 2018 at 10:37
  • $\begingroup$ Yup that's minimal $\endgroup$ Commented Apr 21, 2018 at 11:10
  • $\begingroup$ 1x5, 1x6, 1x7, 1x8, 1x9 all minimal. At least 11 more to find. $\endgroup$ Commented May 9, 2018 at 10:00
  • $\begingroup$ I think I got (almost) all of them now. $\endgroup$ Commented May 21, 2018 at 18:17
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20's a lot, but here's a few to get it started.

$3\times 6$ tiled with a $2\times 4$

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enter image description here

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$3\times 6$ tiled with a $1\times 4$

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enter image description here

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$4\times 4$ tiled with a $1\times 3$

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enter image description here

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$6\times 6$ tiled with a $4\times 4$

.

enter image description here

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  • $\begingroup$ Yup all four are optimal $\endgroup$ Commented Apr 17, 2018 at 3:42

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