C onnect6 / 六子棋 A N ew C hallenging G ame and its Latest Developments 一種新的挑戰性遊戲及其最新發展

1. Connect6 / 六子棋A New Challenging Game and its Latest Developments一種新的挑戰性遊戲及其最新發展 I-Chen Wu / 吳毅成 Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 台灣新竹國立交通大學資訊工程系 October 28, 2008

2. 人腦戰電腦 紅面棋王敗北! (08/24/2007) 民視新聞（http://www.connect6.org/2007824L09M1.wmv ）

3. 個人資料 • 現職： • 國立交通大學資訊工程系教授 • 鈊象交大聯合研發中心主任 • 台灣六子棋協會理事長 • 學歷： • 台灣大學電機工程學系學士(1982) • 台灣大學資訊工程研究所碩士(1984) • Ph.D., Computer Science of Carnegie Mellon University (1993)

4. 研究兩大主軸 • 網路遊戲 • 發展網路遊戲平台： • 研發高速的Java AWT系統[Wang and Wu 2007] • 研發P2P傳輸下載系統。 • 鈊象交大聯合研發中心：並有其他技術移轉！ • 棋牌類人工智慧 • 發表新的棋種：六子棋 • 發展六子棋程式：交大六號(NCTU6) • 兩度獲得奧林匹亞競賽金牌 • 發展其他人工智慧程式：如電腦象棋、數獨驗證系統等。

5. Outline • Introduction to Connect6 • Characteristics of Connect6. • Fairness of Connect6 (六子棋) • Compared with Gomoku (五子棋) or Renju (連珠棋). • Threat-based winning strategies for Connect6 players or programs. • Impact of Our Research • Report of some Tournaments • New challenges in the future.

6. 三大特點 • 規則簡單 • 變化複雜 • 遊戲公平

7. 規則簡單 • 黑先下１子 • 之後各下２子 • 連成６子者勝 一般化K子棋： • Connect(k,p,q) • 六子棋為：Connect(6,2,1)

8. Generalized Connection Games • K-in-a-row (K子棋): Connect(m,n,k,p,q) • Rules in Chinese: • 有黑白兩玩家，黑先下q 子，之後輪流下 p 子。 • 誰最先在棋盤橫向、豎向、斜向形成連續的同色 k子者為勝。 • 棋盤大小是mxn。 • When m= & n=, also called Connect(k,p,q) • Example: • 井字遊戲：Connect(3,3,3,1,1) • 五子棋: Connect(15,15,5,1,1) • 六子棋: Connect(19,19,6,2,1) 或 Connect(59,59,6,2,1)

9. 變化複雜 (Game Complexity) • 六子棋的複雜度非常高。可從目前常用的兩個複雜度標準來看。 • State-Space Complexity: （所有盤面狀態數） • 六子棋複雜度：10172 • 與圍棋複雜度相同 • Game-Tree Complexity: （遊戲樹展開的大小） • 圍棋複雜度： 10360 • 日本將棋複雜度： 10226 • 象棋複雜度： 10150 • 六子棋複雜度：（ 10140～ 10188）象棋與日本將棋之間 • 最早估計一般走30步，則 (300*300/2)30 ~= 10140 • 但目前高段棋士，常走到 40步以上。則約為 ~= 10188

10. 西洋棋一般公認複雜度第四 象棋一般公認複雜度第三 資料來源： Herik的論文[2002] 圍棋一般公認複雜度最高 比五子棋複雜度低 百年前日本即有職業連珠棋院 日本將棋一般公認複雜度第二高 Connect6 10172 10140 ~ 10188

11. 遊戲公平 • Key: Balanced (平衡) • One player always has one more stone than the other after making each move. • 每當一方下出一步（兩子），該方一定比對方多出一顆子。 • B: #1, W: #0 • B: #1, W: #2 • B: #3, W: #2 • B: #3, W: #4 • B: #5, W: #4 • …

12. Outline • Introduction to Connect6 • Characteristics of Connect6. • Fairness of Connect6 (六子棋) • Compared with Gomoku (五子棋) or Renju (連珠棋). • Threat-based winning strategies for Connect6 players or programs. • Impact of Our Research • Report of some Tournaments • New challenges in the future.

