Sums of Games

1. . Sums of Games AE1APS Algorithmic Problem Solving John Drake

2. Sums of games • Previously we introduced matchstick games, using one pile of matches • It is possible to have more than one pile of matches • First a pile must be chosen, then matches may be removed from the chosen pile according to some rule (which may differ from pile to pile) • The game is thus a combination of two games • Combinations of games is such a way are known as sum games

3. In General • We have two games, each with its own rules • Let us call the two games, the left and right games • A position in the sum game is the combination of the position in the left game and the position in the right game • A move in the sum game is a move in one of the left or right games

4. Example – A graph • A graph represents a game. • The positions are represented by nodes. • The moves are represented by edges. • Imagine a coin placed on a node.

5. Example – A graph • A graph represents a game. • The positions are represented by nodes. • The moves are represented by edges. • Imagine a coin placed on a node.

6. Example – A graph • A graph represents a game. • The positions are represented by nodes. • The moves are represented by edges. • Imagine a coin placed on a node.

7. Example – A graph • A graph represents a game. • The positions are represented by nodes. • The moves are represented by edges. • Imagine a coin placed on a node.

8. An example sum game

9. Sum games • In the sum game, two coins are used • One coin being placed over a node in each of the two graphs • A move is then to choose one of the coins, and move it along an edge to another node • A move changes the position of one of the coins

10. An example sum game

11. Labelling Positions in Sum Games • The two games are unstructured, thus we have to use brute force to determine if they are winning or losing positions • The left game has 15 positions, the right game has 11 positions • How many positions are there in the sum game?

12. Computational Effort • There are 15*11 positions (165) • Brute force is not a desirable approach • We now look at how to compute a strategy for the sum of two games • We find that the computation effort needed to find winning and losing states is not the product, but the sum of the computational effort for the individual games.

13. A simple sum game, Matchsticks. • Suppose there are two piles of matches • A move is to choose a pile of matches, and remove at least one match from that pile • The game is lost when a player cannot remove any matches.

14. The single matchstick game • Given a pile of matches, remove at least one • The winning positions are the positions where there is at least one match • The winning strategy is to remove all the matches • The losing position contains no matches • Draw the state transition diagram

15. Sum game strategy • The single game strategy cannot be applied directly to the sum game • Why not?

16. Strategy • Symmetry between left and right allows us to identify a winning strategy. • Let m, n represent the number of matches in the two piles. • In the end m=n=0. • This suggests m=n identifies losing positions. • Choose the larger pile and restore the property that m=n. i.e. maintain the invariant

17. Initial Property • The property n!=m is the precondition for the winning strategy to be applied • Equivalently m<n or n<m. • If m<n , we infer 1 <= n-m <=n • So n-m matches can be removed from the pile with n matches. • After the assignment n:=n-(n-m), the property m=n will hold.

18. More Simple Sum Games • Suppose K matches can be removed (K is fixed) • This restriction disallows some winning moves • We are not allowed to remove m-n matches when K<m-n • The property m=n no longer characterises losing positions.

19. Example K=1 • Example K=1 • Lets set K=1 • (2, 0). • A player is forced to move to (1, 0) and the opponent can then win the game.

20. Symmetry Restrictions • It seems that the strategy of restoring symmetry no longer applies • Worse, if the number of matches we are allowed to remove from the two piles differs

21. Symmetry Restrictions • Even worse, the left and right games may be completely different!!!

22. One Pile game with K • With one pile of matches, where M matches at most can be removed, • The winning strategy is to continually restore the property that the remainder after dividing the number of matches by M+1 is 0 (i.e. m mod (M+1) == 0) • The number, m mod (M+1) == 0, determines if the position is winning or not.

23. Two Pile Game • The one player game suggests the symmetry as: m mod (M+1) = n mod (N+1) • We will prove this in later lectures. • In the final position (0,0), this property is satisfied.