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Riddle of 4 persons crossing a bridge over a river at night within 17 minutes

Bridge crossing riddle: 4 persons to cross a bridge in 17 minutes

It's an emergency in bridge crossing riddle - 4 persons have to cross a bridge in 17 mins

Bridge Crossing Riddle: 4 persons to cross a bridge at night in 17 mins. Two persons can cross at a time with crossing speeds of each different. Read on...

The riddle

An aged man, his wife and 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 can cross at a time,
  • With only one torch between them, 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.

Exclude their thinking time from 17 minute time limit.

Can you help them?

Sure you can, because you have 10 minutes to find a safe way to cross.

Best will be to imagine yourself as one of the four.

Solution to Bridge crossing riddle: First stage of analysis and understanding more about what you can and cannot do

This is the stage of Problem specification and Initial mental trial.

You won't waste time in random thinking. You will solve by forming the first conclusions to pave the way for the next steps,

Conclusion 1: After each crossing, one will remain on the safe side and another will return with the torch to the danger side.

Conclusion 2: As they are four, they can cross the bridge in minimum 3 forward crossing journeys and 2 return journeys (in the last forward journey, 2 members will cross over).

Conclusion 3: Total time for a journey is same as 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 you do for the family to cross in 17 minutes?

Without going too deep, you make a simple mental trial.

You assume total time taken should be minimum if the quickest 1 minute member crosses over with each of the other three in three forward journeys and returns with the torch twice.

This scheme attempts to minimize the return journey time with no consideration for the forward journeys.

Adding up the 2, 5 and 10 minutes for the three forward journeys and further adding 2 minutes for the two return journeys of the 1 minute member, you are disappointed to find the total time as,

$2 + 5 + 10 + 2 = 19$ minutes.

It will overshoot the 17 minute time limit by 2 minutes. This won’t work.

The approach must be more innovative.

Solution to bridge crossing riddle: Second stage of finding the optimum member combinations for the safe crossing

You will find the optimum member combinations for the safe crossing by repeated question, analysis, answer and deductive chain of reasoning. This is the powerful QAA technique.

With more organized thought, you ask yourself,

Critical question 1: Which combination of the two members together on a journey will SAVE MAXIMUM TIME in a crossing?

This approach is different. It attempts to find the optimum member combination. Solution should include the optimum member combination with maximum time saving. It is a reasonable assumption.

And it is easy to find the journey combination with maximum time saving,

Conclusion 4: If 5 minute and 10 minute members cross together and 5 minute member comes back with the torch, onward journey time saving is maximum (as much as 10 minutes) but time wasted in the return journey will also be large. If you can minimize this wastage, you would have your solution of safe passage.

To use this maximum time saving forward journey combination then, you ask the next critical question,

Critical question 2: How to reduce the return journey time even when 5 and 10 crossed together in the forward journey?

Answer to this question turned out to be unexpected,

Conclusion 5: Of course, 5 and 10 have to cross over together because that would save maximum forward journey time (of 10 minutes). What if the 1 minute member had already crossed over and now would return with the torch! This is sure to save maximum time on the backward journey also!

Conclusion 6: So, 5 and 10 must cross over in the second journey. This cannot be the third journey because then either 5 or 10 would have had to return with the torch on an earlier journey that would be wasteful.

This is the key breakthrough made by a series of revealing questions and the answers.

Deciding now on this combination, the next conclusions follow,

Conclusion 7: 1 and 2 would cross over together on the first journey, 1 stays back and 2 returns with the torch.

Solution is easy now. Chalk out the details of the first journey in the solution as,,

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 will be,

Second forward and return journey: 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 will 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.

The four member family also could cross over away from the danger with success because they could think through the problem with systematic and innovative approach. Congratulations to them!

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