Tetris puzzle
I learned this puzzle years ago from this tweet by Edward Kmett, who credits Scott Garrabrant. (Note: My presentation of the puzzle below is slightly harder; the tweet will spoil a bit of it.)
You're playing Tetris (on a normal, width-10 board). There are two blocks on the board, in the bottom-left and bottom-right corner:
..........
..........
..........
..........
..........
#........#
Is it possible to completely clear the board with at most one of each of the seven Tetris pieces?
As spoiler padding, please enjoy this mostly-vibecoded pure-CSS tetromino.
Proof of impossibility
There are two blocks on the board already. Seven distinct Tetris pieces would give you 28 more blocks for a total of 30. So you can only clear at most 3 lines.
Clearing 1 line is impossible since you can't fill the 1×8 space with distinct Tetris pieces.
Clearing 2 lines is impossible since you'd need to add exactly 18 blocks, which isn't divisible by 4.
Finally, clearing 3 lines is impossible through a checkerboard argument. If it were possible, you'd need to add exactly 28 blocks, or all seven Tetris pieces. However, if you color the board and the Tetris pieces like a checkerboard, you see that:
- the initial blocks have opposite colors;
- the T tetromino will always have two more blocks of one color than of the other;
- the six other tetrominoes will always have equally many blocks of both colors; and
- any line you clear will always have equally many blocks of each color.
Thus, the board state after dropping all seven Tetris pieces must have two more blocks of one color than of the other, so it can't be empty.
Therefore, it's impossible.
How do you feel about this proof?
The solution
The above proof is wrong, but finding and then breaking the flawed proof and then the flaw is very helpful for finding the solution.
The checkerboard argument is flawed, because it's possible to clear a line beneath a block, which causes it to drop and change its checkerboard color.
In fact, it's necessary to cause an odd number of blocks to drop to break the checkerboard constraint we discovered above. Since there are always an even number of blocks on the board at any given time, the only way to do this is by clearing the second line first while there are an odd number of blocks on both the first and the third line.
Once you realize you need to do this parity-changing drop, the puzzle becomes fairly easy. We can make such a drop as quickly as possible with the I, J, and L pieces:
J.........
JJJLLLIIII
#..L.....#
Afterwards, the remaining pieces easily clear the board.
JOOSSTTTZZ
#OOLSSTZZ#
Just for fun (?)
How many of the 7! orders of Tetris pieces is it possible to clear the board with?