tetrominos

The T-tile get's it's name from it's "T" shape in the 2nd rotation. Again I use this idea to come up with the nonsense word Tad, for this piece. The Tad has 4 unique rotations. This piece can be both helpful and tricky depending on the situation. It helps to fill in specially shaped holes, and it can be rotated in different ways to accomplish different fill-ins.

This piece has some interesting Mirror properties. If we reflect the "norm" through the y-axis we get the 180 position. If we reflect the norm through the x-axis we get the same shape ! In otherwords the Tad has axis symmetry despite having 4 rotations. Because reflections result in either the same tile, or the same tile rotated we can say that the Tad is it's own mirror.

The Tad is remarkable for being the first polynomino with only one orthogonal symmetry.

(Note: when I refer to the norm, I am always referring to the first rotation displayed on the far left. The other rotations can be called by the angle number as 90,180, and 270 )

Pointer Notation: ^(^/>)

TETROMINO NO. 5 : ZAW ( The Z-tile )

The Z-tile get's it name from it's 2nd rotation. Likewise, I came up with Zaw which seems similiar to the word zig zag. The Zaw has 2 unique rotations, yet it lacks any relavent symmetry ! When reflected through either axis it becomes it's mirror the Saw.

This is one of the infamous stacking tile twins. When playing Tetris they often end up stacking together if there is a long sequence of just Zaw or Saw tiles. This can be risky and lead to premature game overs. This tile also doesn't "sit" very well on a flat surface as it automatically leaves a tricky gap. Only it the case of holes does this piece come in handy. This piece certainly is not preferable.

Pointer Notation: ^>^

TETROMINO NO. 6 : BASIL ( The O-tile )

The "Box" (as I usually refer to it in game) is the first square polynomino after the 1x1. It is also a rectangular polynomino of dimensions 2x2 . Basil is simply a nonsense word adapted from Box, in using the letter B in its name. The Basil has only 1 rotation, just like the Tetril.

Basil's aren't particularly helpful, but they neither are they too problematic. One generally regards this piece with indifference. It sits well on any flat surface, and it can fill large holes. On the rare occassion that neither can be found ( uneven terrian ) then this piece proves a pain since it almost certainly will leave either a gap or burrow.

Pointer Notation: ^>v

TETROMINO NO. 7 : SAW ( The S-tile )

The S-Tile get's it's name from it's S-shape in the norm. Likewise I named it "Saw". It has 2 unique rotations, and a unique mirror, the Zaw. It is non-symmetric having neither orthogonal or slant symmetries.

The Saw just like the Zaw is a trouble maker in game play and has a tendency to create annoying gaps or stacks.

Pointer Notation: >^>

So now we are familiar with the 7 pieces, before continuing let's just have a quick overview of their properties collectively, and in relation to the simpler polynominoes...

Note on Pointer Notation:

You may have noticed that No.4 the Tad is described in Pointer Notation as ^(^/>). What does this mean? When I first introduced pointer notation I only explained the purpose of the arrows. The arrows can be used exclusively to describe a polynomino provided the polynomino can be described by a single snake like path. Unfortunately not all pieces are like this, some of them "branch". The T-tile is the first non-snake tile or branch tile.

To describe branching pointer notation uses parathesis. You can think of this as offering multiple options for the next movement. The notation ^(^/>) can therefore be read as ...

"move up then move either up or right"

This describes the Tad exactly. We use the parathesis when a move can be carried out in more than one way, and we use the slash "/" to seperate optional paths. As we will see, this notation get's more cumbersome as the complexity increases.

PROPERTY COUNTS FOR TETROMINOES

First let's get some terminology straightened out. Tetris is based on treating the Tetrominoes as if they are fixed into only being in orthogonal rotations. "Orthogonal" refers to being lined up with the horizontal and vertical axes, as opposed to diagonals which could be any angle other than orthogonal . As such I am treating all polynominoes in the same way.

Mirror symmetry can therefore occur only along 2 axes.

When we reflect it along a horizontal axis we will call this X-reflection ( because the x axis is horizontal ).

