चेतावनी ! भूलकर भी गणित के इस सवाल को solve मत करना | 3n+1 problem | Collatz conjecture
This video comes with a warning: Do not try to solve this math problem.
You will be tempted. This problem is simply stated, easily understood, and all too inviting. Just pick a number, any number: If the number is even, cut it in half; if it’s odd, triple it and add 1. Take that new number and repeat the process, again and again. If you keep this up, you’ll eventually get stuck in a loop. At least, that’s what we think will happen.
Take 10 for example: 10 is even, so we cut it in half to get 5. Since 5 is odd, we triple it and add 1. Now we have 16, which is even, so we halve it to get 8, then halve that to get 4, then halve it again to get 2, and once more to get 1. Since 1 is odd, we triple it and add 1. Now we’re back at 4, and we know where this goes: 4 goes to 2 which goes to 1 which goes to 4, and so on. We’re stuck in a loop.
Or try 11: It’s odd, so we triple it and add 1. Now we have 34, which is even, so we halve it to get 17, triple that and add 1 to get 52, halve that to get 26 and again to get 13, triple that and add 1 to get 40, halve that to get 20, then 10, then 5, triple that and add 1 to get 16, and halve that to get 8, then 4, 2 and 1. And we’re stuck in the loop again.
The infamous Collatz conjecture says that if you start with any positive integer, you’ll always end up in this loop. And you’ll probably ignore my warning about trying to solve it: It just seems too simple and too orderly to resist understanding. In fact, it would be hard to find a mathematician who hasn’t played around with this problem.
Even though every number that’s ever been tried ends up in that loop, we’re still not sure it’s always true. Despite all the attention, it’s still just a conjecture.
Yet progress has been made. One of the world’s greatest living mathematicians ignored all the warnings and took a crack at it, making the biggest strides on the problem in decades. Let’s take a look at what makes this simple problem so very complicated.
To understand the Collatz conjecture, we’ll start with the following function:
f(n)= n/2 if n is even
3n+1 if n is odd
You might remember “piecewise” functions from school: The above function takes an input n and applies one of two rules to it, depending on whether the input is odd or even. This function f enacts the rules of the procedure we described above: For example, f (10) = 10/2 = 5 since 10 is even, and f (5) = 3 × 5 + 1 = 16 since 5 is odd. Because of the rule for odd inputs, the Collatz conjecture is also known as the 3n + 1 conjecture.
The Collatz conjecture deals with “orbits” of this function f. An orbit is what you get if you start with a number and apply a function repeatedly, taking each output and feeding it back into the function as a new input. We call this “iterating” the function. We’ve already started computing the orbit of 10 under f, so let’s find the next few terms:
f (10) = 10/2 = 5
f (5) = 3 × 5 + 1 = 16
f (16) = 16/2 = 8
f (8) = 8/2 = 4
A convenient way to represent an orbit is as a sequence with arrows. Here’s the orbit of 10 under f:
10 → 5 → 16 → 8 → 4 → 2 → 1 → 4 → 2 → 1 → …
At the end we see we are stuck in the loop 1 → 4 → 2 → 1 → …
Now to know more watch out this full video.
Thanks for watching.
Godel incompleteness theorem - • गणित के इस सवाल ने दी भगवन को चुनौती Gödel...
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चेतावनी ! भूलकर भी गणित के इस सवाल को solve मत करना | 3n+1 problem | Collatz conjecture
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