The Trominoes
THE TROMINOES
Trominoes are 3rd order polynominos, formed by joining 3 unit squares together by common sides. At this point you might realize that we can find the Trominos by considering how to attach an extra Nod to a Doon. How many distinct ways can we do this? It turns out that we can only produce 2 distinct Trominoes under my restrictions. ( If we rule out the operation of rotation than we can actually count as many as 6 distinct rotations of all Trominoes. You can see this in the diagram below .)
Here are the illustrations of the 2 Trominoes and all of their rotations ...
Tromino No. 1 : Tropel
Tromino No. 2 : Treeb
I call the 2 Trominoes Tropel and Treeb ( I have seen some minor terminology used to refer to the Monomino and Domino, but after this unique names become much scarcier. In any case I will be using my own unique naming convention and cataloging order from here on in. I am coining the terms Tropel and Treeb here ).
I came up with the word "Tropel" by combining "Triple" and "Topple". The topple refers to a Tall tower toppling over to do it's height.(Somewhat of an exageration seeing as it's only 3 Nods high)
"Treeb" has no special significance, it might be thought of as combining the "tri" prefix with "Pheeb" a shortened version of the female name "Phebe"
TROPEL
I'm classifying Tropel as the first Tromino. It's standard position in my system is the vertical position. Like the Doon, Tropel has 2 orientations, the vertical and horizontal. Essentially Tropel is simply a rectangle with the ratio 1:3. Like Doons Tropels are easy to combine together. It has XY-symmetry but no slant symmetries.
TREEB
I am considering Treeb the 2nd Tromino. Treeb is the very first Polynomino to have 4 unique rotations ! All Polynominos will either have 1,2, or 4 rotation types depending on the symmetry they exhibit. Treeb has no orthogonal symmetries but it does have 1 slant symmetry (-45 degrees).
INTRODUCING POINTER NOTATION
It would be interesting to be able to have a notation by which we could describe any polynomino, other than a literal construction. I have invented a simple "pointer notation" which seems to be the easiest way to do this.
In it I only use 7 ascii symbols (That's all we will need). They are ...
^v<>/()
To translate a polynomino into arrow notation, first choose a initial cell (square). To describe the shape of the polynomino, simply use the arrows to describe the sequence of motions from the intial cell to cover the entire figure. The arrows are used in the obvious way ^,v,<,> standing for up,down, left, right respectively. In this simple way we can describe both Tropel and Treeb...
Tropel = ^^
Treeb = ^>
( There is also the trivial case of Doon = ^ )
This notation is fairly convenient for the simpler Polynominoes. It get's alittle more tricky as the complexity increases. For now I won't go into more detail into the notation as this is sufficient to describe the polynominoes up to the 3rd order.
THE GAME OF TRIS
We now have 2 unique pieces, surely a game of "Tris" should be more interesting than Monis or Domis. This is partially true.
Tris should be played on a 8x15 board. Tropel's and Treeb's should be choosen to fall at random with equal probability. One of the first things to consider is what would occur if there was a seemingly endless streak of one type of Tromino. Would it force a "game over" ? Let's say that the game kept supplying you with endless Tropels. How could you ensure to make rows to keep the game going indefinitely? The simple answer is just to fill each of the 8 columns with vertical tropels, to gain a triple row (Note that for every nth order game the maximum number of simultanteous rows will be exactly n).
The Game with endless Treebs is alittle more tricky. One way to accomplish a perpetual periodic game is to form 2x3 's using pairs of Treebs. A simple construction would simply line of 4 of these 2x3's vertically ( A cycle thus requires 8 Treebs). First the bottom row and then 2 more rows would be eliminated. Another way using the 2x3 as a building block is illustrated below ...
STAGE I STAGE II STAGE III
In Stage one we first place 2 2x3's horizontally along the base. We then fill in the empty space with an extra 2x3 vertically. 2 rows cancel and we are left with only 2 green Monads on the right. Again we fill the empty space in the base with 2 more 2x3's horizontally ( To avoid cancelation leave one of the purple Treebs out in one of the horizontal 2x3's ). Insert another vertical 2x3 and add the missing purple piece. The result is 2 more cancelled rows and we are left with a 2x2 box of Monads which are all green except for the lower left corner. Again we insert 2 more horizontal 2x3's and 2 more rows cancel and the board is clear again. Repeat this pattern for a perpetual periodic game of Tris (with endless Treebs ). This cycle requires 16 Treebs.
