## 3 couple river crossing puzzle needs systematic and reasoned thinking

3 couple river crossing puzzle: King, minister, washerman and their wives to cross river in a boat that can carry only 2 at a time. A tantalizing puzzle.

### The puzzle

A King, his minister and washerman have to cross a river with their wives in a boat that can carry at most 2 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 of the six can row the boat.

**Question:** How quickly in minimum number of boat trips can the six cross the river without violating any restriction?

And also, if you are interested, a bonus question for you,

**Bonus question:** In how many different ways (trip combinations) can the six cross the river?

There is no recommended time to solve the puzzle.

Please try to solve before going ahead.

We'll explain the solution using analytical reasoning approach.

### Solution to 3 couple river crossing puzzle

Let's look at the puzzle using logic and reason, step by step.

#### First stage analysis: 3 couple river crossing puzzle: Constraint analysis and identifying the core problem in the puzzle

In any problem solving, it is very important to list out all the constraints that you can find out and identify the most important one. In our puzzle the constraints or restrictions are,

- The boat can carry at most 2 passengers in 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 minimize the number of trips, you would generally try to carry 2 passengers in 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 in fact may help you to identify best possible course of action quickly.* - 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 it more difficult for trip planning.
*These two constraints analyzed together should help in identifying, what we call, the core problem in the puzzle.*

The **favorable condition** is—each of the six 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 would travel with any of them.**

It follows that,

Either the washerman and washerwoman must travel together or travel alone.

Now with this knowledge, we are ready to explore further how to transport this couple to the other bank.

#### Second stage analysis: 3 couple 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 cross-over at the first trip itself, and dropping the wife, the washerman comes back!

Yes, it is feasible, **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 cross-over with his queen and return after dropping her on the other side.

**Step 2.** King and queen cross-over 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 males on the first bank and three females on the opposite bank.

What next?

#### Third and final stage analysis: 3 couple river crossing puzzle: Deciding the last two trips

Till now in three trips 3 wives have been transported across without violating any conditions.

In the **critical fourth trip**, the **only option available is for the king and minister to cross the river together**. After all, none of them would travel with the washerman.

Naturally, it is insensible that after reaching the opposite bank one of the two would 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 down, 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 at any time 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 couple 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 and is getting down.

**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 ready to get down.

**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 queen and minister with his wife. The washerwoman just reached the first bank.

**Fifth and last trip:** Reaching the first bank, the washerwoman now picks up her husband with a sigh of relief, rows across with him and both get down. No condition is violated.

The pictorial representation of the five trips is shown below.

**Minimum number of trips for successful crossing must be 5.** This is because, 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 across carried full capacity of 2 persons.

### End note

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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