1. Winning Strategies of Games Played with Chips

2. I got a interesting game Now we show the game 1 2 3 4 5 6 7 8 P 1 =4 P 2 =6 P 3 =8 Rule 1: Two players take turns to move one chip to the next empty square to its left. Rule 2: the player who places a chip on square 1 wins.

3. 3 Question Question 1. Is there a general winning strategy for any given position of chips ? Question 2. Is there a general winning strategy when we play with more chips on the strip ? Question 3. Is there a general winning strategy when we play the similar games with different rules ?

4. Four difference games Games A. the player who places a chip on square 1 wins; Games B. the player who places a chip on square 1 loses; Games C. the player who finishes up with chips on 12...k wins; Games D. the player who finishes up with chips on 12...k loses.

5. Notation We define f A (P)=0 if the first player has a winning strategy for Game A and f A (P)=1 otherwise. We also define W A as the set of all winning position arrays for the first player and L A for the second.

6. General idea (picture) Direct graph formed by W A and L A

7. Two useful formulae f A (P)=1 - max{ f A (P[ i ] )|1  i  k} f A (3P 2 …P k ) = f A ((P 2 - 1)(P 3 -1)… (P k - 1)) Using this two formulae, we can compute the value of f A recursively.

8. About Nim Game (N w ) Show the example

9. Game A & N w We define F A (Q)= f A (P) where Q i = P i - i for all i, after cancel all even numbers in Q, we define O be the set of the cardinalities of all identical odd number in Q,and define Φ(O) = b(O 1 ) ⊕ b(O 2 ) ⊕ b(O 3 )… ⊕ b(O r ) f A (P)=F A (Q)=N W (O) 1If every bit of string Φ(O) is 0 0 else Show the example

10. The relation among Games A, B and C f c (P 1 P 2… P k ) = f B ((P 1 +1 )(P 2 +1 ) … (P 3 +1 )) = f A ((P 1 +2 )(P 2 +2 ) … (P 3 +2 ))

11. Game D & N l N w & N l f D (P)=F D (Q)=N l (O)= N W (O) if Q i ＞ 1 for some i ; 1- N W (O) if Q i  1 for all i. The relation between Game D and Nim Game (N l ) (similar to the relation between Game A and Nim Game (N W ))

12. A surprising connection among games A, B, C, D and two types of Nims F A (Q) = F C (Q) = N w (O), F D (Q) = N l (O), F B (Q) = N w (E)