Since ℊ and f are different functions, numbers have different orbits under ℊ than under f. The function ℊ is similar to f, but for odd numbers it just adds 1 instead of tripling them first. But to see why it’s hard to prove for every number, let’s explore a slightly simpler function, ℊ. It’s easy to verify that the Collatz conjecture is true for any particular number: Just compute the orbit until you arrive at 1. And while no one has proved the conjecture, it has been verified for every number less than 2 68. So if you’re looking for a counterexample, you can start around 300 quintillion. The Collatz conjecture states that the orbit of every number under f eventually reaches 1. If your arithmetic is right, you’ll get there after 111 steps. The starting values of 9 and 19 are fun, and if you’ve got a few minutes to spare, try 27. Try a few more examples and you’ll see that the orbit always seems to stabilize in that 4 → 2 → 1 → … loop. Similarly, the orbit for 11 under f can be represented asġ1 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 → 4 → ….Īgain we end up in that same loop. Here’s the orbit of 10 under f:ġ0 → 5 → 16 → 8 → 4 → 2 → 1 → 4 → 2 → 1 → …Īt the end we see we are stuck in the loop 1 → 4 → 2 → 1 → …. We’ve already started computing the orbit of 10 under f, so let’s find the next few terms:Ī convenient way to represent an orbit is as a sequence with arrows. 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. 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 3 n + 1 conjecture. 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. To understand the Collatz conjecture, we’ll start with the following function: Let’s take a look at what makes this simple problem so very complicated.
#HARDEST SIMPLE MATH PROBLEM CRACK#
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. Despite all the attention, it’s still just a conjecture. But the Collatz conjecture is infamous for a reason: Even though every number that’s ever been tried ends up in that loop, we’re still not sure it’s always true. My friends and I spent days trading thrilling insights that never seemed to get us any closer to an answer. I couldn’t ignore it when I first learned of it in school. In fact, it would be hard to find a mathematician who hasn’t played around with this problem. And you’ll probably ignore my warning about trying to solve it: It just seems too simple and too orderly to resist understanding. The infamous Collatz conjecture says that if you start with any positive integer, you’ll always end up in this loop. 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. Or try 11: It’s odd, so 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. 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. Take 10 for example: 10 is even, so we cut it in half to get 5. At least, that’s what we think will happen. If you keep this up, you’ll eventually get stuck in a loop. Take that new number and repeat the process, again and again. Just pick a number, any number: If the number is even, cut it in half if it’s odd, triple it and add 1. This problem is simply stated, easily understood, and all too inviting. This column comes with a warning: Do not try to solve this math problem.
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