Amazing approach to hardest mathematics problem.

Hey, there's no problem in this universe which had no solution. The need is to just nicely analyse the problem and approach to the solution in the right way.

The "Putnam competion" is one of the toughest exam based on maths problems. It is a six hour long test and have just 12 questions each of 10 marks. It's difficulty level could be understood by seeing the median marks of student appearing in this test that is 1 or 2 out of 120. hence it is a pretty hard test. The hardness level of problem goes on incresing as you go from question 1 to 6. It means the sixth question is the hardest.
Here I am going to share a problem which came in a while back in this exam and it was the sixth problem.

Q. If you choose four random points on a sphere, and consider a tetrahedron with those points as its vertices, what is the probability that the centre of the sphere is inside the tetrahedron?

You should try this question once yourself.

And if you got stuck on it, then here's a video of it's solution in the easiest way.
THIS VIDEO IS AMAZING!!!

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3n+1 conjecture!!!?? What is it?? This will make you think really hard.

Let's start with a game. Think of a positive number (say n). If it is even then replace it with it's half (n/2). If it's odd then replace it with three times of itself plus one (3n+1). for e.g., let's take 15. since its odd, so it becomes 15*3+1 i.e., 46 now 46 changes to 46/2 i.e., 23 agin 23 changes to 23*3+1 i.e., 70 continuing this we get a sereis.... 15,46,23,70,35,106,53,160,80,40,20,10,5,16,8,4,2,1.....---> so this continues. You would be thinking, this is not hard.....Beacause it's not the problem. The problem is that if you start with any of the positive number eventually it ends up with 1. Try with some numbers like 11 - 11,34,17,52,26,13,40,20,10,5,16,8,4,2,1 7 - 7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1 13 - 13,40,20,10,5,16,8,4,2,1 This problem drives mathematicians crazy because there don't seems to be any clear pattern. There are certain amazing numbers like 8192 that collapse down to 1 very quickly i.e in only 13 step...

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