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So, I just bought a Tetris Light like this one.

Tetris Light

With all 7 unique tetrominos, I can build a tower whatever I like and then turning them on at night to give me a good sleep, mm hm. (What a geek!)

I just realized that $7 \times 4 = 28 = 1 + 2 + 3 + ... + 7$. Therefore, with those tetrominos, I'd like to build a tower like this (white-blocks part).

Tetris Tower

Hmm, I have a difficulty building this tower. Can you help me to do that? You may rotate them but it's preferable not to flip ("mirror") them.

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Can I help you build the tower?

No: it is impossible. Color the tower like a checkerboard:
enter image description here
There are 16 black squares and 12 white squares. Besides the purple T, each piece will cover two squares of each color, no matter how it's placed. But then the purple T will have to cover four black squares, which is impossible.

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  • $\begingroup$ I almost answered at the same time but you beat me to it, so I removed it. $\endgroup$
    – Nautilus
    Commented Feb 9, 2018 at 7:34
  • $\begingroup$ Yep, indeed, this is the only correct answer :) $\endgroup$
    – athin
    Commented Feb 9, 2018 at 7:45
  • $\begingroup$ The other way round, can you say that all builds are solvable when hey have an equal number of black and white squares when checkered? $\endgroup$
    – Tweakimp
    Commented Feb 9, 2018 at 14:06
  • $\begingroup$ @Tweakimp, no, a single long line has as much black as white and is unsolvable. $\endgroup$
    – EagleV_Attnam
    Commented Feb 9, 2018 at 14:44
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    $\begingroup$ @Tweakimp actually, in this case no build is solvable if it has an equal number of black and white squares - with the 6 shapes besides the purple T, you will always cover 12 white and 12 black squares. Then the purple T will either cover 3 whites and 1 black, or 1 white and 3 blacks. So a necessary condition (but not sufficient) is to have 15 squares of one color and 13 of the other. $\endgroup$
    – Rob Watts
    Commented Feb 9, 2018 at 16:40

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