Usenet Newsgroups : rec.puzzles Hall of FameThe rec.puzzles Hall of Fame is a compilation of over 500 of the most popular puzzles that have been posted and discussed in the rec.puzzles newsgroup. In most cases a detailed solution has been provided. Many of these puzzles also appear in Braingle's own collection. 

Title: Cliff Puzzle 4: Time in a Bottle From: cliff@watson.ibm.com If you respond to this puzzle, if possible please include your name, address, affiliation, email address. If you like, tell me a little bit about yourself. PLEASE ALSO directly mail me a copy of your response in addition to any responding you do in the newsgroup. I will assume it is OK to describe your answer in any article or publication I may write in the future, with attribution to you, unless you state otherwise. Thanks, Cliff Pickover * * * Consider a chain of bottles (B) each connected to one another by a thin tube. A marble is placed in bottle 1. Each tube contains a oneway valve so marbles can only go from left to right in the tubes which are symbolized with "" marks: 1 2 3 4 B  B  B  B  The tubes are thin so it takes 1 hour of constant random shaking to get the marble from B1 to B2. Likewise for each bottle. I have not fully described the bottle collection. Each bottle has a backward 1way tube to bottle 1. I've tried to diagram these with "*" symbols. Each time the marble enters bottle B(N) it has a 50% probability of going back to bottle 1 via these tubes. ****<******** * * ***<***** * * * * * * * * * 1 2 3 4 B  B  B  B  Stop And Think 1. In how many hours will you expect to get the marble out of bottle 10 after placing the marble in bottle 1? 2. Is there a general formula for the amount of time required to get the ball out of bottle N into bottle N+1 given a probability P of backwards motion (given as 50% in this problem)? 3. In how many hours will you expect to get the marble out of bottle 10 after placing the marble in bottle 1 given two backward tubes for each bottle instead of one backward tube?
Solution