6
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ The 1x5 into 5x6 is minimal, your 1x6 into 6x10 is nice, but there's a smaller one $\endgroup$ – theonetruepath Apr 21 '18 at 0:19
  • $\begingroup$ @theonetruepath ah, of course. I think I found it. $\endgroup$ – Glorfindel Apr 21 '18 at 10:37
  • $\begingroup$ Yup that's minimal $\endgroup$ – theonetruepath Apr 21 '18 at 11:10
  • $\begingroup$ 1x5, 1x6, 1x7, 1x8, 1x9 all minimal. At least 11 more to find. $\endgroup$ – theonetruepath May 9 '18 at 10:00
  • $\begingroup$ I think I got (almost) all of them now. $\endgroup$ – Glorfindel May 21 '18 at 18:17
3
$\begingroup$

20's a lot, but here's a few to get it started.

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

.

enter image description here

.

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

.

enter image description here

.

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

.

enter image description here

.

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

.

enter image description here

$\endgroup$
  • $\begingroup$ Yup all four are optimal $\endgroup$ – theonetruepath Apr 17 '18 at 3:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.