Bridge Over Troubled Students

July 13, 2010 66 comments

There are four men who want to cross a bridge. They all begin on the same side. You have 17 minutes to get all of them across to the other side. It is night. There is one flashlight. A maximum of two people can cross at one time. Any party who crosses, either one or two people, must have the flashlight with them. The flashlight must be walked back and forth, it cannot be thrown, etc. Each man walks at a different speed. A pair must walk together at the rate of the slower man

Man 1: 1 minute to cross
Man 2: 2 minutes to cross
Man 3: 5 minutes to cross
Man 4: 10 minutes to cross

For example, if Man 1 and Man 4 walk across first, 10 minutes have elapsed when they get to the other side of the bridge. If Man 4 returns with the flashlight, a total of 20 minutes have passed, and you have failed the mission.

Communicated by Jeff Miller.
bridge

Combinatorics, Medium

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Warning: Solutions May Be Discussed in the Comments

Written by Steven Miller on March 16, 2011. Reply

Usually not as then people will be able to see the answer, and that removes a lot of the fun.
Written by Steven Miller on March 16, 2011. Reply

For which riddle? Sorry for the delay in answering — the system hadn’t told me there was an attempted post! My email is sjm1 AT williams.edu
Written by Steven Miller on March 16, 2011. Reply

Correct answer! You could also switch who comes back first. I normally don’t post answers as then others see the answers, and when answers are online people tend to go to them first without the joy of trying the riddle. You can always send me answers at sjm1 AT williams.edu.
Written by Fred on July 15, 2011. Reply

that’s so awesome article , thank you 🙂
- Written by Steven Miller on July 18, 2011. Reply
  
  Glad you enjoyed it. //s
Written by malar on July 28, 2011. Reply

as of i know, it takes 19 mins to cross… how is it possible within 17 mins
- Written by Steven Miller on July 28, 2011. Reply
  
  Hint: 5 and 10 need to walk together….
Written by Kofi on September 23, 2011. Reply

Man 1 and Man 2 go 2mins
Man 1 returns 1min
Man 3 and Man 4 go 10min
Man 2 returns 2mins
Man 1 and Man 2 go 2mins

Gives total of 17mins
- Written by Steven Miller on September 23, 2011. Reply
  
  To Kofi (posted on 2011/09/23 at 12:03 pm): correct!
Written by Anonymous on October 11, 2011. Reply

still can not find the answer
- Written by Steven Miller on October 11, 2011. Reply
  
  Email me for a hint/soln
  - Written by Steven Miller on October 30, 2011. Reply
    
    to Anonymous
    123.238.226.142
    Submitted on 2011/10/30 at 5:03 am
    CORRECT, WELL DONE! Not posting as it’s the soln.
Written by jess on November 6, 2011. Reply

It takes 20 minutes! Ok….how do youdo this one?
- Written by Steven Miller on November 6, 2011. Reply
  
  I’ll email you a hint
Written by Soroush on November 11, 2011. Reply

I found out how to do it in 19 minutes… this one is difficult.
Written by Soroush on November 11, 2011. Reply

Oh wait… I just found out how to do it in 18 minutes… but I still can’t figure out how to do it in 17 minutes!
- Written by Steven Miller on November 11, 2011. Reply
  
  email me at [email protected] if you want a hint. ..s
Written by Clay on December 5, 2011. Reply

it took me a while before i got it
- Written by Steven Miller on December 5, 2011. Reply
  
  there’s actually two solutions, but they’re essentially the same
Written by Ron Freeze on December 9, 2011. Reply

Man 1 and 2 go across = 2minutes
Man 1 goes back across = 3minutes
Man 1 and 3 go across = 5minutes
Man 1 goes back across = 6minutes
Man 1 and 4 go across = 16minutes
Everyone is across in 16 minutes
- Written by Steven Miller on December 9, 2011. Reply
  
  You have the wrong values — the third person takes 5 minutes to go over, the fourth takes 10. This gives a total time of 19 minutes I believe.
Written by Steven Miller on December 19, 2011. Reply

unknown (72.45.58.222 Submitted on 2011/12/18 at 2:16 pm): well done, not posting as this is the soln
Written by Steven Miller on January 11, 2012. Reply

J-crew: correct, well done. not posting as it’s the soln. (You can also email solns to be at [email protected]). //s
Written by Anonymous on January 12, 2012. Reply

1 and 2 go…1 comes back…5 and 10 go…2 comes back…1 and 2 go again. 17!
- Written by Steven Miller on January 12, 2012. Reply
  
  To: Anonymous: Submitted on 2012/01/12 at 2:02 am: Correct, well done! Not posting as it’s the soln.
  - Written by Steven Miller on January 12, 2012. Reply
    
    you can do it with binomial coefficients, but this isn’t the right approach (though the answer happens to be close…).
Written by big brain on January 29, 2012. Reply

first number 2 and 5 (5min) number 2 come back (7min)he take number 1 (9min)
number 1 come back (10min) he take number 4 (2omin)

hahhahaha
- Written by Steven Miller on January 29, 2012. Reply
  
  nope, but your second attempt was correct (not posting as its the soln)
Written by awesome on February 3, 2012. Reply

easy, i think
man one 1 min
man two 2 min
man three 5 min
man four 10 min

1 and 2 go, 2 min
1 comes back, 1 min
1 and 3 go, 5 min
1 comes back, 1 min
1 and 4 go, 10 min

in total=19 minutes!!!
- Written by Steven Miller on February 4, 2012. Reply
  
  yes, but there’s a way to do it in just 17 minutes! email me at [email protected] if you want a hint.
Written by Anonymous on February 10, 2012. Reply

im confused
- Written by Steven Miller on February 11, 2012. Reply
  
  email me at [email protected] for a hint ..s
Written by E on April 2, 2012. Reply

