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Inspired by this question series, which was inspired by this question. They give rise to beautiful pictures (at least in the eye of the beholder mathematician) and some nice generalizable solutions.

Together, they make eight questions of the type 'take a pentomino and some rectangle-shaped tiles, and try to fit them in a (bigger) rectangular box'. There are twelve pentominos; so four of them are left, and it's easy to see that it doesn't really make sense to ask the question for I, L and P because they trivially tile a 1x5 or 2x5 rectangle:

enter image description here enter image description here

enter image description here

It turns out that the Y pentomino (see above) also tiles a rectangle without help of other rectangles, but the solution is a bit trickier and a nice puzzle, certainly doable by hand (that's where the tag is for). So the actual puzzle here is:

What is the smallest rectangle which can be tiled by Y pentominos?

Rotations and reflections of the pentomino are allowed, so you actually have 8 forms to work with.

One observation/hint (which, as expected, wasn't necessary, but might help future readers who want to solve this puzzle by themselves):

The total area of the rectangle must be divisible by 5. Therefore, at least one of the sides must be divisible by 5 as well.

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  • $\begingroup$ For those unfamiliar with the terminology, it would be helpful to have an image of a Y-pentomino, so we know what we're working with... $\endgroup$
    – GentlePurpleRain
    May 22, 2018 at 19:16
  • $\begingroup$ Why it is called as a Y pentomino...instead of any other letter pentomino..(for example L)? $\endgroup$
    – Mea Culpa Nay
    May 23, 2018 at 16:02
  • $\begingroup$ @MeaCulpaNay i.stack.imgur.com/xP35M.png $\endgroup$
    – Glorfindel
    May 23, 2018 at 16:14

3 Answers 3

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I brute-forced a solution that I think is the smallest possible. I stumbled upon it while trying to fill a bigger rectangle:

5 x 10

enter image description here

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    $\begingroup$ Dang, beat me to it. I have a nice image I can edit in though. $\endgroup$
    – gnovice
    May 22, 2018 at 20:26
  • $\begingroup$ Can you show that it's the smallest? $\endgroup$
    – noedne
    May 22, 2018 at 21:23
  • $\begingroup$ @noedne The only smaller configurations there are, given the shape of the pentomino, are 4×9, 5×9, and 4×10. As far as I can tell from spending a minute with pencil and paper those are all impossible. (That's about enough rigor for a comment, I think.) $\endgroup$
    – sk29910
    May 22, 2018 at 23:00
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    $\begingroup$ @sebastian_k $4\times9$ is impossible, not being divisible by $5$. How did you arrive at these candidate configurations? $\endgroup$
    – noedne
    May 22, 2018 at 23:06
  • $\begingroup$ Oh man, I didn't even think that far. I was only thinking about the length of the edges. Yeah, that eliminates 4×9 right off the bat! $\endgroup$
    – sk29910
    May 23, 2018 at 3:07
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As @TwoBitOperation, I used paint to represent a possible answer:

I found a 10*5 solution

enter image description here

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    $\begingroup$ Oh, too late :( $\endgroup$
    – franx93
    May 22, 2018 at 20:59
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I brute-forced an answer by hand in MS-Paint. Figured I'd share. The smallest I can get it is

15 x 10.

I do not know if a smaller answer exists

More like Y-did-I-spend-so-long-on-this pentominoes

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  • $\begingroup$ Nice! There is a smaller solution, though. $\endgroup$
    – Glorfindel
    May 22, 2018 at 20:13
  • $\begingroup$ I'll keep at it then! Thanks for the spoiler edit $\endgroup$
    – TwoBitOperation
    May 22, 2018 at 20:14

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