AE1APS Algorithmic Problem Solving
John Drake
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Invariants – Chapter 2
River Crossing – Chapter 3
Logic Puzzles – Chapter 5
Matchstick Games - Chapter 4
◦ Matchstick Games – Winning Strategies
◦ Subtraction-set Games
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Sum Games – Chapter 4
Induction – Chapter 6
Tower of Hanoi – Chapter 8
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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
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We will work with matchstick games
We identify winning and losing positions in
a game
A winning strategy is therefore maintaining
an invariant
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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
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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.
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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).
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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?
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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
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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)
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Node 0 is losing, as there are no edges
from it
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Nodes 1 and 2 are winning, as there is an
edge to node 0
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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
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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
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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
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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.
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Some variations on the matchstick game:
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There is one pile of matches, each player is allowed
to remove 1, 3, 4 matches
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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.?
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What are the winning positions and what
are the winning strategies?
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{1, 3, 4} Subtraction subset
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We can remove 1 or 3 or 4 matches.
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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.
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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.