3 couples river crossing puzzle needs systematic and reasoned thinking
3 couples to cross a river in a boat that can carry at most two at a time. Husbands are jealous and no lady will be with a male other than her husband...
A King, his minister and washerman have to cross a river with their wives in a boat that can carry at most two at a time. The husbands are jealous and without her husband, no lady can be left on any side of the river with any other male.
To make matters more complicated, the washerman or washerwoman can travel together, but not with any of the other two couples.
Everyone can row the boat.
Question: How quickly in least number of boat trips can the three couples cross the river violating no restriction?
Recommended time to solve: 30 minutes.
If you are interested, a bonus question for you,
Bonus question: In how many ways (trip combinations) can the three couples cross the river?
No recommended time to solve this part of the puzzle.
Try to solve before going ahead.
We’ll explain the solution using analytical reasoning approach.
Solution to 3 couples river crossing puzzle
Let’s look at the puzzle using logic and reason, step by step.
First stage analysis: 3 couples river crossing puzzle: Constraint analysis and identifying the core problem
In solving any problem, it is very important to list out all the constraints and identify the most important one. Here, the constraints are,
- The boat can carry at most 2 passengers on a trip—you have to remember this physical constraint and take it into account when planning the trips. But you cannot manipulate this constraint. Its use is thus low level. In fact, to reduce the number of trips, you might decide to carry 2 passengers on every trip.
- No family member of king and minister will travel with the washerman or his wife—this restricts your options and so also the possible passenger combinations in the trips. This constraint may help you identify best possible course of action.
- Without her husband, no lady can be left on any side of the river with other male(s). This constraint, along with the second constraint, makes trip planning more difficult. These two constraints analyzed together should help in identifying the core problem in the puzzle.
The favorable condition is—everyone can row the boat.
The core problem in the puzzle is the most difficult task to carry out.
What is the most difficult task in ferrying the six passengers? Whom to take across is most difficult?
You might have guessed—transporting the washerman and the washerwoman to the other side of the river is the most troublesome task, as none of the other four will travel with any of them.
Either the washerman and washerwoman must travel together or travel alone.
With this knowledge, we are ready to explore further how to transport this couple to the other bank.
Second stage analysis: 3 couples river crossing puzzle: Chain of Reasoning towards solving the core problem of transporting the washerman and his wife
Step 1: If it is the most difficult task, why won’t they crossover at the first trip itself, and dropping the wife, the washerman comes back!
Yes, it can be done, as the ladies can stay together on the opposite side of the river.
This reasonable first step clears up next steps that must be taken. As the ladies can stay together, next, the king will crossover with his queen and return after dropping her on the other side.
Step 2: King and queen crossover and king returns after leaving the queen on the other side. Following in the footsteps of the king, the minister also crosses over with his wife next and leaving his wife on the other side returns to the first bank.
So far, so good. Three men on the first bank and three women on the opposite bank.
Third and final stage analysis: 3 couples river crossing puzzle: Deciding the last two trips
Till now on three trips, 3 wives have been transported across violating no conditions.
On the critical fourth trip, the only choice available is for the king and minister to cross the river together. After all, none of them will travel with the washerman.
Naturally, it is insensible that after reaching the opposite bank one of the two will return to the first bank. Then who would return with the boat?
The solution is waiting to be seen.
As the king and minister get off the boat, the washerwoman boards the boat, rows back to the first bank, picks up her husband and happily crosses over to the opposite bank.
No condition is violated, as all the pieces of the puzzle fitted together perfectly.
Let’s jot down the trip details and status after each trip.
Trip details and status review after each trip: 3 couples river crossing puzzle
First trip: The washerman with his wife crosses over and after leaving his wife on the other bank, rows back to the first bank.
Status review after the first trip:
On the first bank, you have King and minister couples and washerwoman on the opposite bank. The washerman has just reached the first bank in the boat.
Second trip: King takes queen and rows across. Leaving the queen on the other bank he rows back.
Status review after the second trip:
On the first bank you have minister and his wife, and the washerman. On the second bank are the washerwoman and the queen. The king has reached the first bank in the boat.
Third trip: Minister with his wife crosses over and leaving her on the other bank rows back.
Status review after third trip:
On the first bank now you have the king and washerman, on the opposite bank three wives with the minister just reaching the first bank in the boat.
Fourth trip: The minister ferries the king across, both get down on the other bank and this time the washerwoman rows back to the first bank.
Status review after fourth trip:
On the first bank, the washerman waits alone, on the second bank are the king and the minister their wives. The washerwoman just reached the first bank.
Fifth and last trip: Reaching the first bank, the washerwoman picks up her husband with a sigh of relief, rows across with him and both get off the boat. No condition is violated.
The pictorial representation of the five trips is shown.
Least number of trips for successful crossing must be 5: On each of the first four trips only one person could be left on the second bank so that the second person on the trip can row back. Last two persons cross the river on the fifth trip.
Note that each trip from first bank to the second carried full capacity of 2 persons.
Carefully go through the chain of reasoning and try to disprove, improve or absorb. Also, try to solve the puzzle with different passenger combinations.
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