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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 V pentomino plus rectangles

Tiling rectangles with W pentomino plus rectangles

The goal is to tile rectangles as small as possible with the X 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 X-pentomino will tile. Example shown, with the $1\times 1$, you can tile a $3\times 3$ as follows:

X plus 1x1

Now we don't need to consider $1\times 1$ any longer as we have found the smallest rectangle tilable with copies of X plus copies of $1\times 1$.

I've only found two other solutions. I tagged it 'computer-puzzle' but some people can probably work both of these out by hand.

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    $\begingroup$ Why post all of these at once? Why not just one by one so everyone can focus on one before moving to the other? I thought it was spam at first.. $\endgroup$
    – votbear
    Commented Apr 17, 2018 at 2:39
  • $\begingroup$ It should be easy enough to differentiate from spam. Also I thought allowing people to look at "all together" might be a way of allowing more than one or two people to dominate the answers. I might be wrong of course. $\endgroup$
    – theonetruepath
    Commented Apr 17, 2018 at 2:43
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    $\begingroup$ I'd argue otherwise. These puzzles require a lot of effort to solve. Splitting everyone's attention between 6 similar puzzles is prooobably not a good idea. Why put in all that effort for something that most people will overlook? $\endgroup$
    – votbear
    Commented Apr 17, 2018 at 2:51
  • $\begingroup$ Still waiting for the same puzzle with I pentaminoes... $\endgroup$
    – xhienne
    Commented Apr 17, 2018 at 12:11
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    $\begingroup$ @xhienne The I pentomino would have infinitely many solutions. Just put it next to a 1 x n for any n and you have a solution. $\endgroup$
    – Riley
    Commented Apr 17, 2018 at 12:25

3 Answers 3

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I think I've found a smaller one for 1x3:

a 15x10 = 150 solution:

enter image description here

This one has a rather interesting generalization (see the third spoiler block there) for a different pentomino and rectangle size.

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  • $\begingroup$ As this puzzle is now solved (unless somebody comes up with an even smaller 1x2 or 1x3 solution, or one for another size (I'm not sure how exhaustive your tiling program is)), how do you want to continue? Are you going to accept the hardest solution (usually the last) or do you want to have a separate community wiki post with all solutions? $\endgroup$
    – Glorfindel
    Commented May 3, 2018 at 8:40
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    $\begingroup$ My program is exhaustive, and searches rectangles in order of increasing area. It's still running, but hasn't found anything new for 'X' for about a month now. It's entirely likely that there aren't any more, given the intransigent shape of 'X'. I will award the answer now on this one. My criterion is for 'highest area covered by first poster of minimal tilings' so it goes to you on this one. $\endgroup$
    – theonetruepath
    Commented May 3, 2018 at 20:41
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Well, here's one of them

We can tile a $5\times 6$ rectangle using the X pentomino and $1\times 2$ rectangles.

.

enter image description here

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  • $\begingroup$ Yup that's optimal. Plus 1 for the handwritten answer! $\endgroup$
    – theonetruepath
    Commented Apr 17, 2018 at 3:13
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    $\begingroup$ @theonetruepath What software did you use in your images? $\endgroup$
    – Riley
    Commented Apr 17, 2018 at 3:13
  • $\begingroup$ I tile with a program I wrote many years ago, it produces text files likethis: oXo XXX oXo (imagine those stacked up) then I have a program written by Andrew Clarke called Render that turns these into .BMP files which I finalise in LView Pro 2006 $\endgroup$
    – theonetruepath
    Commented Apr 17, 2018 at 3:29
  • $\begingroup$ Andrew Clarke's polyform site:recmath.com/PolyPages/index.htm $\endgroup$
    – theonetruepath
    Commented Apr 17, 2018 at 3:34
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I'm not sure if it is optimal but I found a solution for $1 \times 3$

We can tile a $13 \times 12$ rectangle using six X-pentominoes and $42$ $1 \times 3$ rectangles

Sorry about the faint diagram, I drew it by hand and scanned it

enter image description here

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  • $\begingroup$ Nice answer, but not optimal. A bit of lateral thinking will improve this answer to the smallest area rectangle. $\endgroup$
    – theonetruepath
    Commented Apr 29, 2018 at 19:18

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