13. Fairness of Gomoku • Rules in the free style (no restriction). • After B move, B has one more stone than W. • 每當黑方下出一步後，比白方盤面多一顆子 • After W move, W only has the same number of stones as B. • 然而每當白方下出一步後，盤面子數卻只能與黑方打平。 • Allis [1994] proved that B wins. • Allis [1994]等人證明出先下必勝 。

14. Fairness of Renju • Japanese Professional Renju Association (1903) imposed some new rules to restrict the play of B for professionals. • No double 3, double 4, and no overline. • 限制雙活三、雙死四、長連 • Called Renju (連珠棋) • Problem still. • Professionals still said that B wins. • Wagner、Virag [2001] proved this.

15. Side Effect One • More and more complex rules. • RIF: Renju International Federation. (國際連珠棋協會 ) • New Rules by RIF [1998]. • New restriction in opening. (限制許多開局規則 ) • Call for new rules in 2003. • Vote new rules in 2006. • But, vetoed by Japanese RIF.

16. Side Effect Two • Smaller board [Sakata and Ikawa, 1981] (棋盤變小 ) • Sakata 及 Ikawa said: • Increasing the board size raised B advantage. Restrict the board size to 15x15 • But, a smaller board lowers the complexity of the game and thus makes it easier to solve the game.

17. More about Fairness • Definitions of fairness • Strategy-Stealing Argument • Breakaway (脫離戰場 ) and fairness • Fairness of Connect6 • Fairness of Connect games

18. Definitions of Fairness • Herik et al. 2002： • A game is considered fair if it is a draw and both players have a roughly equal probability of making a mistake.” • “一個遊戲是公平的話，那麼它必須是個平手的遊戲，且雙方有相同的犯錯機率。” • Problem: • hard to have a perfect model for calculating the probability of making a mistake • On the contrary, it is relatively easy and possible to show when a game is unfair.

19. Unfairness • Definitely unfair, • if it has been proved that some player wins the game. • For example, Go-Moku (in the free style) • Monotonically unfair, • if it has been proved that one player does not win the game. • For example, for Connect(k,p,p) or Connect(m,n,k,p,p), based on the so-called strategy-stealing argument. • Empirically unfair, • if most players, in particular professionals, have claimed that the game favours some player. • For example, before Go-Moku was solved, Go-Moku was empirically unfair.

20. Potential Fairness • Potentially fair, • if it has not yet been shown or claimed to be definitely unfair, monotonically unfair, or empirically unfair. • Properties: • A potentially fair game for the time being may not remain potentially fair in the future. • If a game remains potentially fair any longer, it could have a higher chance to be fair.

21. Strategy Stealing Argument • Raised by Nash in 1949 • The argument shows that W does not have a winning strategy in Connect(m,n,k,p,p). • Proof: • Assume by contradictory that W has a winning strategy, say S. • B makes the first move at random. • Then, B simply follows S (stealing W’s strategy) which leads to a win. • If the strategy S requires B to place on the squares where B placed earlier, B chooses other squares at random again, instead. • Thus, B still remains following the strategy S to win the game, contradictory to the assumption.

22. Breakaway and Fairness • A breakaway move: (脫離戰場 ) • place stones far away from the major battle field • An initial breakaway move: • The first move by W (after the first move by B) is also a breakaway move. • Fairness and Breakaway: • If W makes an initial breakaway move without a penalty, then the game is played like Connect(k,p,p) with W playing first.  Such games are monotonically unfair or

23. Breakaway of Connect6

24. Connect(9,6,4) • Breakaway

25. Empirical Analysis • For most δ = k–p = 3 games, most are empirical unfair according to our experiments. • B (W): Informally proved. FB (FW): Favors Black (White)

26. Maker-Breaker • Strategy stealing argument: [Csirmaz 1980, Pluhar 1994] • Connect(m,n,k,p,p),  monotonically unfair (白不會贏) • So, for combinatorial analysis, some researchers proposed Maker-Breaker Model: • W is not allowed to win. • Theorems: • Let values k and p satisfy a condition, roughly like δ = k − p = O(log2p / log2 log2p). For all q, 1 ≤ q ≤ p, B wins Connect(k,p,q). • Let values k and p satisfy a condition, roughly like δ = k − p = Ω(log2p). For all q, 1 ≤ q ≤ p, both B and W tie for Connect(k,p,q).

