1. Search in OOXX Games J.-S. Roger Jang (張智星) MIR Lab, CSIE Dept.
National Taiwan University
2018/11/19

2. Introduction to OOXX Games (井井井井字棋)
OOXX Games (井井井井字棋 or 井4字棋)
Invented by Adrien Wu (吳聖福) in 2018
Rules
Two players, O and X, alternately place two tokens of their own, on a 5X5 board.
O always plays first to put two O’s on the board.
X always plays second to put two X’s on the board.
After 11 plays, the game is ended by putting the last 3 X’s to make the board full.
How to determine the winner
One point for a player is achieved if we have at least 4 tokens of this player in each of the 5 rows, 5 columns, and 2 diagonals.
The player with more points wins the game.

3. An Example of OOXX Game
Result
O: 4, X: 3  O wins!

4. Problem Definition
Given a half-filled board, determine who is the winner (or a tie) if both adopt the best strategy to play.
Three possible results
O win
X win
Draw

5. Examples of Best Strategy (1/3)
Next play by O
Both O and X have a score of 3
O cannot score any more  No way for O to win
The best bet for O is to take the gray positions, which leads to a tie.
If O didn’t choose the gray positions, X will win.
So the best strategy for O is the above two gray positions.

6. Examples of Best Strategy (2/3)
Next play by O
The best bet for O to score 3 is to take the gray positions
X can get an addition score by take the yellow position.
No matter how O play, s/he will lose the game.
So the best strategy for O is actually any two positions (since s/he will lost the game anyway).

7. Examples of Best Strategy (3/3)
Next play by X
If X takes the gray positions (which leads to the board in the previous slide), s/he will win.
If X takes the positions marked with *, s/he will win too.
So the best strategy for X is not unique.

8. Hint: How to Determine the Result?
Given a current half-filled board, search for the next possible board and record its result.
Suppose the current board is O’s move, s/he will adopt the best strategy to determine the result of the current board:
If any of the next boards is “O win”  “O win”
Else if any of the next boards is “Draw”  “Draw”
Else  “X win”

9. Hint: How to Speed Up Computation?
During the search, it is likely that you run into a board that has been computed before.
You can use <unordered_map> to record the result of a given board, so you can easily retrieve it later.

10. Hint: How to Save Memory?
Each board has 25 elements, and each element has 3 states (O, X, and empty)
 Use 2 bits for each element
 Use a long long (64 bits) for a board to save it to an unordered map

11. … … … … Game Tree O X O win O win O X C 2 7 =21 O X C 2 5 =10 X win
Draw
O win
O win O X
C 2 7 =21 … …
X win
X win
X win
X win
X win
X win
Draw
O win O X
C 2 5 =10

12. Hint: How to Speed Up More?
Comments from 吳聖福

13. MinMax Rule: Game Tree of Tic-Tac-Toc
O’s move!
1: O wins
0: Draw
-1: O loses X X O
Max O X O -1 -1
X’s move! X X O X X O X X O
Min O O O O O X O X O O X O 1 -1 -1 1 X X O X X O X X O X X O X X O X X O
Max O O X O O X O O X X O O O O X O X O X X O O X O O X O X O X 1 1 X X O X X O X X O X X O
Min O O X O O O O O X O O O X O O X O X X O O X O X

14. Example: a-b Pruning in a Game Tree (1/3)
Source
Max
Min
Max
Min 8 7 3 9 9 8 2 4 1 8 8 9 9 9 3 4

15. Example: a-b Pruning in a Game Tree (2/3)
Source
Max
Min
Max
Min 3 17 2 12 15 25 2 5 3 2 14

16. Example: a-b Pruning in a Game Tree (3/3)
Source

17. Exercise 1: a-b Pruning in a Game Tree
Display how the search can be reduced via a-b Pruning
Max
Min
Max
Min 5 7 3 9 6 5 2 4 1 8 1 6 4 9 3 4

18. Exercise 2: a-b Pruning in a Game Tree
Display how the search can be reduced via a-b Pruning
Max
Min
Max
Min 8 7 3 9 2 5 5 4 1 8 6 1 4 9 3 4