Is there a winning strategy in Tetris? | Hacker News Reader

Is there a winning strategy in Tetris?

mathoverflow.net

94 pointslainon36 comments

True story: the Cambridge Maths Society, The Archimedeans, had a journal called Eureka (of course). It happily published interesting but not important bits of mathematics. One student wrote a proof of an impossible sequence for any width board. It's even impossible if you have perfect knowledge of the sequence. All you need is to drop Ss and Zs in an irrational order.

I think this got published in 1990, but I can find little evidence even if the journal's existence online.

I think you mean this one? Of course it's also possible that multiple people have published multiple proofs.

https://www.researchgate.net/publication/2347389_How_to_Lose...

The proof itself is quite self-evident: given a perfectly randomly block generator, and infinite play time, eventually you're going to produce a very long sequence of left- (or right-) facing S-blocks. Since those are impossible to eliminate (there's always one block left open on a row) eventually you lose.

This actually isn't true for modern variants of Tetris. A correct implementation of the random number generator requires that the 7 pieces be shuffled into a bag and then drawn randomly, ensuring an even distribution, and disallowing long runs without a desired piece. (Maximum of 12.) The bag is re-shuffled every 7 blocks in the sequence.

https://tetris.wiki/Tetris_Guideline

https://tetris.wiki/Random_Generator

Thus, in a game of Tetris adhering to these official guidelines, the longest sequential run of S and Z pieces would be 4, in the following sequence:

[5 Random Blocks] [Z and S in any order] [Z and S in any order] [5 Random Blocks]

I'd be very curious to see a proof of an impossible sequence which adheres to the official rules, as that would be quite a feat indeed!

In that case it would probably still be possible, but the proof would have to be found by example. It seems like an advanced version of the Eight Queens puzzle: which sequence of bags guarantees a losing board state?

>I'd be very curious to see a proof of an impossible sequence which adheres to the official rules, as that would be quite a feat indeed!

It sounds like, given enough piece previews, you can play without ever losing: http://tetris.wikia.com/wiki/Playing_forever

> [5 Random Blocks] [Z and S in any order] [Z and S in any order] [5 Random Blocks]

Any dab hands at regex out there? I think this is the regex to match, but something feels wrong about it

    (([^sz]){5}[sz]{2})([sz]{2}([^sz]){5})

Not that one. It's from 1996, the one I'm thinking of was 1993 latest and I think 1990. There was also a previous article that demonstrated that the player always won for 3-omino Tetris (not a massively interesting result).

As observed elsewhere, neither work according to official Tetris rules.

I misunderstood "board" as blackboard and thought that a Cambridge student had written a proof that, for any arbitrarily wide blackboard, there was a sequence which it was impossible to write on the board.

This seems like the sort of theorem that a college student might be inclined to prove.

Search for "tetris championship" on YouTube. Some great tetris gameplay and you can pick up on a lot of the strategy pros use.

Ex: https://youtu.be/DdfRQjb5o9k

I once played a game of Tetris on the Nintendo DS (about 10 years ago now) where I maxed out the lines… then the levels… and then maxed out the score.

I played this same game for weeks, mostly on the train to and from work. And wouldn't you know it, but Tetris DS has a bug that won't let you save a score once you pass 100,000,000 or whatever the highest possible score is.

Tetris got a lot easier when they added the ability to infinitely rotate a block for as long as you want at the top of the pile.

There's a related annoying question for which I've never found a conclusive answer: for the standard tetris board (width 10, height 22), is there a strategy for the player that guarantees that they will be able to fill at least one line? or conversely is there a strategy for the game (supplying the pieces) to make the player die without having filled at least one line?

Yes, a strategy does exist which can force at least one line in a well of width 10 (and different strategies for 4, 6 and 8). I did a tedious brute-force calculation which demonstrated this a few years back. The code is not particularly presentable and the calculation takes about 11 days on a modern machine (it runs in a single thread), but here: https://github.com/qntm/tetris

The limiting factor is headroom (the depth of the well). For wells of depth 0, 1 and 2, the AI clearly wins, but once the depth increases enough a strategy becomes possible. My belief is that such a strategy exists for all even well widths, although proving this is annoyingly hard.

Hi, thanks a lot for pointing this out to me! However https://qntm.org/tetris#sec6 does not have any cases for width 10 where the player wins. Are you implying that you have run more calculations than described in this article, and that the player can win for width 10 with a sufficient high depth?

