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> Hold that T piece! A parity experiment...
Blitz
post Feb 22 2013, 06:21 PM
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Hold the T pieces! This thread is the result of an experiment where I tried to abuse the parity properties of the T tetomino as much as possible to see if it resulted in a more flat field or not. If you have not read this thread, http://minodistrict.forumotion.ca/t47-paul-s-t-parity-notes you may want to read it in order to understand what I'm talking about. Actually, you MUST read that first to understand this.

I was wondering what it would look like, if your field had even parity during the entire game, or if it's even possible to keep even parity throughout an entire game. The results are quite interesting.

In order to keep even parity throughout the entire game, you would have to remove all elements that can cause uneven parity.

Things that can change the parity of a field is:
Skimming (dealth with by doing only tetrises)
Receiving garbage lines (dealth with by playing a mode without garbage)
Placing a T tetromino (We might have a problem here)

For every problem there is a solution.
In this case, the problem is that no matter how you place your T tetromino, it will cover an uneven number of black and white squares (if you imagine a matrix filled with squares).
So when you place the first T tetromino, you can think of it as you just placed it in a +1 or -1 position. Let's say +1 means it covers 2 more black than whites, and -1 means it covers 2 more whites than blacks. To get even parity, you must place your next T tetromino in a -1 if your last was a +1, or place it in a +1 if your last was a -1.
Thinking about this while actually playing a game can quickly get complicated, so I have found a way more simple solution that solves this problem.

By using hold, you can force yourself to avoid situations where you have to place a single t piece, and instead create situations where you have the ability to place 2 t pieces in a row and form "blocks" made from 2 T tetrominoes joined together. If you do this, you can see that one will reverse the parity effect of the other causing the field to have even parity at all times.



Can you create any kind of "blocks" with the T pieces?
NO. You can't. The examples above will always result in even parity. You can rotate and mirror them as much as you want, and they will still have even parity. But there are some cases that does not allow you to keep the parity, but instead they will make the parity even worse.



Think of the T pieces as having one "centre" and 3 "arms". From the example above, we can see that if the "centre" of one T piece, is connected to any of the "arms" of the other t piece, it will create worse parity.

Now that's enough theory. Let's see how this works out in reality.

Here are 2 100 line and one 40line game. Notice how I hold the T pieces, and place 2 in a row creating "blocks".





I am not an expert at stacking , but these results are a lot better than my normal average. Based on these results I think a field with even parity will provide you a lot more options for good stacking than fields with uneven parity.

Also read this post AND THIS!
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Integration
post Feb 22 2013, 07:27 PM
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I stack 5:3 parity in Cultris.

IPB Image

If you do non-center 4-wide with 3 residual, you have 3:1 parity in your 4 combo columns and 4:2 parity in the other 6 columns.

Also this:




Conclusion: Uneven parity wins!
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post Feb 23 2013, 12:04 AM
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Blitz, this was a very enlightening post. I never realized parity could be applied to Tetris, or the implications it would have on stacking. You've set up a good argument for your theory, that even parity results in better stacking. I wonder what other results would arise if you pushed this research further Wink.png


QUOTE(Integration @ Feb 22 2013, 07:27 PM) *


Conclusion: Uneven parity wins!


You've referenced the "Playing Forever" technique as proof that uneven parity is better for stacking. Indeed, the technique's highlight is that one could theoretically stack forever, thereby making it the "most optimal" stacking method. But look carefully. The right side of the field is flat and even, with J/L/O being used. Meanwhile, the left side is intentionally jagged and uneven, for the sake of simplicity and repetition. The key of this technique is the strong recurring pattern, which relies on a very flexible assortment of pieces, thus giving it the "infinite" effect. But notice how when the player is forced to use a T in this technique, he stacks two T's as complements before using another S or Z. The fact he must do so is very similar to what Blitz himself does in his 100L video. To maintain even parity and facilitate smoother stacking. In other words, the Playing Forever technique seems to actually follow the principle of even parity, rather than odd.

I'm in bed with the flu so I'm not willing to actually do the research to validate my statements, but meh my two cents lol.
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StS
post Feb 23 2013, 01:50 AM
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Massive lol at Integration's post, using Blitz's own video at an attempt to (albeit jokingly) refute his hypothesis.