What if we are presented with alternating pieces of Tropels and Treebs ? There is actually a simple way to use this sequence. Create a series of 4 vertical 2x3's. Make sure that 2 are constructed out a pairs of Tropels and 2 are contructed out of pairs of Treebs. When the pieces fall in alternation simply construct the appropriate 2x3's until you create a solid 8x3 wall, which will result in the cancelation of a triple row. This basic 2x3 strategy can be adapted to handle other repeating sequences of Trominoes. Tris is therefore a relatively tame game. None the less, despite it's seeming triviality, Tris turns out to be a mathematically interesting game. It plays alot like an easy version of Tetris.
In every turn there is a total of 42 possible moves ( assuming you can choose freely between a Tropel and Treeb), and if the pieces automatically are selected there is 14 possible moves with a Tropel and 28 possible moves with a Treeb ( I'm not including "slide-in" cases in the number of possible moves). Compare this to Monis with only 3 possible moves, and Domis with 9. The complexity has definitely increased. Although it is difficult to say precisely, we can guess that a machine performing random moves in Tris has a fairly good chance of losing rather quickly. It might require a moderately clever program to play Tris reasonably well. One thing we could tell it to do would be always to stack Tropels vertically and in pairs to create vertical 2x3 blocks.
For a human this game at least provides alittle variety, but other than the challenge faced by increasing the speed of the game, the game is still alittle too simple. Using 2x3's it's fairly easy to develop basic stratedgies and keep the game going (assuming one doesn't make careless mistakes ).
SOLVING TRIS
what would it mean to "solve" Tris. In order to solve a game, we have to devise a stratedgy such that, no matter what infinite sequence of pieces we are presented with, we can determine algorithmically what move to make to always avoid a game over. Believe it or not, Monis and Domis are so simple that they are already solved ! Since there is only 1 possible piece to choose from, there is only 1 possible infinite sequence, and once we can solve this, we have solved the game.
So can we solve Tris ? Although one might first think this should be easy, given how easy the game is to play, that it must also be completely solvable. It turns out however that this is not the case. Although the 2x3 strategies are essential, this alone can not solve the game. Using the vertical 2x3 strategy for example... consider the first 8 pieces that fall. If there is an even number of Tropels and Treebs, then they can always be constructed into 2x3 blocks and all cancel in a triple row. Regardless of the order these fall in, we can fit them in the appropriate spots and always cancel, but what happens when the number of Tropels and Treebs are both odd ! In this case there will always be a left over piece we can't fit. This in turn leads to complications and makes it difficult to devise a singular strategy for all situations. As far as I know Tris has not yet been solved. Even this simple game proves surprisingly complex mathematically.
It is however pretty easy to come up with solutions to repeating sequences. A repeating sequence is kind of like the repeating digits in the decimal expansion of a fraction. We simply define a finite sequence of pieces and repeat it indefinitely.
For example, consider the repeat sequence of Tropel,Treeb,Treeb. We can use the piece numbers to shorten this sequence to 122. The question is, given an infinitely repeated sequence of 122 can we come up with a periodic sequence of moves to keep the game going indefinitely. I have found a solution requiring 24 pieces. This however is only a small part of solving Tris in it's entirety. In order to do this we would need to have a way to determine the best move given any piece and situation. This still an open game.
If Tris presents challenges for being completely understood, what hope to we have to solve Tetris or higher order games ?
Poly(3)
Poly(3) = 2 , because there are 2 distinct Trominoes, namely, Tropel and Treeb. Poly(3) is the first to be greater than 1, and exhibit at least some variety (albeit not much of an improvement).
At the moment the "growth" rate of Poly(n) seems to be exceedingly slow. Even it's argument is growing faster than it's output ! So far we have the sequence 1,1,2, ...
WHAT'S NEXT
Now things get interesting. Next comes 4th order polynominoes leading to one of the worlds most popular puzzle games, Tetris. You can click on Tetrominoes to learn about the Tetris pieces or you can click Polynominoes to return to the homepage to select any page ...