When we reflect it along a vertical axis we will call this Y-reflection ( because the y axis is vertical )

So every polynomino can have 1 of 4 possible symmetries. Either it is symmetric in regards to both X and Y, it is symmetric only to one X or Y, or it is non-symmetric (not symmetric in either axis )

Given these ideas we can now study the Polynominoes up to Tetrominoes by classifying them by properties. Here is a table which identifies the various properties of all the polynominos we have looked at so far...

Poly-order Seq No. Tile Name No.of Rotations X-symmetry Y-symmetry Unique Mirror

1 1 Nod 1 YES(1.1:1) YES(1.1:1) -

2 1 Doon 2 YES(2.1:1) YES(2.1:1) -

3 1 Tropel 2 YES(3.1:1) YES(3.1:1) -

3 2 Treeb 4 NO(3.2:4) NO(3.2:2) -

4 1 Tetril 2 YES(4.1:1) YES(4.1:1) -

4 2 Jed 4 NO(4.3:3) NO(4.3:1) Led

4 3 Led 4 NO(4.2:3) NO(4.2:1) Jed

4 4 Tad 4 YES(4.4:1) NO(4.4:3) -

4 5 Zaw 2 NO(4.7:2) NO(4.7:1) Saw

4 6 Basil 1 YES(4.6:1) YES(4.6:1) -

4 7 Saw 2 NO(4.5:2) NO(4.5:1) Zaw

By examining this chart we can identify a number of trivial facts. First and foremost we have so far looked at the first 11 polynominoes in order.

Of these there are only 5 with Total symmetry (both symmetries) and 5 non-symmetric pieces ( no symmetries ). Only 1 piece has only a single symmetry ! In fact the Tad is unique in being the first polynomino to have only a single axis of symmetry.

Treeb is the first non-symmetric tile, and the Jed and Led tiles are the first pair of unique mirrors.

It would be interesting if the symmetry itself could indicate something about unique mirrors. There are only 2 sets of twins out of all the polynominos we have studied so far. All 4 of these tiles are non-symmetric. Might that be a criteria? Do all non-symmetric tiles have unique mirrors ? Unfortunately Treeb disproves this. Treeb is non-symmetric yet it is it's own mirror in both reflections. How do we account for this? How is Treeb qualitatively different than L,J,Z, and S that makes it it's own mirror.

One feature that sets it apart is that it has a diagonal symmetry ! L,J,Z and S do not.

What about single rotations ? There are only 2 polynominoes so far with a single rotation. Not surprisingly both have total symmetry. These two features should always go hand in hand since a Total symmetric tile must contain no distinguishing marks giving it no distinguishability in rotation. Also only having 1 rotation means no interesting features so that symmetry can create distinctive tiles.

There are 5 polynominoes with 2 rotations. What is striking here is that they either are Totally symmetric or non-symetric. Can a 2 rotation tile have a single symmetry ? I doubt that.

Lastly we have 4 polynominoes with 4 rotations. All are non-symmetric except for the odd Tad with it's single symmetry.

Now let's talk specifically just about Tetrominoes. There are 3 4-rotations, 3 2-rotations, and 1 1-rotation. There are exactly 4 non-symmetric tiles, 2 total-symmetric tiles, and 1 semi-symmetric tile ( X-symmetry ) By cross referencing symmetries and rotations we can make some inferences about what properties a polynomino can and can not share ...

Property PairsRotations/Sym No. of Tetrominos with Properties

1 / 00 0

1 / 10 0

1 / 01 0

1 / 11 1

2 / 00 2

2 / 10 0

2 / 01 0

2 / 11 1

4 / 00 2

4 / 10 1

4 / 01 0

4 / 11 0

It is probably true that most of the zeroes are impossible property pairs. That is, no polynomino can have such property pairs.

For example, is there any polynomino with 4 rotations with total symmetry ? This seems clearly impossible. Others may actually be possible even though we do not yet have an example. Take 4 / 01. This means 4 rotations with only Y-symmetry. If we can have 4 rotations with only X-symmetry (T-tile) , why not just Y-symmetry? Yet we haven't found any such tile yet.

