Four persons trying to cross the bridge in 17 minutes must think systematically to find the innovative way
An aged man, his wife with their daughter and son have to cross to the other side of a deep river at night over a frail old bridge.
One of them takes 1 minute to cross while second one takes 2 minutes, third takes 5 minutes and the fourth takes 10 minutes. They have to think hard for a safe way to cross because,
- Only two at a time can cross,
- They have only one torch between them so that one member must return with the torch for the next crossing, and,
- Unless all four of them safely cross over to the other side within 17 minutes, they will be in very great danger.
Their thinking time is excluded from 17 minute time limit.
Can you help them?
Surely you can, because you have 10 minutes to find a safe way to cross.
Best would be to imagine yourself as one of the four.
Solution: First stage of analysis and understanding more about what you can and cannot do - Problem specification and Initial mental trial
You don't waste time in thinking randomly and start solving by forming the first conclusions that would help in next steps,
Each time after two of you cross over, one would remain there and the other would have to return with the torch to the danger side.
As you are 4, there have to be 3 forward crossing journeys and 3 return journeys.
Time taken in a journey would be the time taken by the slower member. For example, 2 minute and 5 minute members would take 5 minutes to cross together.
This is the problem specification of constraints.
What can be the first thought for a safe crossing?
This is where you try out an apparently promising course of action that satisfies all the conditions except the time limit. This is your initial mental trial or experiment.
If the quickest member who takes 1 minute, crosses over with each of the other three in three journeys and comes back with the torch twice, total time taken should be the minimum.
Adding up the 2, 5 and 10 and further adding 2 minutes for the two return journeys of the 1 minute member, you are disappointed to find the total time to be,
2 + 5 + 10 + 2 = 19 minutes.
It would take 2 minutes more than 17 minute time limit. This option is invalid.
The approach must be more innovative.
Solution: Second stage of finding the safe member combinations for the journeys by repeated question, analysis, answer and deductive chain of reasoning
Thinking systematically, you ask yourself first,
Which combination of the two members together in a journey would save maximum time in a crossing?
This is a critical and natural question to ask at this stage.
Instead of analyzing time taken by ALL THE JOURNEYS TOGETHER, ANALYSIS IS BROKEN DOWN TO SINGLE JOURNEYS.
This is again natural—the problem breakdown technique.
And while you analyze, you think of maximum benefits that a combination for a journey can have.
It is easy to find the journey combination with maximum time saving,
If 5 and 10 cross together and 5 comes back with the torch, onward journey time saving is maximum but time wasted in the return journey would also be large.
Realizing that, for safe crossing you have to use this maximum time saving forward journey combination, you ask the next critical question,
How to reduce the return journey time even when 5 and 10 crossed together in the forward journey?
You are now a little surprised to find the answer to this question to be something you didn't think of in the beginning,
Of course 5 and 10 have to cross over together because that would save maximum forward journey time. What if 1 had already crossed over and now would take back the torch! This would surely save maximum possible time on the backward journey also!
And so 5 and 10 must cross over in the second journey. It cannot be the third journey because in that case either 5 or 10 would have had to return with the torch in an earlier journey that must be avoided.
This is the key breakthrough. You have made this breakthrough by,
Raising a series of important revealing questions and finding the answers easily because focus of thinking is split up into small pieces by the chain of reasoning.
This is deductive chain of reasoning.
Deciding now on this combination confidently, the next conclusions automatically follow,
1 and 2 would cross over together in the first journey, 1 stays back and 2 returns with the torch.
Mentally forming the details of the first journey you get,
First forward and return journey: 1 and 2 cross over together, 1 stays back, 2 comes back with the torch. Total time taken 4 minutes.
Danger side has: 2, 5 and 10. Safe side has: 1.
It is time to send 5 and 10 together on the second journey as planned.
Result after this planned second journey would be,
Second forward and return jouney: 5 and 10 cross over together, both stay back, 1 returns with the torch. Total time taken: 11 minutes. Cumulative time elapsed: 15 minutes.
Danger side has: 1 and 2. Safe side has: 5 and 10.
And result after the third journey of 1 and 2 crossing over would be,
Third forward journey: 2 was waiting on the danger side, 1 returns with the torch, and 1 and 2 cross over safely in 2 minutes with cumulative total time of 17 minutes.
This is your solution for the family for their safe crossing over.
It is good yo know that all four members of the family could actually cross over safely away from the danger because they could think through the problem systematically and innovatively.
If someone is under great pressure for solving a problem, mind can do wonders!
Puzzles you may enjoy
Riddle of 4 persons crossing a bridge over a river at night