Questions, questions, and more questions!

Recently, I attended a camp called Anvika conducted by the IIST, Thiruvananthapuram for students from across the country. We had a very interesting class there - where we spent an entire 2 hours trying to find as many questions based on an experiment he showed us as we could. While the experiment in itself wasn't too interesting (in my opinion, atleast), this thought of asking questions sparked within me and I had a couple of very interesting questions over the camp which I  tried answering along with some of my friends there.

On one of the nights, we saw a satellite in the sky one night, and it was moving quite fast. In fact, it seemed to practically cover a third of the visible sky in about a minute. The following thought came to my mind: if the satellite covers the entire sky in about 3 minutes, does it mean that it completes a revolution around the Earth in just 6 minutes?! Even accounting for errors, this would mean that the satellite covers a revolution of the Earth really fast, right?

I discussed this with a couple of friends there and they were quite surprised too. Google, however, seemed to suggest that most satellites took at least 90 minutes to 2 hours to complete a revolution. We were quite sure it wasn't a plane so there must have been some fault in the argument.

One possible issue was the following: does the satellite really cover close to half the earth when it covers the visible sky? Well, if you consider the earth to have no rotation, then in that reference frame the moon and sun do cover around half the earth when they cover the entire sky. So when a satellite covers the entire visible sky, it should cover close to half the earth too, right?

But if it did cover half the earth it would definitely be back within 15 minutes or so. Even waiting for half an hour or so, we did not see it come back. 

After a good 10 to 15 minutes, we finally had an explanation. I'm not going to tell you though ;)

On another day, we were playing with small sticks (stems of small plants that were lying around - but more like wands for us) like 5 year old kids, and so that everyone got a stick - we were breaking the sticks from the middle by applying a downward force on opposite sides of the stick. The question that arised though, was the following: is it better to apply the force at two points near the centre, or to apply it near the ends?

The first guess is torque: but this is definitely not torque in the normal sense. The 2 downward forces balance out the torque on the stick (let's just consider it as a rod). Elasticity is another idea that we may think of, but that would have made a lot more sense if we had been pulling outwards from the ends.

A more plausible explanation was that you can consider the rod as the union of infact two rods: from each end to the centre. Now the torque explanation does make some sense: and if you do the math you see that you end up with a $2F$ force at the centre upwards due to the two downward $F$ forces you apply and this is independent of where you pull from. But... we were not convinced. Firstly, is a stick really like a rod? Our stick was bending too, so an idea we had is to say something like: if you pull from the ends the force gets distributed through out. 

The craziest part was that we couldn't even experimentally find the answer - some sticks were too hard to break, and for the ones that we could break: some of us found the centre method easier while others found the end points method easier. Human factors like grip do come in, so a more valid question to ask would be: what would happen if a machine did it by applying the same force on both sides?

What do you think?

Another one of these questions that has intrigued me in the past was when I was first learning the Pigeon hole principle in grade 8 or so. The problem was as follows:

You are given 5 numbers. Show that you can choose some three out of these five so that the sum of those three numbers is divisible by 3.

This on it's own is not so special - and is quite reasonable to prove. But the next problem I saw was

You are given 17 numbers. Show that you can choose some nine out of these seventeen so that the sum of those nine numbers is divisible by 9.

Wow! Something is definitely on over here. What even are these numbers 5 and 17 doing? If you think long enough you'd realise that $5 = 3 \times2 -1$ and $17 = 9 \times 2 - 1$. 

So, maybe the following general statement held?

You are given $2n-1$ numbers. Show that you can choose some $n$ out of these $2n-1$, so that the sum of those $n$ numbers is divisible by $n$.

I spent the next two weeks desperately trying to prove this, making all sorts of observations: for instance that if the statement holds for $n = a$, $n = b$ then it holds for $n = ab$. I used this to reduce the problem for $n$ being a prime, and found that it worked for $n = 2,3,5,7$. But the work was getting messier and I didn't see how the method could be generalised.

It was only after a month or so that I found out that this is a pretty famous theorem: Erdős–Ginzburg–Ziv theorem. And it was only after more than a year that I understood the proof ;)

Being able to ask questions is a really underrated skill. Most breakthroughs in science and mathematics happen because someone decided to ask why on a statement that was deemed to be obvious. 

Not convinced? 

I’m sure most of you have heard of the number line. Right? But how many of you have heard of the number circle? Believe it or not, most of us see a number circle almost daily in our lives - a clock! This number circle, in fact, gives rise to modulo congruency - and it’s no exaggeration to say that your credit card password is safe (Cryptography!) only because someone decided to stop and question why there couldn't be a number circle.

Elsewhere, someone thought that we need not restrict ourselves to one dimension - and constructed the number plane of complex numbers. For years people thought that it’s absolutely stupid to ask the question: what is the square root of -1? But today that same $\sqrt{-1}$ is a part of Schrodinger's equation, the fundamental equation of Quantum Mechanics, and perhaps the most important and mysterious equation in physics today.

It’s really important that when we come across something - we pause and think. We question what we've seen, and try to work out the reasoning. Even if it doesn't make you the next Nobel prize winner, I'm sure that these questions will open you up to entirely new worlds and solving them will give you satisfaction enough to make all the time spent worth it :)