River crossing puzzle 3 monkeys and 3 humans to cross a river
3 monkeys and 3 humans are to cross a river in a small boat. The boat can carry at most 2 on a trip. Monkeys will eat up humans if more in number.
Only the humans and a big monkey can row the boat and several trips permitted. The two small monkeys can't row.
Question: How can the six cross the river safely in least number of boat trips?
And also, if you are still interested, a bonus question for you,
Bonus question: In how many ways (trip combinations) can the six cross the river?
No recommended time to solve the puzzle.
We’ll explain the solution using analytical reasoning approach.
Solution to the River crossing puzzle 3 monkeys and 3 humans to cross a river
Our natural tendency is to approach solving a problem in a random way. Try to avoid it and think analytically and systematically.
Let’s look at the puzzle using logic and reason, step by step.
First stage analysis of River crossing puzzle 3 monkeys and 3 humans: which is the biggest constraint in solving the puzzle?
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 hold only 2 members on a trip. Remember this physical constraint and take it into account when planning the trips, sure. But you cannot manipulate this constraint. Its use is thus low level. In fact, to minimize the number of trips, you will try to carry 2 passengers on every trip.
- Only humans and the big monkey can row the boat—again this constraint must be kept in mind and used carefully for planning the trips. But do you think it’s a critical constraint? The main contribution of this constraint is to identify the fact that, to ferry the two small monkeys, 2 trips are certainly required.
- At all times, number of monkeys must not become more than the number of humans on any bank of the river. Otherwise, the monkeys will eat up the humans! This must be the most critical constraint—it is a matter of life and death. It will help you not only plan the current trip, but also plan for the return journey and the next trip.
Now, let us plan for the least number of steps required.
Second stage analysis of River crossing puzzle 3 monkeys and 3 humans: Planning for the first trip
Two small monkeys are to be ferried across on two trips—they can’t row. So best way to start is to take 1 small monkey on the first trip itself.
Leaving the small monkey with no humans on the other bank won’t cause any problems. The number strength of the monkeys on the first bank will be reduced and degree of safety of the humans on this bank increased.
What about the rower? Whom will you assign rowing the small monkey across and then returning alone?
There are two options, either the big monkey or a human can take up the task.
We will prefer the big monkey, primarily because we are always of the opinion that the inferior animal should do the bulk of labor.
Jokes apart, putting the big monkey on the first trip will reduce the number strength of monkeys on each bank (the big monkey has to return alone).
There being no other active constraint in this first trip, this should be the most reasonable decision.
So be it.
First trip: The big monkey and a small monkey cross the river. After dumping the small monkey on the other bank, the big monkey rows back to the first bank.
Status review after the first trip:
On the first bank, you have 3 humans and 1 small monkey and on the second bank just 1 small monkey. The big monkey has just reached the first bank in the boat.
Third stage analysis of River crossing puzzle 3 monkeys 3 humans: Planning for the second trip
You cannot go on increasing the number of monkeys to two on the other bank.
A time will come soon when 1 human is to be ferried across. With 2 monkeys on this bank, he will be eaten up as soon as he gets off the boat.
We have used consequence analysis—foreseeing the consequence of ferrying a second small monkey across in the second trip.
We must go on BALANCING THE NUMBER OF HUMANS AND THE NUMBER OF MONKEYS on both sides of the river. To do that,
The second trip must include a human.
Not only have we used consequence analysis technique, we have also used critical load balancing strategy, critical load being relative number strength of the humans and monkeys on either shore.
So we know for sure that on the second trip, a human has to be put across the river on the opposite bank.
Now to decide, who is to take the human across? You have only one choice now.
Two humans can’t leave the bank. The 3rd human will be alone with two monkeys still present.
The big monkey must take the first human to the other bank on the second trip and after leaving the human on the other bank, return to the first bank.
This settles the second trip details.
See that load balancing of numbers of monkeys and humans is active not only on two banks, but on the boat as well.
Second trip: The big monkey and one human cross the river and leaving the human on the other bank the big monkey returns to the first bank.
Status review after the second trip:
On the first bank, you have 2 humans and 1 small monkey and on the second bank 1 human and 1 small monkey. The big monkey has reached the first bank in the boat.
Fourth stage analysis of River crossing puzzle 3 monkeys 3 humans: Planning for the third trip
This is the critical step.
It is for sure that 1 small monkey cannot be allowed to be ferried across—that will cause outnumbering of humans on the other bank.
Certain conclusion is—one human must be one of the two passengers on the third trip. This is the only way to balance the critical load on both the banks.
Question is—who will be the co-passenger?
We are sure that from this point on you could complete the rest of the steps to the solution.
So stop now and try to solve from here.
Okay, hope you have tried.
Let’s take up the reasoning thread again.
You cannot leave the lone human with two monkeys on the first bank. So you have only one choice of putting both humans on the boat for the third trip.
Third trip: Two humans cross over, one human calmly gets off, outnumbering the lone small monkey on the other bank and the second human rows back. Simple.
Status review after third trip:
On the first bank now you have the big monkey and 1 small monkey, on the opposite bank 2 humans and 1 small monkey with the 3rd human just reaching the first bank in the boat.
Fifth stage analysis of River crossing puzzle 3 monkeys 3 humans: Planning for the fourth and later trips for the solution
As we have told you, from this point onward, it is easy.
The human rowing back cannot get off the boat. He will at once be outnumbered.
Now he must carry out the labor of transporting either the small monkey or the big monkey across, say the big monkey to the other bank. Putting the big monkey safely on the opposite bank, the 3rd human rows back to the first bank to ferry the last monkey left, the small monkey, to the opposite bank. Both get off the boat.
So fourth trip details will be,
Fourth trip: The 3rd human ferries the big monkey across, puts him on the other bank and rows back to the first bank.
Status review after fourth trip:
On the first bank, the hapless small monkey waits alone, on the second bank 2 monkeys and 2 humans coexist amicably and the 3rd human reaching the first bank.
Fifth and last trip: Reaching the first bank, the 3rd human now picks up the small monkey lovingly, rows across with him and both get off the boat. Story over with all happy.
The pictorial representation of the five trips is shown.
BM stands for Big Monkey, SM for Small Monkey and HM? Must be for human, isn’t it?
Least number of trips for successful crossing must be 5: On each of the first four trips only one animal could be left on the second bank so that the second animal that traveled with it can row back. Last two animals cross the river on the fifth trip.
Note that each trip carried full capacity of 2 animals (well, humans are also an animal, isn’t it?).
End note
Carefully go through the chain of reasoning and try to disprove, improve or absorb. Also, if you are still interested, try to find whether you can do the job using different passenger combinations.
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