A typical game of
Tetris has 10 columns and 7 pieces. Consider a variation with fewer columns and only one piece, which repeats indefinitely.
For some pieces and column widths it is trivial to see that an infinite game is possible. For example the I tetromino with any column width or any other of the
tetrominoes with an even column width.
For other pieces, an infinite game is possible, but not trivially so. Consider the T tetromino in three columns, an infinite game is possible by following a pattern:
The first piece is horizontal with the stem of the T pointing down.
The second piece is vertical, with the stem of the T pointing to the right and the T is pushed to the left edge.
The third piece is vertical, with the stem of the T pointing to the left and the T is pushed to the right edge. 

For each piece and number of columns listed below, find a strategy for each which allows for an infinite game:
1. 2.
### in three columns ### in four columns
# #
#
3. 4.
### in four columns ##### in five columns
# #
#

Notes:
1  Using the reflection of an asymmetrical piece is not allowed.
2  Use classic gravity: when a row is filled, it is removed and all rows above move down, but no fragments of one partially filled row fall into another partially filled row.
1.
Let the first two drop without rotation, leaving a stack of 2 on the left, as the horizontal long arm of the piece completely disappears in each case:
2 
1 
++
Place the third one on the right in normal letterL formation with the short arm on the bottom:
 3 
23  which becomes  3 
133 23 
++ ++
The next one will be an upsidedown L on the right:
 44
 34  44
234 which becomes  34
++ ++
The fifth and sixth will come straight down:
555
544 becomes then 666
 34  34 634 disappears completely,
++ ++ ++
ready for a new start.
2.
The first piece goes to the bottom left with the stem pointing up:
 1 
 1 
111 
++
Then the second piece with stem pointing down, to the right of this:
 222
 12 
 12 
111 
++
The third piece:
3 
333  3 
3222 333 
 12  becomes  12 
 12   12 
111  111 
++ ++
The fourth:
 4  
3444  
3334 becomes  4
 12   12 
 12   12 
111  111 
++ ++
The fifth:
   
5   
5554 5 
512  becomes 512 
 12   12 
111  111 
++ ++
The sixth:
   
 6  
5666  
5126  6
 12  becomes  12 
111  111 
++ ++
The seventh:
   
   
7   
7776 becomes 7 
712  712 
111  111 
++ ++
The eighth:
   
   
 8  
7888  
7128 becomes  8
111  111 
++ ++
There are only 8 members of this cycle as the next drop of an upward stem (upside down T) on the left:
   
   
 9   
 9  leaves  9 
9998  9 
111  111 
++ ++
which actually replicates the result of the first drop, though the configuration is a combination of pieces from two pentominos. So play continues as in drop number 2.
3.
The first piece is placed:
 
 
 1 
 1 
 111
++
then pieces 2 and 3
 333  
2223  333
 123 leaves  123
 12   12 
 111  111
++ ++
and so forth:
444   
4333  
4123 leaves 444 
 12   12 
 111  111
++ ++
   
 555  
4445 leaves  555
 125  125
 111  111
++ ++
   
666   
6555 leaves  
6125 666 
 111  111
++ ++
two steps here, 7 and 8:
7   
7888  
7778 leaves  
6668 7 
 111  111
++ ++
   
 9   
 9  leaves  9 
7999  9 
 111  111
++ ++
The last configuration is the same as after the first drop, so play continues as in drop number 2.

Posted by Charlie
on 20071219 19:00:59 