How do these properties effect game play, and can these properties be used to count up polynominoes in an orderly way? Can we at least find formulas to determine numbers of nth order non-symmetric tiles, or total-symmetric tiles? All of this is still speculative.

TETRIS

So now we are back where we started, considering the game of Tetris. Tetris is played on a 10x20 board using random infinite assortments of the 7 unique tetrominoes.

Tetris, unlike the predecessor games of Monis, Domis, and Tris, is much more varied. At this point a human must begin to develop various strategies. Random moves would almost certainly end the game rather swiftly. The chances of creating a row in this manner would be slim, as there are many ways to leave gaps and burrows if one is careless.

Making a program that plays Tetris well might be a considerable challenge for the average programmer. No single strategy covers all situations, when one plays one usually has to vary their strategy depending on how the pieces fall.

Tetris is well known today thanks to the fact that it became a hit video game on the original Nintendo. Why did such a puzzle game become so popular? What's the appeal? Usually this game is described as "addictive". This could be because of the way it stimulates the brain. It seems that our minds find the task of fitting pieces together of enough importance to be recognized as worth doing, and therefore stimulating.

One might then assume that any such game would become a success. Why not then market a slew of variations of Tetris? Would Monis, Domis, and Tris also have the same appeal. Why don't we see other nth order games, why only 4?

The reason I believe, has to do with the particular mix of variety and simplicity that Tetris employs, and which other nth order games do not. Monis, Domis, and even Tris are simply too simple to be interesting for more than 3 or 4 minutes. Though the interest level does increase gradually through the 3 games. Tetris however takes a quantum leap from the first 3 games in terms of appeal. Working with 7 pieces seems to open up a world of possibilities. There are a total of 124 possible moves one can make with all of the 7 pieces (Not even considering slide-in cases)! That is a big leap from 3,9, and 42 ! Yet one can identify several practical ways to fit pieces together in the game. The game presents a good challenge, but is not so difficult to become mind-numbing or impossible.

One might then come to the conclusion that higher order = more playability. If Tetris is fun, how great of a game would the 5th order, or even 6th order games be?! Unfortunately, although 5th order games exist, they tend to be alittle frustrating. True, there is greater variety in both pieces and possible moves, and one is granted a larger board, however the difficultly increases rather rapidly. The 5th order game, sometimes called Pentris, borders precariously between challenging and impossible. One has a sense of the inevitability of the stack gradually rising. The game, it must be said, seems slightly unstable. Why is this? With the greater variety one might assume there are more ways to link up pieces and ways to create rows. In actuality it's the pecularities and particulars of the pieces themselves that make them more difficult to assemble coherantly. Gaps and burrows almost appear by neccesity, and one is forced to tolerate them to a certain extent. A 6th order game would only be more problematic. Thus we find that 4 seems to be a natural middle ground, and the order which proves the most pleasing as a game.

Since Tetris is a well known game, let's see if we can deal with the 7 possible endless streaks that could occur ...

STREAKS

We can define a streak as a sequence of the same piece for 2 or more turns. Generally speaking, streaks are not so good. They kill the variety, and can cause problems.

For the sake of the game of Tetris it might be worth considering what would happen in the case of the ultimate streak, ... an infinite sequence of the same piece from the beginning of the game. Would any such streak force a game over ? Actually all the streaks are quite simple to deal with.

We only really have to consider 5 of the 7 possible streaks. This is because we can use the same strategy for Led as Jed and Zaw as Saw.

Let's first consider the Tetril streak. This is simple. Simply line up 10 vertical Tetrils for a quadruple row.

An endless streak of Basil's is also very simple. Simply lay out the 5 Basil's without stacking them for a double row.

What about Led's and Jed's? This is also very simple. First pack pairs of Led's into 2x4 blocks, and then have 5 of these spaced out without stacking. You'll get a single and then triple row.

What about Zaw's and Saw's ? This is also fairly simple, but alittle more tricky to explain the pattern. Below is a diagram showing a procedure to get the streak into an endless loop.