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338

It is impossible. Proof: Let the $7\times 28$ area be painted with black and white squares in a checkerboard pattern. Every piece will cover $2$ black and $2$ white squares, except the T-piece, which covers $3$ of one color and $1$ of another. Since there are $7$ T-pieces, a tiling that uses every piece cannot cover the same number of black and white ...


48

TLDR: I'll fill the board and prove that the solution is unique. First, let's start by: I'll paint those green: Let's repeat those steps a few more times, using orange, blue, red and purple, in precisely that order: Now, We can easily fill the topmost white squares by that reasoning. They can't be filled in any other way: Now, let's look at: And by ...


22

I think that this tiling is a valid Tetris stack:


17

It can be done with Some explanation: Since


14

Can I help you build the tower?


9

Since a single line doesn't do damage, it is possible to do To achieve this, there are a couple of requirements: To get these pieces one after the other For this to occur so that the final piece also clears the board, we get these constraints on the number of pieces $X$: Given these, the smallest $X$ that satisfies both requirements is This is a small ...


8

Shown here and in a tile formation Interpreted the even when rotated part of the question as just stating that a tetromino cannot touch the same colour tetromino regardless of orientation.


7

It is and the following strategy does it: See for formal proof. There is


5

A simple solution involves Stack the first five arranged the same direction (on the side of the T with the bottom facing either left or right. In my example, the bottom is facing the right). This eliminates the second row (shown with Xs through them). Next, flip the T 180 degrees so the bottom is facing the other direction and fill the gaps as such. This ...


5

It looks like This works all nice and fine, There may, of course, be something obvious I'm missing. For the J: With the L, there are some problems that are very similar to the Z case, Here's the T: And finally, the S: And for completeness' sake, the solutions already found by others: I-piece: O-piece:


5

A possible solution can be reached by building For clarification, here is a picture.


4

This is my solution (just incrementally adding only valid pieces):


4

Just to make sure, is this a valid answer for I Tetromino?


3

As an upperbound, I can attack as little as with by following configurations:


3

Just to point out the obvious one:


2

Here is a link to a Mathematics and Computer Science journal published by Kaitlyn M. Tsudura from Saint Mary's University in Halifax, Canada. It summarizes the results of studies published by John Bruztowski and Heidi Burgiel The studies were basically to determine whether it was possible to make a Tetris game only winnable but they found that it is ...


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