4 Games and Fuses
• U:\APPS\Ghostgum\gsview
• Shared folder U:\2nd Year Share\Computer Science
Courses\Semester One\Algortithmic Problem Solving
• Book: Algorithmic Problem Solving by Roland Backhouse.
• Other books: any puzzle book or you can Google logic puzzles,
problems, brainteasers, conundrums. Or search for the content of
the problems e.g. river crossing problems.
• Office hour Tue 130-230, Fri 1130-1230. Office AB 325.
• If you have any “nice” problems please let me have them.
• Any questions, please ask me?
FUSE CLOCKS
• A fuse is lit at one end and will burn for a specific amount
of time i.e. n minutes.
• The rate at which a fuse burns is irregular i.e. it cannot
be lit in the middle with any confidence that the amount
of time taken to burn will be half.
• Two n minute fuses both take n minutes to burn, so the
average rate is the same.
• But at any moment the instantaneous rates may differ.
• A fuse is a length of wire which take a certain amount of
time to burn along its length, not to be confused with an
electrical fuse which is thin and burns out if too many
amps flow thought it.
Example
• With a single n minute fuse, we could light it at
one end and it would take n minutes to burn.
Alternatively we could light it at both ends
simultaneously and it would take n/2 minutes to
burn. Also, given two fuses, we can light the
simultaneous so that they are burning together,
or light one after the other (i.e. we can burn
fuses in serial or partially in parallel). This way
we can create new time periods.
• Imagine we have 2 60 minute fuses. How can
we build a 45 minute fuse?
Solution
• Solution. Light the first fuse at one end
and the second fuse at both ends. When
the second fuse has burnt out, light the
other end of the first fuse. When this has
burnt out it makes a total of 45 minutes.
How many
• How many other clocks can you make with a pair of
fuses?
• Remember you cannot have a negative time period (e.g.
you can burn two fuses in parallel lighting the ends at the
same time, giving n-m minutes but we assume this is a
positive number). Clocks can be constructed by starting
one clock after another, starting two clocks together but
starting the timing when one clock has completed. We
can also halve the time of a clock by lighting it at both
ends. These three construction techniques correspond to
addition, subtraction and halving of times.
HOURCLASS CLOCKS
• An hour glass is a glass timer which contains
sand.
• The sand runs under gravity from one end to the
other.
• Unlike a fuse it can be reused (i.e. it can be
turned upside down at any point).
• In this game you are given 2 hourglasses one of
7 minutes and the other of 11 minutes. Your task
is to use these 2 hourglasses to measure times
of 15 minutes and 24 minutes. Can you do that?
Simple two player games.
• The goal is to have some method (i.e. algorithm)
do decide what to do next.
• The key to winning is to recognise the invariants.
• We introduce a number of matchstick games.
• We identify winning and losing positions in a
game.
• A winning strategy is therefore maintaining an
invariant.
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 lasts matches
is the winner.
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
• 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
• Suppose there is one pile of matches, and
you can remove 1 or 2 matches.
• How do we identify winning and losing
positions?
• The losing positions are a multiple of 3.
• The remaining positions are winning
positions.
Some Variations
• Some variations on the matchstick game.
• There is one pile of matches, each player is allowed to
remove 1, 3, 4 matches.
• What are the winning positions and what are the winning
strategies.
• 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.
winning strategies
• Draw a state transition diagram p44
• Nodes are labelled with the number of
matches remaining.
• Edges are labelled with the number of
matches removed on that turn.
• We can now label the nodes as winning or
losing.
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).
• Note that this seems like a circular definition
• 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.
• Draw thick lines indicating winning positions and winning moves.
P45
• A clear pattern emerges; losing positions are where the number of
matches is a multiple of 3.
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.
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. Think of the situation is OXO.
Formally
• The terminology we use to describe the winning
strategy is to “maintain invariant” property that
the number of matches is a multiple of 3.
• Formally we can write this as
•
(the number of moves remaining before the
game ends is even)
• equals
• (the position is a losing position)
• I.e. we alternate between winning and losing
states or positions.
Subtraction Subset
• {1, 3, 4} Subtraction subset
• Now we can remove 1 or 3 or 4 matches.
• Go thru the process of labelling states as
winning or losing and build up the table on
page 49.
• Draw the diagram.
Winning Strategy
• Calculate the remainder r after dividing by
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.
• Suppose a daisy (a flower) has 16 petals
arranged symmetrically around its center.
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.
The Coin Problem.
• Two players have an unlimited supply of
circular coins of varying diameter (both
players have the same set of coins). The
players take it in turn to place a coin on a
rectangular table, such that it does not
overlap any coins already on the table and
does not overhang. The winner is the one
who puts the last coin on the table. Who
should win and what is the winning
strategy?
EXPLORER 1
• An explorer walk one mile south, one mile
east and one mile north, but find that he
has returned to the point where he started.
Why is this? Where was he standing?
Explorer 2
• The point of this problem is that we may
find one solution then stop, but in fact
there is an infinite set of solutions. Also
ask why it is colder at the poles than at the
equator (there are two reasons at least1.
further away, 2, light glances 3 light has
further to travel thru the atmosphere.)
COLOURING THE FACES OF A
CUBE
• Colouring cubes
• How many different ways are there to
colour a cube.
• I think there are too many just to count, so
you need a more principled way to get the
solution.