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.

If the piece drops as it is shown in the problem, its horizontal parrt would be removed and only a single cell in the middle column would stay.

Start with dropping 5 'horizontal pieces' , labelled A through E and two verticals F and G

  FEG
  FDG
FFCGG
 FBG
 FAG

Remove the middle line and we have a 3 by 4 square in the central columns.

Again 5 horizontal in the centre and two verticals on the outside.

     F
     E
G  D  H
G  C  H
GGBHH
GAAAH
GAAAH
  AAA
  AAA

We have:

    A
    A
A  A  A
A  A  A  
  AAA
  AAA

One horizontal and two verticals.

  CBD 
  CAD
CCADD
ACACA
ACACA  
  AAA
  AAA

Which produces again the 3 by 4 square

 

Edited on December 20, 2007, 2:19 pm

Posted by Hugo on 2007-12-20 14:18:37