"Tetris" has to be well-defined mathematically before it can be confronted as a problem. Rules governing piece rotation behaviour, piece spawning, piece randomization, wall kicks, loss conditions and even well dimensions need to be laid out first.

I have great news for you! https://tetris.wiki/SRS

In classic Tetris you can slide the piece briefly after it lands, so you can always keep the game going.

A strange game. The only winning move is not to play.

How about a nice game of chess?

To reference: https://youtu.be/xOCurBYI_gY?t=15m57s

It's fun how program learning to play teris pauses the game just before loosing.

Source of the reference is War Games (1983). https://www.youtube.com/watch?v=6DGNZnfKYnU

I've always thought that the mechanics of Tetris are pretty straightforward and it is always possible to arrange incoming pieces without any gaps. The main difficulty for a human player is coming from the increasing speed, not the pieces not fitting together.

Tetris is NP Hard. Also Hatris: https://www.youtube.com/watch?v=EIBqEUvvDi0

Not sure if your comment was to say "I thought tetris was easy in a turn based environment, turns out I was wrong" or if you were contradicting the linked thread without basis

I think your disagreement comes from a problem of definition. This paper (linked in the mathoverflow thread) appears to demonstrate playing Tetris optimally is NP-hard, even in a turn-based environment; that's because playing Tetris optimally involves getting lots of Tetrises, not just surviving indefinitely.

http://publications.csail.mit.edu/lcs/pubs/pdf/MIT-LCS-TR-86...

In contrast, it's not clear to me that playing Tetris indefinitely with the piece randomizers used in current Tetris games is NP-hard.

If you only get z shapes (with the same orientation). Is it then possible to avoid gaps? This is not clear to me at all.

You can go infinite if only supplied Ss. You'll have intermediate holes which will be filled in after the above row is cleared

Ah yes, of course!

Independently of that in Tetris the pieces (tetrominoes) are drawn from a bag: http://tetris.wikia.com/wiki/Random_Generator

Only in some versions. E.g. Puyo Puyo Tetris on the Nintendo Switch (to quote the most recent commercially-available Tetris I can think of) doesn't use that randomiser. I may or may not have played a few very poor games before I caught on to what was going on...

Puyo Puyo Tetris draws from a bag of 2 sets of tetriminos, so every 14 pieces you'll have gotten two of each, afaik

Also you'll never get three of the same piece in a row*

*almost 100 % sure of this

Haven't happened to me in a game yet and my rating is around 4480

But are those pieces selected at random or not? Probability of getting even 10 z shapes in row is comparable to probability of winning a lottery (2.01599×10^-8, if I got Wolphram Alpha query right)

For the purposes of finding a "winning strategy" we have to have a solution for every piece sequence, regardless of probability.

If you only get Z-shapes (of the same orientation) forever, then no. The game's not winnable.

Proving the least favourable distribution of z-shapes to other shapes which would allow a game to continue indefinitely would be really interesting. :)

In fact you can just line five of the Z shapes up along the bottom, over and over again. This technique works indefinitely.

Now, if there's an occasional S shape thrown into the mix, you have a problem.

As additional clarification if you align the piece upward to look like this then you're fine:

   #
  ##
  #

You'll end up with

   # # # # #
  ##########
  # # # # #

which becomes

   # # # # #
  # # # # #

which you can just repeat over and over again forever (if your variant of Tetris includes gravity even this collapse to nothing).

Certainly the first thing that's apparent when watching pros play is their speed. However, after playing years of Tetris myself (though nowhere near pro level), I've really come to appeciate that speed is only a small fraction of what makes pros pro.

* Many multiplayer games give bonuses for combos (multiple consecutive line clears), T-spins, and Tetrises. Setting these up efficiently can require reading several pieces ahead and takes a ton of practice. An example where both players do a fairly elaborate setup, one with combos and the other with T-spins: https://www.youtube.com/watch?v=mASSHXLTuU4

* 20G mode, made famous by games like TGM, drastically reduces the ability to manipulate the pieces. Even surviving takes new strategies (you're forced to play a lot of strange moves), much less making tetrises like the pros do. https://www.youtube.com/watch?v=aoU0DQh7zdU

* Some multiplayer games give difficult garbage, I'm thinking of Cultris 2 specifically. Clearing garbage efficiently, known as "downstacking", requires much more careful piece placement than regular play. It's hard to appreciate from video, but if you compare pros downstacking to a normal person, you can see they use fewer pieces to clear the same amount of garbage, which is a real competitive advantage. https://www.youtube.com/watch?v=OgK4aJ_3zCw

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