## Thursday, September 3, 2009

### Traveling Ant

Here is one of numerous Gardner puzzles:

Imagine elastic rope 1 meter long, an ant start moving with speed of 10 centimeters per second from one end of the rope. He moves steadily with constant speed. Every second someone stretches the rope to increase it's length by 1 meter uniformly. Will ant ever make it to the other end of the rope?

this one is rather interesting, try to prove your answer!

Nick said...

Yes he does. it take s while though

Proof:
Use the followng notation
P(i) is the percentage of the rope travelled lust before the stretch at i seconds.
B(i) is the lenght travelled before the stretch at i seconds
A(i) is the lengh Travelled after the stretch at i seconds

We have that P(1)=0.1 since the ant has travelled 10% of the rope inthe first 10 seconds

For i>1 we have that

1: P(i)=B(i)/i

(the length travelled divided by the length of the rope)

In turn

2: B(i) = A(i-1)+0.1

The length travelled just before the streatch at time i is equal to the lenght travelled at just after the stretch at time i-1

In turn

3: A(i-1)=P(i-1)*i

The distance travelled after the stretch at time i-1 is the percentage travelleed at time i-1 multiplied by the length of the rope just after the stretch at time i-1

Combining 1 and 2

we get

4: P(i) = (A(i-1) + 0.1)/i

Cobining 4 and 3 we get

5: P(i) = P(i-1) + 0.1/i

this gives

P(i) =0.1*(1 + 1/2 + ... +1/i)

As i tends to infinity, the sum above tends to infinity also, so there is a point when P(i) is greater than 1. THis occrs after 12367 seconds

Vrungel said...

Very good analysis !