Transcript Three jealous couples

Three jealous couples
Problem statement
• Three couples (husband and wife) wish to
cross a river. They have only one boat that can
carry at most 2 people. The husbands are so
jealous that none is willing to allow their wife
to be with another man if they themselves are
not present.
• How can all 3 couples get across the river.
Symmetries
• There are a couple of symmetries that might
help us solve the problem.
• We could get each couple across individually.
• We could get all the women across first.
• We could get all the men across first.
Notation
• We use H, W, C to denote husband, wife and
couple (H+W).
• 2H means two husbands
• 1C, 2H means one couple and two husbands.
• 1H, 1W mean a husband and wife who are not
a couple.
• Note we do not name people e.g. Alice, Bob,
Clare… as it is only the number that is relevant
States
• A state describes a situation where each
person is on the left or right river bank.
• 3H||3W means 3 husbands on left bank and 3
wives on right bank
• 1C, 2H || 2W denotes one couple and two
husbands on the left and two wives on the
right.
Action
• An action is a state transition (verb)
• An action is a person(s) being transported
across the river.
• 3H|2W|1W means two wives are crossing the
river.
• Not that this does not say if the boat is going
left or right
• Is this abiguous.
Invalid States
• This notation allows valid and invalid states to
be identified easily.
• 1C, 1W||1C, 1H is invalid, why?
• 3H|3W| is invalid, why?
Problem statement
• Start state 3C||
• Final state ||3C
• A solution is sequence of actions from the
start state to the final state.
• {3C||}
• S?
• {||3C}
Sequence of actions
• An action results in a state change.
• If p and q denote states, and S is a sequence
of actions
• {p}
• S
• {q}
• A sequence of actions S will take us from state
p to state q.
Sequence of actions e.g.
• {2C, 1H||1W}
• 3H|2W|1W
• {3H||3W}
//state
//action
//state
Problem Decomposition
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{3C||} S0 {||3C} //start
We can decompose this into the following.
{3C||} S1 {3H||3W},
{3H||3W} S2 {3W||3H},
{3W||3H}S3 {||3C} ,
What do you notice about S1 and S3
S1
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{3C||}
1C, 2H | 2W|
{1C, 2H||2W}
1C, 2H|W|W
{2C, 1H||1W}
3H|2W|1W
{3H||3W}
S2
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We can break S2 down into two parts
{3H||3W}
T1
{1C|1C|1C}
T2
{3W||3H}
What can we say about T1 and T2.
T1
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{3H||3W}
3H|1W|2W
{1C, 2H||2W}
1C|2H|2W
{1C||2C}
And T2 is just the reverse.