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} //state 3H|2W|1W//action {3H||3W}//state

Problem Decomposition {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 {3C||} 1C, 2H | 2W| {1C, 2H||2W} 1C, 2H|W|W {2C, 1H||1W} 3H|2W|1W {3H||3W}

S2 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 {3H||3W} 3H|1W|2W {1C, 2H||2W} 1C|2H|2W {1C||2C} And T2 is just the reverse.