Otdfbt

In Tetris 99, Mr. T loves performing All Clears, which happen when a piece clears all lines in the playing field. Being a gentleman, he also tries to minimize the total damage he sends to his opponents in the process.

Rules:

• 
  

  The total damage sent by a piece is the sum of all applicable bonuses:

  
  
  
  • An All Clear adds 4 damage.
  
  
  • 
    

    When a piece clears multiple lines, a bonus is applied based on the number of lines cleared:

    
    
    

    +-------+--------+
    | Lines | Damage |
    +-------+--------+
    | 2 | 1 |
    | 3 | 2 |
    | 4 | 4 |
    +-------+--------+
    

    
    
  
  
  • 
    

    When a piece clears a line and all of the last $n$ pieces also cleared lines, an $n$-combo bonus is applied based on $n$:

    
    
    

    +-------+--------+-------+--------+
    | Combo | Damage | Combo | Damage |
    +-------+--------+-------+--------+
    | 1 | 1 | 6 | 3 |
    | 2 | 1 | 7 | 4 |
    | 3 | 2 | 8 | 4 |
    | 4 | 2 | 9 | 4 |
    | 5 | 3 | 10+ | 5 |
    +-------+--------+-------+--------+
    

    
    
  
  



• To generate the random piece sequence, the 7 tetriminos are placed in a bag and randomly drawn without replacement. This process repeats for the entire game.




• Mr. T does not use hold or soft drop (pieces are never slid or rotated under each other), and no one ever attacks him. In practise, this means that all the pieces are dropped from above, in the order they occurred.

Now, suppose Mr. T just performed an All Clear.

1. How little damage could he have sent to his opponents this game?




2. How few pieces could he have used to achieve this minimum?

Since a single line doesn't do damage, it is possible to do

no other damage apart from the 4 damage from the all clear.

To achieve this, there are a couple of requirements:

* to avoid the multirow bonus, the last piece must be a horizontal line, and

* to avoid the combo bonus, the next to last piece must not be needed to clear an earlier row, which means that it also must be a horizontal line.

To get these pieces one after the other

The bag must get refilled just before the final piece.

For this to occur so that the final piece also clears the board, we get these constraints on the number of pieces $X$:

$X equiv 1 ;(bmod; 7 ;)$
$X times 4 equiv 0 ;(bmod; 10;)$

Given these, the smallest $X$ that satisfies both requirements is

15 pieces.

This is a small enough number that it should be relatively easy to find an entire game leading up to the "all clear". Here's a possible game I found after a quick search:

I made sure this position is reachable without combo bonuses by ensuring that each layer has at least two pieces that do not extend to a lower layer. The yellow dashed lines separate the different bagfuls from one another.

(A later, less quick search seems to indicate that I may have been very insightful and/or lucky in my search, since the possible games reaching this particular all-clear don't seem to be that common at all.)

As an upperbound, I can attack as little as

$5$ damage

with

$10$ pieces in total

by following configurations:

link