1&4 – 10
1 – 11
2&3 – 16
2 – 18
1&2 – 20
Didn’t think I could do that.
- Written by Steven Miller on April 2, 2012. Reply
  
  But this takes 20 minutes — there’s a way to do it in 17. Email me at sjm1 AT williams.edu if you want to chat more. //s
Written by James on April 10, 2012. Reply

wow this is a very hard solution… wish i do really know the answer lol this is pretty HARD
- Written by Steven Miller on April 11, 2012. Reply
  
  email me at sjm1 AT williams.edu — tried emailing y ou but it bounced. //s
Written by Marc on May 6, 2012. Reply

what is the answer to bridge over troubled students and how do you work it out
- Written by Steven Miller on May 6, 2012. Reply
  
  let me send a hint first ..s
Written by Steven Miller on June 1, 2012. Reply

Ritish: correct (and correct on other one); not posting as these are the solns
Written by Chris on June 6, 2012. Reply

The key to this riddle (to me) is the following statement: ‘Any party who crosses, either one or two people, must have the flashlight with them’.

Answer:

First, 4 and 3 cross the bridge. Then, 4 takes the flashlight to go get 1. However, while 4 is crossing to get 1, 1 is walking towards 4 (this is allowed because the stem states that the flashlight must be with one of them, not that they must be right next to each other) and then 1 walks 4 towards the other end (i.e., towards 3). Once, 1 walks 4 to the other side of the bridge, he (man 1) takes the flashlight and walks towards 2, while 2 walks towards 1. When they meet, 2 and 1 walk towards the end of the bridge. This process takes roughly 13 minutes.
- Written by Steven Miller on June 6, 2012. Reply
  
  the flashlight is very weak and only works if you’re right next to it. //s
Written by Katie on June 12, 2012. Reply

This is really confusing!!!
- Written by Steven Miller on June 13, 2012. Reply
  
  I’ll send a hint
Written by Steven Miller on October 31, 2012. Reply

anonymous: correct (sjm1 at williams.edu)
- Written by brenda on December 1, 2012. Reply
  
  plz tell me the ans i can’t figue it out and i want to ask my friends. i can’t ask them without knowing the ans
  - Written by Steven Miller on December 1, 2012. Reply
    
    email to you bounced – email me at sjm1 AT williams.edu
Written by John Jackson on May 27, 2014. Reply

We are really enjoying these but we are stuck here, is there a way to go part of the way across without the flashlight?
- Written by Steven Miller on May 28, 2014. Reply
  
  nope — must go all the way
Written by Steven Miller on August 5, 2014. Reply

[email protected]: your email doesn’t work, your soln is correct.
Written by Dwayne on August 24, 2015. Reply

Do all four have to end on the other side?
- Written by Steven Miller on August 24, 2015. Reply
  
  yes
Written by anonymous on January 26, 2016. Reply

do you also add the time it takes for man x to get back across w/the flashlight?
- Written by Steven Miller on January 26, 2016. Reply
  
  yes
Written by Jayashree on April 4, 2016. Reply

We have a classic problem for children similar to this, in India. It goes like this- a man has to ferry a tiger, a goat and a bundle of hay across a river. He can ferry only one at a time, or the boat would sink. He can’t leave the goat and the tiger together and take the hay, or the tiger would eat the goat. Also , the hay and the goat can’t be left in a bank alone. How would he ferry all three? Children love this one.
- Written by Steven Miller on April 4, 2016. Reply
  
  you eventually realize there are a few very popular problems with many
  variants!
Written by Prakash Rajendran on July 1, 2016. Reply

Thank you for the problem and its solution which I noticed on the comment column. Could you please tell me how to connect to binomial expansion or any algebraic expression.
Thanks
- Written by Steven Miller on July 3, 2016. Reply
  
  Sure — drop me an email ([email protected]), but I think this comment was meant for a different riddle.
Written by Steven Miller on December 25, 2016. Reply

Please write in English
Written by tasiras on February 7, 2017. Reply

what is the answer
- Written by Steven Miller on February 7, 2017. Reply
  
  email me at [email protected] for a hint
Written by Cher on March 3, 2017. Reply

Hi I’m a homeschool teacher. Amy I please have the solution to this problem?
- Written by Steven Miller on March 13, 2017. Reply
  
  Once you solve a problem email me at [email protected] (or email me for a hint). I’ll then provide, if you wish, the password for the student/teacher corner, which has a lot of aids to use these in classes.
Written by Clark on June 8, 2017. Reply

Is there an answer key? Thanks
- Written by Steven Miller on June 8, 2017. Reply
  
  Yes. I keep it hidden as it’s valuable to work on the problems first and if there are posted solutions often the temptation is too great…. If you are a teacher, once you solve a problem I’m happy to give you access to the student/teacher corner, which has discussions on how to use these in classes.
Written by James Livingston on April 20, 2018. Reply

I love using these riddles with my students.
- Written by Steven Miller on April 20, 2018. Reply
  
  so glad to hear. once you send a correct soln I’m happy to send the password to the student/teacher corner, which has supplemental material.

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