One big monkey, two small monkeys and three humans are to cross a river in a boat
Three humans, one big monkey and two small monkeys are to cross the river in a boat that can carry at most two.
Only the humans and the big monkey can row the boat.
Most alarmingly—at all times, number of monkeys must not become more than the number of humans on any bank of the river, because in that case the monkeys will eat up the humans!
Question: How quickly in minimum number of boat trips can the six cross the river safely?
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?
There is no recommended time to solve the puzzle—you may try at your heart's content whenever you feel like. It'll surely be fun.
We'll explain the solution using analytical reasoning approach.
Solution to the puzzle: 3 monkeys and 3 humans crossing a river
Our natural tendency is to approach solving a problem or puzzle 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: which is the biggest hurdle or constraint in solving 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 are,
- The boat can hold only 2 members in a trip—you have to 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 would generally try to carry 2 passengers in 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 feel 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's a matter of life and death. It will help you not only plan the current trip, but also planning for the return journey and the next trip.
Now as we know the relative importance of the constraints clearly, let's exercise our brains further to flesh out the actions that we can or must take by analyzing the task of planning the minimum number of steps required.
Second stage analysis: Planning for the first trip
As ferrying two small monkeys across is an essential and important task, let's decide the best way is to start with 1 small monkey in the first trip, as,
Leaving the small monkey with no humans on the other bank won't cause any problems as well as the strength of the monkeys in numbers on the first bank will be reduced thus increasing degree of safety of the humans on this bank.
What about the rower? Whom would you assign the task of rowing the small monkey across and then coming back alone?
There are two options, either the big monkey or a human can take up the task.
We would 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 boat will reduce the number strength of monkeys on the first bank (the big monkey has to return alone).
There being no other active constraint in this first trip, this seems to be the most reasonable decision.
So be it.
First trip: The big monkey with a small monkey crosses over and after dumping the small monkey on the other bank, 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 and is getting down.
Third stage analysis: Planning for the second trip
Now what to do? You cannot go on increasing the number of monkeys to TWO on the other bank, because a point of time surely would come soon when 1 human has to be ferried across and as soon as he gets down he would immediately be eaten up (there being 2 monkeys on this bank in this scenario).
We have used what we call, consequence analysis—foreseeing the consequence of ferrying a second small monkey across in the second trip.
In short, 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.
This is deductive reasoning based on reasoning and problem solving techniques.
So we know for sure that in 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 option now.
Two humans can't leave the bank leaving the 3rd human alone with two monkeys still present.
The big monkey must take the first human to the other bank in the second trip and after leaving the human on the other bank, return back to the first bank.
This settles the second trip details.
Observe that load balancing of numbers of monkeys and humans is active not only on two banks, but as a consequence 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 rows back 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 ready to get down.
Fourth stage analysis: 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 would result in outnumbering of humans on the other shore.
Certain conclusion is—one human must be one of the two passengers in 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 would be able to 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 option of putting both humans on the boat for the third trip.
Third trip: Two humans cross over, one human calmly gets down 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: Planning for the fourth and subsequent trips for the solution
As we have told you, from this point onward, it is easy.
The human rowing back cannot get down. He will immediately 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 down 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, where both get down.
So fourth trip details would be,
Fourth trip: The 3rd human ferries the big monkey across, puts him down 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 down. Story over with all happy.
The pictorial representation of the five trips is shown below.
BM stands for Big Monkey, SM for Small Monkey and HM? Must be for human, isn't it?
Minimum number of trips for successful crossing must be 5. This is because, 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 a kind of animals, isn't it?).
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 a different passenger combinations.
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