Best Answer superprismatic, 03 December 2013 - 01:47 AM

Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account. As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends. Of course, you can also enjoy our collection of amazing optical illusions and cool math games. If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top. If you have a website, we would appreciate a little link to BrainDen. Thanks and enjoy the Den :-) |

Guest Message by DevFuse

Started by BMAD, Nov 26 2013 04:08 AM

Best Answer superprismatic, 03 December 2013 - 01:47 AM

Spoiler for Here's the gist of my not-very-clever method of solving this:

Go to the full post
9 replies to this topic

Posted 26 November 2013 - 04:08 AM

You have N cars that are all traveling the same direction on an infinitely long one-lane highway. Unfortunately, they are all going different speeds, and cannot pass each other. Eventually the cars will clump up in one or more traffic jams.

In terms of N, what is the expected number of clumps of cars?

Posted 26 November 2013 - 12:37 PM

by simulation i get...

Spoiler for

Posted 27 November 2013 - 05:27 AM

I find that answer surprising if true.

Let A be the slowest car in the group. Since every car behind A will eventually catch up to A and be stuck behind it, there will be one traffic jam with A and all subsequent cars, and howevermany traffic jams involving the cars in front of A. If there are M cars in front of A then there can be no more than M+1 traffic jams, and that's only in the special case where every car in front of A is traveling slower than the car in front of it (if you count a single unimpeded car to be a traffic jam). In general there will be less than M+1. So if car A is randomly distributed among the pack, M will on average be about N/2 and the total number of traffic jams should be less than that.

Let A be the slowest car in the group. Since every car behind A will eventually catch up to A and be stuck behind it, there will be one traffic jam with A and all subsequent cars, and howevermany traffic jams involving the cars in front of A. If there are M cars in front of A then there can be no more than M+1 traffic jams, and that's only in the special case where every car in front of A is traveling slower than the car in front of it (if you count a single unimpeded car to be a traffic jam). In general there will be less than M+1. So if car A is randomly distributed among the pack, M will on average be about N/2 and the total number of traffic jams should be less than that.

Posted 27 November 2013 - 05:05 PM

yes, i tihnk i ran my simulation wrong. trying to think of a way to fix it.

Posted 28 November 2013 - 10:22 AM

Spoiler for approximately

*A leader's credentials do not hang on his wall. They are written by the lives of those he has led.*

Posted 29 November 2013 - 09:44 PM

Spoiler for apparently

Posted 02 December 2013 - 03:34 PM

Spoiler for apparently

I agree. Nice!

Is there a clever derivation?

*A leader's credentials do not hang on his wall. They are written by the lives of those he has led.*

Posted 03 December 2013 - 01:47 AM Best Answer

Spoiler for Here's the gist of my not-very-clever method of solving this:

Posted 03 December 2013 - 01:49 PM

Spoiler for Here's the gist of my not-very-clever method of solving this:

Spoiler for Nice!

*A leader's credentials do not hang on his wall. They are written by the lives of those he has led.*

Posted 03 December 2013 - 07:22 PM

Spoiler for stirling of the second kind

**Edited by BMAD, 03 December 2013 - 07:23 PM.**

0 members, 0 guests, 0 anonymous users