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Can you place five dominoes, five trominoes, five tetrominoes, five pentominoes and five hexominoes inside a 10x10 grid, such that:

  • No two polyominoes overlap
  • No two polyominoes of the same size (by area) touch each other orthogonally (horizontally or vertically)
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  • $\begingroup$ My previous puzzle was too easy so I made this one. $\endgroup$
    – Dmitry Kamenetsky
    Jun 18, 2022 at 23:33
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    $\begingroup$ Maybe we should have a separate "XXominoes inside XxX grid" site. $\endgroup$
    – WhatsUp
    Jun 19, 2022 at 1:14

2 Answers 2

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Unfortunately, this is also too easy.

This solution is trivial:

enter image description here

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  • $\begingroup$ I feel so stupid now. It took my program many hours to find a solution. $\endgroup$
    – Dmitry Kamenetsky
    Jun 19, 2022 at 2:08
  • $\begingroup$ What if we disallow straight polyominoes (except dominoes), can it still be solved? $\endgroup$
    – Dmitry Kamenetsky
    Jun 19, 2022 at 2:10
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    $\begingroup$ I did it without straight polyominoes (except dominoes). $\endgroup$
    – RobPratt
    Jun 19, 2022 at 2:21
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    $\begingroup$ @DmitryKamenetsky Don't feel bad. It can be hard to come up with good puzzles. Just a tip, though: computers usually aren't the best way to judge the difficulty of a puzzle. Computers don't have imaginations or experience, which are important tools for solving things. Humans, on the other hand, have both those traits. $\endgroup$
    – Sylvester Kruin - try Codidact
    Jun 20, 2022 at 15:51
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I can do it with no straight polyominoes (except dominoes):

enter image description here

An obvious upper bound for the maximum number of distinct shapes is $1+2+5+5+5=18$, and...

...this solution attains that upper bound: enter image description here

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  • $\begingroup$ You could try to maximize the number of distinct polyominoes. $\endgroup$
    – Daniel Mathias
    Jun 19, 2022 at 1:57
  • $\begingroup$ @DanielMathias Good suggestion. Done. $\endgroup$
    – RobPratt
    Jun 19, 2022 at 2:25
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    $\begingroup$ Nice, and with distinct fixed trominoes. $\endgroup$
    – Daniel Mathias
    Jun 19, 2022 at 2:41
  • $\begingroup$ Very nice! You've solved every version of this problem :) $\endgroup$
    – Dmitry Kamenetsky
    Jun 19, 2022 at 3:23

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