In multiplayer Tetris, there are a number of opening setups that let you send powerful attacks very quickly. One popular opening is the DT Cannon, which allows the player to very quickly send a T-Spin Double followed by a T-Spin Triple.
The standard DT Cannon setup looks like this:
Here's a video of a guy completing a DT Cannon
In most modern Tetris games, blocks are generated using the "7-bag" random generator. For our purposes, this means that the first 7 pieces in a Tetris game will always contain one of each type.
In the DT Cannon setup, these 7 blocks are always placed first:
You might notice that certain pieces must be placed before others: The
J piece must be placed before the
Z piece, the
L before the
S piece, and the
S before the
T piece. Because of this, the setup is not always possible depending on the permutation of the first 7 pieces. For example, if your first two pieces are
Z, you are unable to place either piece in it's proper position.
However, if you have a
S piece followed by a
L piece, you're still able to continue the setup! Modern Tetris games allow you to hold a single piece, allowing you to swap your current piece with the piece in the hold. In this case, you can place your
S piece in your hold, place the
L piece, and swap out the next piece with the
S piece to place down. Note that if your first two pieces are
Z, it's still impossible to complete the setup, even if you hold the
After the first 7 pieces are placed successfully, it's always possible to complete the rest of the DT Cannon setup.
What percentage of all the possible 7-piece permutations are able to complete the DT Cannon setup?
As this is no-computers, your answer must contain some form of logical explanation or reasoning. However, you may use computers to verify your solution.
Tetris pieces color guide:
I - light blue,
J - blue,
L - orange,
O - yellow,
S - green,
Z - red,
T - purple