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Matchstick Games. AE1APS Algorithmic Problem Solving John Drake. 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

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matchstick games

Matchstick Games

AE1APS Algorithmic Problem Solving

John Drake

outline of the course
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
simple two player games
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
simple two player games1
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
matchstick games2
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
terminology
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.
winning and losing moves
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).
identify positions
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?
winning strategies
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
state diagram
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)
state diagram1
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
winning strategy
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

formally
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
initial situation
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.
some variations
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.?
subtraction subset
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.
winning strategy1
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.
the daisy problem
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.
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