1. AE1APS Algorithmic Problem Solving John Drake

2.  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.  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.  We will work with matchstick games  We identify winning and losing positions in a game  A winning strategy is therefore maintaining an invariant

6.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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 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.  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.  Calculate the remainder r after dividing by 7 i.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.  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.