Puzzle : Ants Walk on a Stick

Twenty-five ants are placed randomly on a meter stick. Each faces east or west. At a signal they all start to march at 1 centimeter per second. Whenever two ants collide they reverse directions. How long must we wait to be sure that all the ants have left the stick?

This sounds immensely complicated, but with a simple insight the answer is immediately clear. What is it?

You need to wait for.....seconds only!
 

Analysing Ants Walk on a Stick

Read the question associated with the walk.

For a moment, let's assume that there are only 2 ants 20 cm away from either end of the stick. Now, after 30 seconds they both will collide with each other & will reverse the direction.

At 50th seconds they will be at the end of the sticks falling off the stick.

So after 80 seconds they will fall off the stick. Now, imagine if ants avoid collision & pass through (or above) each other. Still, both ants would need 80 seconds to leave the stick.

In short, 2 ants' collision & reversal in direction is equivalent to their passing through each other. The other ant continues the journey on the behalf of the first ant & vice versa.

And in case, if they were 100 cm apart, they would need 100 seconds to get off the stick. Again, after collision at halfway mark here, the other ant travels the rest of distance that other ant was supposed to travel.

On the similar note, we can say that even if there are 25 ants on the stick then each ant will cover some distance on the behalf of some other ant. And we need to wait for maximum 100 seconds if 1 of 25 ants is at the edge of the stick. 

All 25 ants together completes the journey of each others in 100 seconds. The ant which is at the edge of stick might complete journey of some other ant which might be only 10 seconds long. But the 100 seconds journey of that ant will be shared by rest of ants. 

Avoid The Collision of Ants

Three ants are sitting at the three corners of an equilateral triangle. Each ant starts randomly picks a direction and starts to move along the edge of the triangle. What is the probability that none of the ants collide?

'THIS' is the probability!

To Avoid The Collision of Ants

What was the problem?

Each ant can decide to go either clockwise or anti-clockwise. That is there are 2 options available for each ant to go. Hence, there will be total 2x2x2 = 8 possible combination of ants' different paths.

Now 2 ants won't collide if & only if all are either moving clockwise or anti-clockwise. In short, out of 8 possible combinations only 2 combinations are there where ants won't collide.

Hence, the probability that ant won't collide is 2/8 = 0.25.  

Time Taken For The Journey

A RED ant is sitting on one side of a table (point X) and a BLACK ant is sitting on the opposite side of the table (point Y).

Now both of them decides to exchange their places and starts crawling. On the way, both of the ants meet and after that, it takes 20 seconds for the RED ant to move to point Y and it takes 5 seconds for the BLACK ant to reach point X.

Find out the total time taken by the RED and the BLACK ant to make the journey.

Here are steps to calculate! 

Source 

Calculation of Time For The Journey

What was the journey?

Let the speed of RED ant is R & that of BLACK ant is B

Let time taken by them to meet be T.

Now we will apply the basic formula of distance:

Distance = Speed * Time.

The RED ant travels R T distance before meeting and 20 R after the meeting.

The BLACK ant travels B T distance before meeting and 5 B after the meeting.

Now as per the question,The distance traveled by RED ant before they both meet will be equal to the distance covered by BLACK ant after they meet. We can say the same for the vice versa case as well.

Thus,

RT = 5B and BT = 20R i.e. B = 20R/T, putting in RT = 5B

R T = [20R/T] * 5

RT = 100R/T

T^2 = 100

T = 10.

Thus the RED ant will require 10 + 20 = 30 seconds to travel the distance.

And the BLACK ant will take 10 + 5 = 15 seconds to travel the distance.