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Current 40L (Nullpo): 35.32s
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post Feb 23 2013, 01:57 AM
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QUOTE(StS @ Feb 23 2013, 01:50 AM) *

Massive lol at Integration's post, using Blitz's own video at an attempt to (albeit jokingly) refute his hypothesis.


Hey! your avatars a Digimon right?? haha brings back memories Wink.png
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myndzi
post Feb 23 2013, 05:56 AM
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Never saw Paul's original post, it's quite interesting. It puts into words something I've known but not nailed down. I was trying to talk about this with perfect clear stacking the other day, and now I have some vocabulary that helps say what I meant to say:

You get 1 T every bag, so if you're stacking a 10x4 perfect clear, you must account for the parity switch. This means making a place to put jagged pieces that fit with 'odd' parity, and making a place to put flat pieces that fit with 'even' parity. The observation about skimming is interesting, and you can take it a bit further:

When a T is stacked in left or right rotation, two pieces can be cleared out of the middle of it, leaving two minos in even parity with each other. This doesn't happen when the T is in spawn or 180 orientation.

It also helps to explain this kind of move:


I have referred to this before as perpetuating a shape, or forcing shapes; you can see by looking at it from a parity perspective that it fixes the 'effective' parity on that side of the field by creating a hole that you don't have to deal with when stacking.

Very interesting observation Paul, and interesting results, Blitz.

One more observation: The usefulness of 'even' parity makes lots of sense since each tetrimino is made of an even number of minos. Combining this with the 'box' theory I described to Mario when talking about Perfect Clears, you can see that you need to manage your parity wisely to clear clean groups of lines, and keeping your 'box' small gives you faster access to your garbage (fewer pieces need placing to access any given column)

I wonder where it'd get me to combine the box concept with parity and the minimum-stack-height thing I described to caffeine with regards to driller downstacking...

I should probably post about those other two things, otherwise people are gonna be like 'wtf are you talking about'?


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baseballboy4296
post Feb 23 2013, 06:00 AM
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As soon as I finished reading this, and before I even saw Panda's post regarding the playing forever method, I thought of it myself. He pretty much summed up what I was gonna put out there. I'm gonna put it into my own words anyways. Sticking Out Tongue.png

The playing forever method really shows this even/uneven idea very well, at least for me. Looking at the side stacked with L minos, J minos, and O minos, it is apparent that the stack is relatively even throughout, because of the fact that each mino covers two of each color, black and white, if one looks at it using the checkerboard pattern idea.

On the side stacked with the S minos, Z minos, and T minos, the stack is made uneven on purpose for the allowance of S and Z minos to be placed. This is executed by placing one flat T mino, creating a jagged stack that can only continue to be stacked with S, Z, and other T minos.

Looking at the T mino's occupation on the checkerboard shows that the only ways it can be placed are such that there is an uneven distribution of colors of squares being covered, no matter what. This is the cause of the unevenness in the stack. The only way (while still following the patterns of the playing forever method) to balance this unevenness is to place one more T mino such that it covers the color that has less squares occupied on the checkerboard.

Now, in order to keep this "jagged stack" present, the above-mentioned "balancing" method must be used.

At a certain point, there are two possible ways to continue stacking the uneven side. One involves flattening the stack with a T mino, or making the stack even by balancing the number of occupied squares, and then making it uneven once again, continuing on with the same exact pattern as one would at the start of the game, and the other involves keeping this jagged formation, while still placing two T minos that balance the square counts for both colors on the checkerboard. Both paths end with the exact same result. (I'm too lazy to fumen these two situations... I'm hoping someone who reads this understands where I'm coming from here and maybe can explain it better.)

In es2mac's post on minodistrict, his example number 5 involves the T parity when stacking in an empty 4-wide well. This is closely related to the method used to stack S, Z, and T minos in the playing forever method.
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Integration
post Feb 23 2013, 06:18 AM
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Honestly, what I said has not much to do with even and uneven parity. I wanted to refer to the following scenario: You want to fill a rectangular shape consisting of an even number of columns (2 wide, 4 wide). Then you count the number of black and white fields on your surface provided you have no overhangs (in this case the difference on the surface is the same as the difference in your stack). Let's say you build a 2 wide. There're 2 different ways to do so: a straight 2 wide or a 2 wide with a small lever.

