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Matchstick Games

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  1. Matchstick Games AE1APS Algorithmic Problem Solving John Drake

  2. Outline of the course • Invariants – Chapter 2 • River Crossing – Chapter 3 • Logic Puzzles – Chapter 5 • Matchstick Games - Chapter 4 • Matchstick Games – Winning Strategies • Subtraction-set Games • Sum Games – Chapter 4 • Induction – Chapter 6 • Tower of Hanoi – Chapter 8

  3. Simple two player games • The goal is to have some method (i.e. algorithm) do decide what to do next so that the eventual outcome is a win • The key to winning is to recognise the invariants • Remember - An invariant is something that does not change

  4. Simple two player games • We will work with matchstick games • We identify winning and losing positions in a game • A winning strategy is therefore maintaining an invariant

  5. Matchstick Games

  6. Matchstick Games • Played with one or more piles of matches • Two players make alternate moves • A player can remove one or more matches from one of the piles, according to a given rule • The game ends when there are no more matches to be removed • The player who cannot take any matches is the loser, i.e. the player who took the last match(es) is the winner

  7. Terminology • This is an impartial, two person game with complete information. • Impartial means rules for moving apply the same to both players. • Complete information means that both players have complete information about the game i.e. they know the complete state of the game. • An impartial game that is guaranteed to terminate, it is always possible to characterise the positions as winning or losing positions.

  8. Winning and Losing Moves • A winning position is one from which we can assure a win. • A losing position is one from which we can never win. • A winning strategy is an algorithm for choosing moves from winning positions that guarantees a win (i.e. we maintain an invariant).

  9. Identify Positions • Suppose there is one pile of matches, and a player can remove either 1 or 2 matches • How do we identify winning and losing positions?

  10. Winning Strategies • Draw a state transition diagram (p.70) • Nodes are labelled with the number of matches remaining • Edges define the transition of state when a number of matches removed on that turn • We can now label the nodes as winning or losing

  11. State Diagram • A node is winning if there is an edge to a losing position • A node is losing if every edge from the node leads to a winning node (i.e. we cannot escape from the losing situation)

  12. State Diagram • Node 0 is losing, as there are no edges from it • Nodes 1 and 2 are winning, as there is an edge to node 0 • Node 3 is losing, as both edges from 3 are to nodes 1 and 2 which are already labelled as winning • A clear pattern emerges; losing positions are where the number of matches is a multiple of 3

  13. Winning Strategy • Beginning from a state in which n is a multiple of 3, and making and arbitrary move, results in a state in which n is not a multiple of 3. Thus removing n mod 3 matches results in a state in which n is again a multiple of 3. NB: Only labelled to 7

  14. Formally • The terminology we use to describe the winning strategy is to maintain invariant property that the number of matches remaining is a multiple of 3

  15. Initial Situation • If both players are perfect, the winner is decided by the starting position. If the starting position is a losing position, the second player is guaranteed to win. Starting from a losing position, you can only hope that your opponent makes a mistake, and puts you in a winning position.

  16. Some Variations • Some variations on the matchstick game: • There is one pile of matches, each player is allowed to remove 1, 3, 4 matches • There is one pile of matches, each player is allowed to remove 1, 3, 4 matches, except that you are not allowed to repeat the last move. So if you opponent removes 1 match you must remove 3 or 4. • What are the winning positions and what are the winning strategies.?

  17. Subtraction Subset • What are the winning positions and what are the winning strategies? • {1, 3, 4} Subtraction subset • We can remove 1 or 3 or 4 matches.

  18. Winning Strategy • Calculate the remainder r after dividing by 7i.e. mod 7 • If r is 0 or 2, the position is a losing position. Otherwise it is a winning position. • The winning strategy is to remove 1 match if r=1, • Remove 3 matches if r=3 or r=5, remove 4 matches if r = 4 or r =6.

  19. The Daisy Problem. • Suppose a daisy (a flower) has 16 petals arranged symmetrically around its centre. Two players take it in turns to remove petals. A move means taking one petal or two adjacent petals. The winner is the person who removes the last petal. Who should win and what is the winning strategy.