Three Tetris Mini-Puzzles

Every man dog and millipede on the planet knows there are seven possible pieces in Tetris.

In each of the following puzzles Joe Bloggs wants to achieve something specific within the first seven moves, without repeating any pieces. Assume that (1) Joe Bloggs can “call” the order in which the pieces appear (2) pieces can be rotated but not flipped and (3) “sliding or rotating underneath” is illegal.

• LEVEL 1: Can Joe Bloggs form a 7x4 block if there are TEN columns?

• LEVEL 2: Can Joe Bloggs complete four lines if there are SEVEN columns?

• LEVEL 3: Can Joe Bloggs complete seven lines if there are FOUR columns?

Level 1 is impossible:
Suppose it were possible. Imagine coloring the final 7x4 block in a black-and-white checkerboard pattern. The T piece has more squares of one color, all other pieces have the same number of black and white squares, but the entire block has the same amount of black and white squares too. Contradiction.

Level 2 is possible:

Level 3 is possible:

• nice work - and bonus points for finishing Level 3 with a Tetris! One can also condense the solution by writing numbers in a grid e.g. 7666/5564/3554/3344/7322/7112/7112. Note that the 7's are connected once the 6th piece is played and three lines are cleared. Jul 3, 2020 at 9:30

Why it's wrong:

I carelessly assumed (without, of course, noticing that I was doing so) that when a line is completed it's always at the bottom of the board. Of course that is very much not necessarily true. This invalidates everything I said about #2 and #3.

Wrong answer preserved below because I don't believe in hiding my stupid mistakes:

Two of the answers are

no, to "Level 1" and "Level 3",

because

if you imagine colouring a 7x4 block (in whatever orientation you please, whatever the total number of columns) in a checkerboard pattern, clearly it will contain equal numbers of black and white squares; six of the seven Tetris pieces also contain equal numbers of black and white squares however they are placed. This immediately shows that #1 is impossible. #3 is also impossible even though lines may be removed one at a time, because each line-removal removes the same number of black and white squares. So if we could do this then there would be 7+4 events (7 "place piece" and 4 "remove line"), all but one of which preserve the white/black balance, starting and ending with white=black=0, which is impossible.

As for the third,

since 7 is an odd number it might seem that with 7 columns we can arrange some sort of colour-imbalancing shenanigans as rows are removed. Not so. Suppose that instead of removing rows we merely allow them to move downward off the edge of the board, and suppose that the colouring of a piece or piece-fragment moves with the piece. Then clearing four rows on a 7-wide board still just means forming a 7x4 block, and that we can't do no matter how we place the pieces.

None of the above is changed

if we give Joe the ability to place the pieces wherever he chooses -- not merely "sliding and rotating underneath" but also leaving them hanging in mid-air. So long as he can't actually place the pieces so that they overlap one another, the checkerboard colouring guarantees that he can't perform any of the feats asked for here.

• I was thinking of an argument along these lines but I am not totally convinced by it on either #2 or #3. Imagine the T-piece being rotated 90 degrees from the picture. If the middle row of 2 were removed by a line then this actually can change the white/black balance because the bottom square remains the same while the top square moving down changes parity. It's a little hard to describe without pictures but does this make sense? Jul 2, 2020 at 11:02
• I started off thinking that there was an equivalence between #2 and #3 and forming a 7x4 block but since lines can be removed "in-between" it's sort of equivalent to forming a 7x4 block where we can "cut" some of the tetris pieces with some restrictions. Jul 2, 2020 at 11:07
• There was an earlier tetris puzzle in which 10 T-tetrominoes were used to complete 4 lines of 10 columns, even though it is impossible to tile a 4x10 rectangle with T-tetrominoes (for more complicated reasons than checkerboard colouring parity). Jul 2, 2020 at 12:34
• Yeah, my answer is wrong, because it assumes that when a line is completed it has to be the bottom one. Oops. Jul 2, 2020 at 12:41
• How is this getting upvotes after I put "Wrong answer" in big bold letters at the top? Oh well, I guess I can't exactly complain. Jul 2, 2020 at 22:17