IPB Image vs IPB Image

straight 2 wide: The black-white ratio will switch between 4:4 and 6:2 (resp. 2:6) dependent on how many T pieces you dropped. 6:2 can be too hilly, but isn't that bad. 4:4 is good, but you must pay attention that your stack isn't entirely flat.

2 wide with lever: If you pay attention to your T placements the ratio will always be 5:3 (resp. 3:5), independent of how many T pieces you dropped. This results in a consistent way of stacking. You still have to watch out to not drop a T piece on a wrong spot or you get a 7:1 (resp. 1:7) ratio. 7:1 ratio means, that there are not even 2 positions where you can drop O, L or J pieces simultanously.



QUOTE(Panda @ Feb 23 2013, 03:04 AM) *
The key of this technique is the strong recurring pattern, which relies on a very flexible assortment of pieces, thus giving it the "infinite" effect. But notice how when the player is forced to use a T in this technique, he stacks two T's as complements before using another S or Z. The fact he must do so is very similar to what Blitz himself does in his 100L video. To maintain even parity and facilitate smoother stacking. In other words, the Playing Forever technique seems to actually follow the principle of even parity, rather than odd.

The key is indeed a reoccurring pattern. Let's ignore the middle 2 columns. Then it can be seen as some special kind of 2 wide, where you want to maintain the 6:2 ratio instead of the 4:4 ratio.
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myndzi
post Feb 23 2013, 06:34 AM
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In brief, the 'box' concept is about the size of the smallest box required to clear out all your minos. For example:



Standard perfect clear build, the box is 10x4. But with the same pieces in a different shape...



The box is 10x5. This is because the single mino on the left can't be filled alone. In fact, the only option for a 10x5 box (ignoring that we can't clear an odd number of lines) is to place an L piece:



Any other piece filling that space would increase the height of the 'box'. That makes this shape as restrictive a placement as this one:



(In terms of perfect clear stacking / staying low to your garbage)

... and therefore that shape is to be avoided if possible.

The other important thing to the 'box' concept is the empty mino count. If you divide the box into more than one set of blank areas, each blank area must on its own be divisible by 4, else you will be forced to stack up higher. For example:



Box size is 10x7 (or 10x8 if you're being strict about it). Even though there are many pieces that can be placed on either side of the O, they cannot completely fill the space without "spilling out" of it. For this reason, you cannot clear the occupied lines without stacking into the next unoccupied line(s).

This is relevant in general play because it means that while your stack may *look* low, it may actually be shaped in such a way as to force you to stack many more pieces than you want to, effectively being "taller".

Here's one example of how this concept applies in everyday play; I run into this on KoS reasonably often.

(I actually ran into a good example while trying to construct one, so here's a fumen that looks stupid but illustrates the point)



This looks like a good fit for a skim, but because of that final J placement, we now have to fill in the right side before we can access the next garbage hole. Ideally we'd use a Z piece, but even that ideal piece would add 3 to the height of our box before accessing the second garbage hole! An L or an I in the 2-high hole opens the right side to an exact-match J placement without raising the box as high. Compare:



Bare minimum 5 lines cleared to access the garbage.



Bare minimum 4 lines cleared to get to the same state, but only 2 cleared to reach the garbage itself.

This is kind of a murky description I guess, but the idea is that you can know to avoid certain placements, such as that initial L, because you can see that they force a significant increase in the height of your 'box'.

Going along with this idea of the box, it can also be seen that you can calculate the number of minos that must be stacked in a given column above a garbage hole before you are able to access it. I believe there's a mathematical way to do it that won't be terribly accurate, but should at least follow the trends properly; I haven't worked it out yet. The point is to count how many lines you have to add due to garbage hole locations and shapes; essentially the size of the "box" above the garbage hole in question. Then, since you know you MUST clear at least that many lines, you know that you can make the box that high without fear -- there's no faster way to access that garbage hole than to clear that many lines to begin with. This helps for downstacking to deeper holes because while instinct tells us to avoid stacking above shallow holes, we don't really know how MUCH to avoid it -- and the deeper the holes are, the more you can and must stack over them. Knowing how high you can safely stack in a given position in conjunction with knowing how to stack there so as not to accidentally increase the height of your box will let you access your garbage as efficiently as possible.


