Crossing the Bridge
Date: 05/12/97 at 19:27:58 From: Ian Comfort Subject: 4 men want to cross a bridge... One of my friends gave me this problem. Try as I might, I cannot seem to get the answer he says he does. Well, here goes: Four men want to cross a bridge. The bridge is of a width such that a maximum of 2 people may cross at a time. One man takes 10 minutes to cross, another takes 5 minutes to cross, another takes 2 minutes to cross, and the last takes 1 minute to cross. I will call these men A, B, C, and D, respectively. Here is the catch. If two people cross the bridge together, they must walk at the pace of the slower one. So if A and B cross, they take 10 minutes. Also, it is night. Each trip requires a flashlight. There is only one flashlight. The men are not allowed to toss the light over the river. How fast can you get all 4 men over the bridge? Here is what I have done: I figured that there must be 5 total trips. Two go across, one comes back with the light, two more go, one comes back again, and then the last two men cross with the light. I also figured that it would be fastest to have D make all the return trips, so I came up with something like this: A and D go across 10 minutes D goes back with light 1 minute B and D go across 5 minutes D goes back with light 1 minute C and D go across 2 minutes -------------------- --------------- Total 19 minutes My friend says he can get them across in 17 minutes, but he won't tell me how. Please help! Ian Comfort
Date: 05/23/97 at 15:23:24 From: Doctor Barney Subject: Re: 4 men want to cross a bridge...Rev. 1 Hi Ian, This is a cool problem. First I read the problem and tried to solve it before I read your answer. I came up with the same solution. But then, as I thought about it longer, I did find a way to get all four men across the bridge with the restrictions imposed in 17 minutes, no tricks. Here's a hint: You are correct that because of the flashlight there will be three trips across the bridge with two people together, and two trips back with light with one person bringing the light back alone. Now, instead of optimizing the back with light trips, see if you can optimize the across trips to create a better overall solution, even if one of the return trips takes longer than 1 minute. -Doctor Barney, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 02/10/98 at 05:33:06 From: Nigel Armitage Subject: Bridge riddle Dr. Math, Can you tell me the answer to this one please? There are 4 men on one side of a narrow bridge. At most 2 men can cross at a time, and since it is night and the bridge is narrow and rickety, they must cross with a flashlight. Unfortunately there is only 1 flashlight, and all 4 must get across the bridge in at most 17 minutes. However, the men take the following lengths of time to cross the bridge: Man No. 1 takes 1 minute Man No. 2 takes 2 minutes Man No. 3 takes 5 minutes Man No. 4 takes 10 minutes When two men walk together, they walk at the pace of the slower one. So if No. 1 and No. 4 cross together, it will take them 10 minutes. How can they do it and all 4 get across in no more than 17 minutes? Best wishes. Nigel Armitage Link (Legal Information Network), UK
Date: 02/10/98 at 06:55:06 From: Doctor Allan Subject: Re: Bridge riddle Hello Nigel, First of all I hope that you agree with me that there must be a total of 5 trips. First two go across, then one returns with the flashlight, then two more go across, one returns with the flashlight, and finally the two remaining men go across. Now if you optimize the trips where the flashlight is returned, you will get something like this: No. 1 and No. 4 go across: 10 minutes No. 1 returns with flash: 1 minute No. 1 and No. 3 go across: 5 minutes No. 1 returns with flash: 1 minute No. 1 and No. 2 go across: 2 minutes ---------- 19 minutes This is the first solution one tries (at least I did!). Probably one should try to optimize the across trips instead of the trips returning with the flashlight in order to get a better result. I will give you a hint: No. 1 and No. 2 go across: 2 minutes No. 2 returns with flash: 2 minutes No. 3 and No. 4 go across: 10 minutes ---------- 14 minutes Now No. 1, No. 3 and No. 4 have crossed, so one of these needs to return with the flashlight and afterwards go across with No. 2 in 3 minutes - do you see how? -Doctor Allan, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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