I received this article in my news feed this year. It’s a great little article – you can read the article and see how desipite moving the individual Tetris pieces around to form whatever shape you wish you cannot create a “clean” rectangle using this tetris lamp.
Seriously. If you want to understand this you should read this article otherwise this sounds crazy
His proof is really nice and even elegant but he is operating under the impression that you can never change the number of blocks. If you were to say buy two sets of the tetris blocks could you create a clean rectangle?
This hinges on breaking a property he discovered, where “the number of black squares must be equal to the number of white”. In the single set case you are forced to deal with the T block which has three dark squares and on light squares, as he diagrammed perfectly using this image:
However deciding that this T has three dark squares and one light as opposed to three light and one is arbitrary. With two sets you have two T’s, so setting one to the “dark side” and the other to the “light side” should give you an equal balance of colors, allowing you to create a clean rectangle.
The Tetrimino that brings balance to the rectangle
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This is great in theory, yet finding the full tetris block is difficult. To do that work I used TetPack, written by COTO, a java program written to find optimal packing of tetriminos. This yielded optimal solutions for a 14 x 4 rectangle and a 4 x 28 rectangle.
Figure 1: Rectangle created using two lamp sets in a 4 x 14 configuration
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Figure 2: Rectangle created using two lamp sets in a 8 x 7 configuration
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Notes
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- Due to the rotationality of tetriminos these also serve as solutions to the 7 x 8 and 14 x 4 case
- Attempts were made to output all optimal solutions was made but I am trash at Java. That could be a future endeavor
- Solving the 2 x 26 case isn’t possible
- Technically the Z and S blocks are different yet due to the way the lamp is made you can turn an S into a Z by mirroring it about the X axis. The same goes for J and L. These were kept in their intended form for this process
- Please actually don’t think this means I want a tetris lamp, much less two