Trivial Hazh games equation

[alan2here]

Hazh is a game I invented for which I have yet to document the rules.

 

It is based on a number of players taking it in turns cyclically to draw symbols onto a grid of a given size, each grid square can hold only one symbol.

 

When one of a set of given spatial pattern of symbols appears anywhere on the grid a player wins.

 

Thease winning conditions are agreed on before the game starts by drawing them symbolically beside the grid.

 

The notation used is designed to group the conditions such that they are perceptually easy to understand, not to be mathamatically easy to describe, it is also irrelevant to creating a generic solution to this problem therefore I will use a more generic but somewhat similar notation in my examples.

 

For example a game of Hazh could be played to mimic noughts and crosses.

 

In this game the grid would be of the size 3x3, two players would be playing, one using a 'O' symbol and the other an 'X'. The winning conditions would be.

 

O wins =

OOO
,

O
O
 O
,

O
O
O
,

 O
O
O

X wins =

XXX
,

X
X
 X
,

X
X
X
,

 X
X
X

 

The grid could be larger, this would automatically allow players to match there shape anywhere on the board. One player could have differnt shapes required to win than another or there could be a combination one player needs that involves another players symbols etc...

 

For additional challenge the board is allowed to be set to wrap as if toroidally.

 

........................................

 

The question I am asking given this scenario is.

 

How trival will a given game will be?

 

This could be answered in part only, for example.

 

How many moves will be taken before a player wins?

 

It will probably be an approximation, should presume all players play well and be answered given less calculation than is required by simulation.

 

Usefull values for answering this question could be, for example.

 

board area.

lowest number of symbols in winning conditions.

etc...

 

For example (* = all players)

 

* wins =

**
,

*
*

 

Would allow any player to win by getting two of his symbol in a left\right or up\down line. This game is trival, when played on a 1x1, 2x1 or 1x2 non wrapping grid as no player can win (I also consider games that end in only a few moves or strongly favour the order of players to be trivial), any other combination results in the player who goes first winning.

 

A simpler verstion of this statement would be.

 

Any game where any player has a win condition involving three or less pieces will result in a trival game.

 

I am aware that playing well is subjective and that the challenge is difuclt or immposible but hopefully intresting.

Edited November 28, 2009 by alan2here

[Mr Skeptic]

Can your grid be multi-dimensional as well? May the winning conditions involve opponent's pieces as well?

 

The answer as to how difficult the game is depends on the attributes of the grid and the winning conditions.

 

If you are a relatively skilled programmer, you could do one of the following.

You could make your Hazh game into a computer game with a rules editor. Players may submit rules they like.

You could make a computer player, for small enough boards, that can think a certain number of moves ahead (if the board is small enough, it could find all possible solutions). This will be fairly simple if the board is small enough. Pitting two computer opponents against each other will give you an idea as to how fair and balanced the game is under those rules.

[alan2here]

Can your grid be multi-dimensional as well?

In the game it can only be 1D or 2D due the difficulty of drawing symbols or placing counters in an environment of 3 or greater dimentions. For the sake of this challenge and Hash theory then yes, it is multidimensional.

 

Suprisingly there are non trivial 1D games.

 

May the winning conditions involve opponent's pieces as well?

As already stated, yes. A winning condition for a certain player may involve unspecified space, his own and other players symbols. When specifying the winning conditions empty space is generally used for unspecified except sometimes where it is unclear where a central character dot is used. Obviously each winning condition involves an infinite amount of unspecified space.

 

The answer as to how difficult the game is depends on the attributes of the grid and the winning conditions.

Yes, of course. It's a lovely game in part because of this flexabilty, you can have a long game or a short game. A game that tends towards draws or not. A game where each player must spend several minutes thinking about their move or one where movies are spotted quickly.

 

If you are a relatively skilled programmer...

I may tackle this at some point. It's not that hard for players or a program to generate rules that generally work quite well, but you can never be sure that a rule will behave as you expect.

 

 

In the game itself a 90 degree rotational symmetry symbol and a horizontal or vertical reflection symbol may be used by the winning conditions to show that other versions are allowed. This is usefull for players but as stated already just complicates things for us as it's just a way of drawing lots of conditions in such a way as not to appear complicated.

 

Also in the game the "* wins =" is drawn as a large verstion of the symbol being defined above and to the left of the winning condition definitions.

Edited November 28, 2009 by alan2here

[Mr Skeptic]

In the game it can only be 1D or 2D due the difficulty of drawing symbols or placing counters in an environment of 3 or greater dimentions. For the sake of this challenge and Hash theory then yes, it is multidimensional.

 

To do additional dimensions, you just add extra copies of the game board. Eg for 3D, a 5x5x5 board would be 5 squares of 5x5 each. More dimensions can be done but it gets even worse. More dimensions don't necessarily make the game more complicated, but they do allow more squares to be closer to each other.

[Zolar V]

regarding the 3 dimensions, you could even make the game board selectable in slices to decide where to put your symbol, and to review the gameboard between player turns for strategy.

 

the slices would be selectable via all three dimensions.

for instance selecting a corner square, the game would ask wich slice you want, x-y-z axis would be the choices. i suppose the easiest way to describe it would be a traslusent rubix cube.

[alan2here]

It's hard enough in 2D.

 

Also, nobody has produced an answer yet.