There have been many cases of scientists finding solutions to problems in their dreams. Some have stumbled upon the required solution by their intuitive approach. I firmly believe that this method leads to discovering new things... things which are hard to explain by pure logic. One such problem is:

There is a 100m long stick lying on positive x-axis with one end on \(x=0\) and there are 101 ants placed on this stick such that there is one ant on \(x=0\), \(x=1\), \(x=2\)...\(x=100\). The first ant is facing right, the second is facing left, the third right, the fourth left and so on. All these ants now start walking in the direction they are facing with speed \(1 m/s\). Whenever 2 ants collide, they exchange their directions immediately (but speed remain the same) and continue moving.Ants fall-off the stick once the move beyond end of the stick. Find the time taken for all the ants to leave the stick.

I am not going to reveal the answer. At first I thought, "this is a very tough question". But then the solution was extremely elegant and the motivation for it was intuition... and not hardcore maths. (Although the actual solution does involve math).

I recently stumbled upon this video and it rekindled in me, the idea of intuition. I will state a mathematical proof here. Can someone explain this in a non mathematical form?

Consider \(S_1=1-1+1-1+1....\)

now \(S_1= 1-1+1-1+1...\) (i.e shift the 1's to the right)

adding both gives \(2S_1=1\Rightarrow S_1=\frac{1}{2}\)

Now consider \(S_2=1-2+3-4+5...\)

now, \(S_2= 1-2+3-4+5...\) (i.e. shift the numbers to the right)

adding both gives \(2S_2=1-1+1-1...=S_1=\frac{1}{2}\Rightarrow S_2=\frac{1}{4}\)

Now consider \(S_3=1+2+3+4...\)

now, \(S_2=1-2+3-4...\)

We have \(S_3-S_2=0+4+0+8+0+12...\)

or \(S_3-\frac{1}{4}=4+8+12....=4\times(1+2+3...)=4S_3\)

\(\Rightarrow -\frac{1}{4}=3S_3\Rightarrow S_3=-\frac{1}{12}\)

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