welcome to the website for the project disx.
use above to navigate.
feel free to leave comments and suggestions.
have a nice day!

welcome to the website for the project disx.
use above to navigate.
feel free to leave comments and suggestions.
have a nice day!

group leader: group members:

raymond chan (3i304) alvin tan (3i302) theodore yoong (3i332)

contact us here

Our project is to research and investigate the game DisX, using the relevant fields of mathematics, namely game theory, and more specifically non-zero-sum game theory. This game is also related to the concept of self-organized criticality. We have also used the concept of coordinates for the notation of movesets in our game.

DisX has a 6x6 game board, with 1 disc on each peg, and an additional 1 disc every other peg, as shown in the diagram on the left.
Each player starts the game with 7 discs, placing 3 every turn. There are a total of 150-54-14=82 discs in the bag.
Avalanche occurs whenever a peg has 4 discs or more. Discs on the peg are then transferred, one each, to the pegs orthogonally adjacent to it. Additional discs are given to the player.
1 disc from the bag is awarded to the player for each avalanche. Any discs that "spill" out of the game board are also taken by the player.
When the bag runs out of discs, the game ends; whoever has the most number of discs wins.

NOTATION
Our notation system and an example of it is shown below.

vertical coordinates - a, b, c, d, e and f

horizontal coordinates - u, v, w, x, y and z

In "bw cy fu 2", the first disc is placed on bw, the second on cy and the last on fu.
After avalanche, there is a net gain of 2 discs.

PROGRAMS
We have written 3 C++ programs to assist our research. The first simulates the game in the blink of an eye, the second calculates optimal moves and the third plays out all optimal games. We have combined the first and second program too, for convenience. The programs are attached here:

1-run prog5.exe; output2.txt shows the optimal moves, output3.txt keeps track of the game itself.
2-run list.exe; games.txt shows all the optimal games.

MODIFICATIONS

Edge length	Board size	Discs placed each turn	Starting no. of discs pp	Discs on board at start	TOTAL DISCS
a	a2	a2/12	(a2+6)/6	(3a2)/2	(25a2)/6
2	4	1/3	1 2/3	6	16 2/3
4	16	1 1/3	3 2/3	24	66 2/3
6	36	3	7	54	150
8	64	5 1/3	11 2/3	96	266 2/3
10	100	8 1/3	17 2/3	150	416 2/3
12	144	12	25	216	600

Possible modifications include:

size of game board - 2x2, 4x4, 8x8
one avalanche affects the 4 pegs around it (or less if it is a corner/side peg). As such, if we decrease the board size, the ratio of area of effect to total area would increase greatly, as shown in the diagrams.

number of discs placed per turn - 1
placing 1 disc per turn will guarantee player 1 will make the first wrong move by making a peg have an odd number of discs. if the game was a fair game (each player has the same number of turns), player 1 would lose since he made the first (and only) bad move.

number of discs needed for avalanche - 4n
discs in bag, number of discs transferred to each peg every avalanche, starting number of discs will be multiplied by n. discs obtained from avalanche remains as 1, and discs placed remains as 3
high points per turn, but slower game - more discs gained from avalanche at sides (from game board), discs from bag gained slower (still 1 disc per avalanche).

STRATEGIES
From our program, we found that player 2's 1st move, axaxay leads to player 1 wins, while axayay leads to player 2 wins. The only difference between the two moves are that they lead to different spread of discs. However, there was no simple strategy found, as observing the spread of discs is very complicated.

OTHERS
Top moves are always positive. The first move is known to be positive - 2 at a corner peg guarantees net 1 disc. There will then be a 2-2 at the sides of the game board, which guarantees 1 point. Discs are then distributed to pegs adjacent to it, which creates move 2-2's. When the cycle is complete, there will be 1 disc placed on the sides in the 4x4 square not at the sides and 2 on the corners. This will lead to many 3's, which obviously allow for a positive move. Discs are then distributed back out where there are mostly 1's and 0's, to become 2's and 1's. The cycle then repeats.
If we ignore one of the above mentioned moves, then the move will still be there until we touch it, so there will still be at least one positive move.

[1] Bak, P., Tang, C. & Wiesenfeld, K. (1987). “Self-organized Criticality: an Explanation of 1/f noise”. Physical Review Letters 59 (4): 381-384. College Park: American Physical Society.
[2] ChessNotation.com --- how to record and study chess games (n.d.). Retrieved from http://chessnotation.com/algebraic.htm
[3] Gutowitz, H. A. (1995). Self-organized Criticality. Retrieved from http://tuvalu.santafe.edu/~hag/internet/node9.html on 4th April 2011.

We would like to acknowledge and thank our mentor for assisting us throughout our project.

alvin
The questions we came up with originally seemed to be rich in prospects and results. However, it turned out that the methodology became tedious, and we had to frequently change our topics. This taught us to plan ahead before executing our project.
raymond
From this project, I have discovered that we have little sense of urgency and inability to manage our time and resources properly. This project has given us many chances to train and practice our time management skills.
theodore
Through this project, I have explored deep into the realm of game theory, and thus I have learnt many new concepts and terms regarding game theory.