**Question 1**
Is there a winning strategy (algorithm to play infinitely) in Tetris, 
or is there a sequence of bricks which is impossible to pack without holes?


Consider generalized Tetris with Young diagrams (for some $n$) are falling down.


**Question 2** Is there winning strategy?
If not - consider some probability measure on Young diagrams (e.g. uniform). What will be "losing speed"? I.e. how fast the height of uncancelled rows will grow for best possible algorithm?


**Question 3** Can one relate such Tetris like questions on Young diagrams with some conceptual/conventional theories where Young diagrams appear - representation theory of symmetric group or something else?



Is there winning strategy in Tetris ? What if Young diagrams are falling?
Heidi Burgiel's first paper "How to Lose at Tetris" (which she wrote towards the end of our time in grad school in Seattle) answers Question 1 in the negative. It was published in the *Mathematical Gazette* volume 81 (1997) 194--200. If you want to see the published version, that volume is still on JSTOR where you can access some number of articles for free, <http://www.jstor.org/stable/3619195>. (Current Gazettes are on the Cambirdge journals site.) As mentioned in the comments, there's a version at the old Geometry Center site, <http://www.geom.uiuc.edu/java/tetris/tetris.ps>.


A related article "Tetris is Hard, Even to Approximate" by Erik Demaine, Susan Hohenberger, and David Liben-Nowell shows the difficulty of dealing even with a finite sequence known in advance. Although it generalizes the board size, it keeps tetris pieces. The document <http://publications.csail.mit.edu/lcs/pubs/pdf/MIT-LCS-TR-865.pdf> is much longer than the published <http://erikdemaine.org/papers/Tetris_COCOON2003/paper.pdf>.


To your other questions: For the simple part of Question 2, I'd think the answer is "no" since any "pentris" or larger Young tableaux / Ferrers diagrams for $n \ge 5$ would be extensions of the Tetris pieces that are already too much for a winning strategy. The details of the Demaine et al. paper(s) may help address the rest of that question.


Towards Question 3, there are skew tableaux which can be thought of as building blocks for a Young tableau. Also, there's the notion of removing rim hooks to get to the $p$-core of a partition.


