March 3rd, 2018, 01:45 AM  #1 
Newbie Joined: Mar 2018 From: poland Posts: 2 Thanks: 0  How to solve this problem? (HARD)
A giant rabbit is tied to a pole in the ground by an infinitely stretchy elastic cord attached to its tail. A hungry flea is on the pole watching the rabbit. The rabbit sees the flea, jumps into the air and lands one kilometre from the pole (with its tail still attached to the pole by the elastic cord). The flea gives chase and leaps into the air landing on the stretched elastic cord one centimetre from the pole. The rabbit, seeing this, again leaps into the air and lands another kilometre away from the pole (i.e., a total of two kilometres from the pole). Undaunted, the flea bravely leaps into the air again, landing on the elastic cord one centimetre further along. Once again the rabbit jumps another kilometre and the flea jumps another centimetre along the cord. If this continues indefinitely, will the flea ever catch up to the rabbit? (Assume the earth is flat and extends infinitely far in all directions.) 
March 3rd, 2018, 04:41 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
Yes.

March 3rd, 2018, 07:37 AM  #3 
Senior Member Joined: Sep 2016 From: USA Posts: 379 Thanks: 205 Math Focus: Dynamical systems, analytic function theory, numerics 
In the limit they will be $\frac{1}{12}$ meters apart.

March 3rd, 2018, 07:54 AM  #4 
Newbie Joined: Mar 2018 From: poland Posts: 2 Thanks: 0 
Thank you, but If could I ask you, HOW did you solve it?

March 3rd, 2018, 08:31 AM  #5 
Senior Member Joined: Sep 2015 From: USA Posts: 1,944 Thanks: 1011  
March 3rd, 2018, 01:14 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
It's an easy question, but first consider what happens if the rabbit leaps a much shorter distance, such as 2 cm. Bear in mind that once the flea is on the elastic, it moves as the elastic is stretched by the rabbit's leap.

May 7th, 2018, 02:17 PM  #7  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,165 Thanks: 867  Quote:
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May 8th, 2018, 04:16 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
If you obtain a formula for those distances, you'll see that the flea does eventually catch up.

May 8th, 2018, 07:34 PM  #9 
Senior Member Joined: Sep 2016 From: USA Posts: 379 Thanks: 205 Math Focus: Dynamical systems, analytic function theory, numerics 
I think CountryBoy is not assuming that each jump made by the rabbit stretches the elastic cord and causes the flea to move further away from the pole. This happens in addition to the distance traveled by the flea. His analysis is correct within that interpretation. However, I think this is not the interpretation intended. Math is precise for a reason, though rarely is that reason so transparent as attempting to get inside the mind of a random internet crazy person to determine what they mean when they say a rabbit leaps 1km. 
May 8th, 2018, 08:24 PM  #10 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
Country Boy stated "So the cord has doubled in length. The flea, that was 1 cm from the pole is now 2 cm from the pole." Hence the flea's movement due to the cord being stretched by the rabbit is being taken into account. This is an old puzzle.


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