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Reason at the bottom

Short version: Is it possible, to tile a rectangle with rectangles in a way to keep a non-moveable hole?

Long version: Considering two (fat) rectangles $R_{1}$ ($A\,\times\,B$) and $R_2$ ($C\,\times\,D$), which may or may not be of similar shape;

Is it possible, to tile the larger $R_{1}$, with an integer number $n$ tiles of shape $R_{2}$, in such way, as to leave a hole, which either

  • A) can not be moved through the tiling at all, or

  • B) only allowing movement of a single stone a time, and only lengthwhise

?

Additional Data:

  • The Hole Your pattern may have a single or multiple holes, but should in general be massive (mostly hole-free). (Simply stacking the excluded solution 1. is not intended.) If you use multiple holes they may not be combined to break the rules. Size and shape of the hole are free to choose, but between half the width and two times the length of $R_2$, and square, rectangular or $L$-shaped is probably a good starting point.

  • Rectangles Both rectangles are fat, meaning their length is at most two times their width (If you find solutions for higher ratios, still feel free to post them) Their sides are not given. They, as well as the (or a) pattern are searched for parts of a solution.

  • the main goal is to find a pattern with $n$ being at most $200$ less than $50$ would be nice. (If you find solutions for higher numbers, still feel free to post them.)

  • Excluded Solutions: The following simplex cases are closed. However feel free to use them in a larger solution of your own:

    1. $R_1$ is square and sized $C+k*D$ (thanks to @quarague) (The solution here is a ring with the hole being $D-C$)

    2. The hole can be moved or combined to stretch over a complete side of $R_1$

EDITs:

  1. The question is not if there is any solution, or if there allways is a solution, but to find a non-trivial solution.

  2. The rectangles are ment to be non-square. (Thanks to @Andrei for pointing it out.)

Reason: (Added as sort of explanation) Consider filling a box with packages. You want to fill the box to the brim. However you want to be able to retrieve each package, one after another. During the ransport, it should not rattle: There is to be no space left. A tight packing, spare for one hole, where to start pulling the packages out. If possible, there should be no way of moving the hole, as to not allow rattling. However, as there might be fairly few solutions for this, there are two loopholes provided:

  1. there might be more than one hole, provided it is still a rather tight packing

  2. a single package may slide, in one direction to fill the hole and open it somewhere else. Just one though, and only along its length, as otherwhise the packing could start to loosen and rattle again.

    I know of tiling rectangles with $2\times 1$-rectangles with a single, or fairly few holes, the question is can one do this with fat rectangles too, and could one achieve non-moveable holes with it.

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  • $\begingroup$ I don't understand what you are asking at all. What does moving the hole accomplish? Also, why isn't any number of small fat rectangles inside a large fat rectangle an answer? Whatever isn't covered by the small rectangle is a hole. $\endgroup$ – saulspatz Aug 21 '19 at 15:01
  • $\begingroup$ @saulspatz The question started as simple as this: consider a box you want to carry some packages of cigarets in. I want a to stack them in a way, that the box will not rattle, but that I can easily retrieve a package. Then I started getting lost within thoughts about geometric constraints. $\endgroup$ – Teck-freak Aug 21 '19 at 15:09
  • $\begingroup$ There is not enough information. It might or it might not, depending on the values. For example, if $c=d=1$, and the bigger rectangle has integer dimensions, and the hole has integer dimensions, and the hole is offset by integer steps in both directions with respect to a corner (assuming axis are aligned), then you have a trivial solution. You see, there are so many assumptions. And I can get a case where you cannot find a solution. $\endgroup$ – Andrei Aug 21 '19 at 15:12
  • $\begingroup$ I think it would be better to pose the question in terms of the cigarette packs, with what constraints on the shapes of the packs and container you have in mind. $\endgroup$ – saulspatz Aug 21 '19 at 15:14
  • $\begingroup$ You can expand your excluded solution 1. by cutting each of the smaller rectangles in half. For example 8 rectangles of size $3 \times 4$ fit into a $10 \times 10$ square with center $2\times 2$ as a hole. I don't think you can arrange the 8 rectangles in any other way. $\endgroup$ – quarague Aug 21 '19 at 15:17
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Interesting problem (especially if you can formulate it so that you don't have to exlcude so many cases).

Here is another scheme (that I am sure you can see can be extended to allow for various numbers of holes):

enter image description here

The tiles can be halved, and quartered, to give more schemes.

Here is another:

enter image description here

Again, you can find more schemes by bisecting or quartering the tiles.

All these solutions employ the ring around the holes, but the rings overlapped rather than stack.

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  • 1
    $\begingroup$ Dear Herman, thanks for that wonderful solution. I'll mark it as solved, but if you happen to find more patterns, I'd be happy to see them. Best regards. $\endgroup$ – Teck-freak Dec 4 '19 at 18:04

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