27.  (=k-p) 我們證明必為平手  = (log2 p) 其他潛在公平的遊戲 六子棋  = 3 我們分析為對某一方有利 p K子棋的公平性分析 • K子棋的公平性分析 • 尚未解的其他潛在公平K子棋遊戲。

28. Outline • Introduction to Connect6 • Characteristics of Connect6. • Fairness of Connect6 (六子棋) • Compared with Gomoku (五子棋) or Renju (連珠棋). • Threat-based winning strategies for Connect6 players or programs. • Impact of Our Research • Report of some Tournaments • New challenges in the future.

29. Threat-based Strategy (迫著戰術 ) • One has t threats (迫著)  If the other needs t stones to defend from connecting 6. • B wins for 3 threats.

30. List of Double Threats （雙迫著）

31. List of Single Threats （單迫著） Single defense point. Further Attacking.

32. 活三

33. 活二

34. 死三

35. 死三活二 單一子形成「單迫著」 還保留另一「單迫著」 。 也可成活二 （兩子成「雙迫著」）

36. Threat-based Search Strategy迫著為主的搜尋策略 • VCF (Victory by Continuous Fours): 所謂的追四勝 • All double threats. • Well known since Connect6 was introduced. • Double threats mixed with single threats. • New developments.

37. Winning Moves • Winning moves withsingle threats. • Winning moves withdouble threats.

38. Null Move Heuristics • Like null moves in Chinese Chess or Weichi, • Need to find threats. • Need to build a relevance zone. • R1-Zone: For one null move. • R2-Zone: For two null moves. • R3-Zone: For three null moves. • A powerful technique to solve game positions. • Theorem 1: Black wins in Connect(6,2,3). White cannot make a breakaway move. （不能脫離戰場）

39. R2-Zone

40. R1(1a)-Zone

41. Winning Openings for Connect6 • Theorem: B wins for the following openings. (17th popular opening)

42. Key of the Winning Move • This move uses a special null-move technique.

43. More Winning Openings for Connect6 • We have found • At least 5 openings with B winning. • 4 of them are popular openings. • We are verifying them before announcing.

44. Top 15 Popular Openings

45. Top 16-30 Popular Openings

46. Outline • Introduction to Connect6 • Characteristics of Connect6. • Fairness of Connect6 (六子棋) • Compared with Gomoku (五子棋) or Renju (連珠棋). • Threat-based winning strategies for Connect6 players or programs. • Impact of Our Research • Report of some Tournaments • New challenges in the future.

47. 六子棋提出後之後續發展(1) • 被登錄到國際線上維基百科全書。 • 成為電腦賽局競賽項目之一 • 奧林匹亞電腦賽局(Computer Olympiad) • 中國象棋電腦博弈錦標賽暨機器博弈學術研討會(Chinese Computer Games Conferences-CCGC 2006) • 六子棋公開賽之舉辦 • 交通大學盃： • 第一屆(2006):人數86人！ • 第二屆(2007):人數92人！ • 群想盃： • 第一屆(2006):人數約60人！ • 俄羅斯六子棋大賽：Andrey主辦

48. 後續發展(2) • 六子棋遊戲網站：如 • www.cycgame.com （群想遊戲網站；中文語系；即時；有許多台灣中國高手） • littlegolem.net （英文語系；非即時；有許多東歐高手） • pente.com （英文語系；即時） • www.brainking.com （多國語系；即時） • 總人次 > 100,000

49. cycgame.com

50. www.littlegolem.net