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tk198
post Feb 23 2013, 08:08 AM
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Myndzi I might be completely missing he point of the last bit of your post but why didn't you place the L vertically(?) on the left side and clear out the two holes? It seems like the more obvious choice when downstacking.
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Blitz
post Feb 23 2013, 12:47 PM
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The example games I played yesterday was done at my max speed, and therefore they were not 100% smooth all the way.
I just played one game, where I slowed down to around 0.5pps. This game turned out a lot better than my other examples. I think this one demonstrates the true potential of keeping even parity during the entire game.


Also it's only 2 days since I first read pauls parity notes, so I have not had time to practise this a lot. Imagine how smooth you could be stacking if you really went deep into this.
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post Feb 23 2013, 09:24 PM
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Blitz, so the only difference between your fumen above and a regular Tetris stacking is in how the T's are placed? Because the T's are the only minos that can affect parity?

Also, you displayed earlier an assortment of two-T-piece combinations that can preserve even parity. How do you decide which of these combinations to apply? Is there a specific preference for one over another depending on the given circumstance and field layout?

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Blitz
post Feb 23 2013, 09:46 PM
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QUOTE(Panda @ Feb 23 2013, 09:24 PM) *

How do you decide which of these combinations to apply? Is there a specific preference for one over another depending on the given circumstance and field layout?


When I play i dont think of it as creating blocks. I just think of it as getting 2 t pieces in a row, then I place them where I think they fit the best. I just make sure they are connected to eachoter. This automatically create one of those blocks.
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post Feb 23 2013, 10:34 PM
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QUOTE(Blitz @ Feb 23 2013, 09:46 PM) *

When I play i dont think of it as creating blocks. I just think of it as getting 2 t pieces in a row, then I place them where I think they fit the best. I just make sure they are connected to eachoter. This automatically create one of those blocks.



What if there was a stacking method that was better than even parity? In your fumen, whenever the player gets a T piece, the piece is either held or is combined with another T in a pre-determined combination. In other words, every move regarding a T-piece is systematic and mechanical, following a strict procedure with no space left for intuition. This systematic approach results in not only faster processing time, but also in a smoother stacking field.

What if we could establish such a systematic approach for not just the T-piece, but every piece in the game? What if there were factors beyond just parity that affects the flatness of the field, and depending on the factor a new systematic approach would be used on a particular piece? Right now, your fumen only takes the factor of parity into account and only has a respective systematic approach for the T-piece, while every other piece is placed down in non-discriminatory freestyle. But if we could somehow establish a systematic approach for every single piece, then imagine the enormous possibilities in speed or stacking.

Of course, as of now, I have no idea what these other factors may be, or if they even exist. But just a thought Sticking Out Tongue.png
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jujube
post Feb 23 2013, 10:46 PM
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QUOTE(Blitz @ Feb 23 2013, 12:47 PM) *


Considering the field to be "flat" when there is no more than one step up or down from one column to the next, I went through the game and recorded each frame at which this measure of flatness was achieved. I threw out the first 10 pieces (during this stage the player should be stacking in a way that leads to sustained flatness later, without concern for immediate flatness), and the last frame because no piece was placed.

CODE
   **11   12                       17   18   19

20   21   22   23   24   25                  29





               53   54   55   56             59

60                  64   65   66   67   68

               73   74   75   76   77

80        82   83             86   87

                              96        98

00   01   02                                 09

10   11   12   13   14   15

                         25   26

                         35   36   37   38   39

40   41   42



                                   67   68   69

               73   74   75   76        78

80   81   82   83   84   85   86   87

          92                       97   98   99

                              06   07   08   09

     11             14   15   16   17        19

20   21   22   23   24

30   31                       36   37   38   39

40   41

                         55   56        58   59

60   61   62   63   64   65

            **

The field was "flat" in 41.2% of the frames from 11 to 272.

Of note is the stretch from frame 17 to 25 where flatness was never broken, followed by only 1 flat frame out of 27. Perhaps there's a price to pay for going after flatness with each piece for too long, although I'm only looking at it abstractly and not criticizing any one placement. There's a more consistent pattern from frame 167 to the end as far as I can tell.
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