5
$\begingroup$

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 #4

Next puzzle in this series Tiling rectangles with Heptomino plus rectangle #7

The goal is to tile rectangles as small as possible with the given heptomino, in this case number 6 of the 108 heptominoes. 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 14 more. I considered component rectangles of width 1 through 11 and length to 31 but my search was not complete.

List of known sizes:

  • Width 1: Lengths 1 to 8, 10 to 12
  • Width 2: Lengths 2, 3, 5
  • Width 3: Length 5

Most of these could be tiled by hand using logic rather than just trial and error.

$\endgroup$

2 Answers 2

7
$\begingroup$

Finally, a more interesting heptomino :) (in the sense that previous ones all had generalizable solutions who looked very much like this hexomino)

Here's the minimal solution for $1 \times 2$:

$3 \times 6 = 18$
enter image description here

and for $2 \times 2$:

$6 \times 13 = 78$
enter image description here

For $3 \times 5$:

$19 \times 22 = 418$
enter image description here

My program found another one for $2 \times 7$:

$21 \times 30 = 630$
enter image description here

a very narrow one for $1 \times 10$:

$6 \times 31 = 186$
enter image description here

another one for $1 \times 11$:

$12 \times 32 = 384$
enter image description here

and another one for $1 \times 12$:

$12 \times 26 = 312$
enter image description here

This is probably the $1 \times 8$ solution you're looking for:

$17 \times 22 = 374$
enter image description here

I like how this one and Jaap's attempt are fundamentally different; this one is 'chaos' and the other one 'order'. It's asymmetric but it can be turned in a symmetric one; there are two ways to tile the irregular shape formed (twice) by the darker shaded polyominos. If you use the same one for both, you get a symmetric solution.

Here is the minimal solution for $1 \times 9$:

$19 \times 22 = 418$
enter image description here

$\endgroup$
5
  • $\begingroup$ Yup 1x1=3x6, 2x2=6x13 both minimal. From 1x1 to 1x7 all would make nice hand tiling puzzles. $\endgroup$ Jun 19, 2018 at 7:07
  • $\begingroup$ 3x5=19x22 is optimal yes. $\endgroup$ Jun 20, 2018 at 7:58
  • $\begingroup$ got an extra dollar sign in the last one $\endgroup$ Jun 20, 2018 at 7:58
  • $\begingroup$ I love the symmetry of all of these - including the asyymetric 2x7! $\endgroup$
    – Chris
    Jun 21, 2018 at 16:58
  • $\begingroup$ Your net area just shot up to 2400, time to award I think $\endgroup$ Jun 22, 2018 at 10:20
5
$\begingroup$

Here are a few more solutions.

$1\times3$

$7\times7$
enter image description here

$1\times4$

$7\times9$
enter image description here

$1\times5$

$6\times9$
enter image description here

$1\times6$

$6\times13$
enter image description here

$1\times7$

$8\times14$
enter image description here

$2\times3$

$13\times14$
enter image description here

$2\times5$

$11\times14$
enter image description here

Edit:

Here is a $1\times8$ solution that is surely non-optimal.

$26\times42$
enter image description here

$\endgroup$
3
  • $\begingroup$ Yup 1x3,4,5,6,7; 2x3,5 all optimal $\endgroup$ Jun 20, 2018 at 3:50
  • $\begingroup$ @theonetruepath I added a 1x8 solution that is surely not optimal. $\endgroup$ Jun 20, 2018 at 6:17
  • $\begingroup$ The 1x8 is a thing of beauty... but not optimal, as you say. My optimal solution is asymmetric, I haven't searched all solutions to see if there's a symmetric one. $\endgroup$ Jun 20, 2018 at 7:56

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.