The Collatz Conjecture

This problem has been a fascinating question to me, and, although it seems very simple, no one has ever been able to solve it for more than 40 years. This problem is known as the Collatz Conjecture, or the 3n+1 problem. In this article I will introduce it to you, and explain why this problem is complicated to solve.

Let’s start with a natural number, which is any number which we can count, like 1, 2, 3, 4, etc. If the number is an even number, we divide it by 2, and if it is an odd number, we multiply it by three and then add 1. In other words, if we call this natural number n, then if n is even, we calculate n divided by 2, or n/2, and if n is odd, then we find 3n+1, which is the reason why this problem is also known as the 3n+1 problem. We have a new number from this calculation. With the new number, we do the same process again, dividing by two if it is even, and multiplying it by 3 and adding 1 if it is odd. We repeat this process infinitely, and the conjecture is that the result will end at one. This means that, although not proven, we believe that all numbers will eventually reach one, no matter what natural number we start with.

This question seems very simple due to the fact that the hardest mathematics in the question is multiplication and division. Compared to some of the other unsolved math questions, this is a lot simpler. At least, that is how it looks. Many believe that this is basically proven as huge amounts of numbers have been tested. However, in mathematical proofs, induction by a lot of examples does not work, as there still may be a counter-example that couldn’t be found. We have to prove it in some mathematical way, or find a valid counter example that proves that the conjecture is incorrect.

Another reason that this problem is difficult is because it seems rather counterintuitive. After all, when the number is odd, we multiply it by 3 and add 1. We are multiplying by three nearly most of the time, so why would it decrease to 1? However, when n is odd, 3n+1 is actually always even, and hence after the multiplication of 3(and adding 1), we actually divide the number right after by 2, so it is still possible to decrease, and we believe that it decreases. 

There are two ways for this conjecture to be false. Firstly, there could be a number that continues to increase towards infinity. Secondly, there could be a number that exists in a loop that does not include one. This would mean that the loop would occur forever, with no numbers in the loop reaching one. No counter-examples have been found yet, nor any proof of the conjecture.

Everything we learn in school in math seems to have an obvious solution, but not in mathematics at a higher level, as we can see in the Collatz Conjecture. Collatz Conjecture may have a proof or a counterexample to prove it, but until now, there is no answer yet for this question, something that anyone passionate about mathematics has to be ready for.

Source:

https://www.youtube.com/watch?v=094y1Z2wpJg

The Moon Cycle

Despite the moon being 384,400km away, it plays a vital role in our lives on Earth. 

Firstly, it has a strong gravitational pull which stabilises our planet's tilt. This prevents extreme weather changes and thus gives us seasons. If we didn’t have these seasons it would make the world unlivable. Additionally, this gravitational pull is what controls the tides, which is essential to keep our marine ecosystems intact and alive. Tides are extremely important as not only do they look pretty but marine ecosystems absorb up to 30% of the carbon dioxide that is produced by human activity, helping to regulate Earth’s temperature and soak up unwanted carbon dioxide from the atmosphere. 

Not only are we extremely reliant on the moon to simply survive but it aids a variety of animals in numerous ways. For instance, some organisms are dependent on the moon for navigation and procreation. These animals generally include amphibians and marine species as their reproduction patterns synchronise with the moon cycles. For example, some corals spawn after the full moon. 

The moon cycles through eight phases in the span of a month. These phases include: new moon, waxing crescent, first quarter, waxing gibbous, full moon, waning gibbous, third quarter and waning crescent. The phases of the moon have been essential in helping us track time as there is evidence of ties between the moon and historical calendar systems.

Throughout the world, people look at the moon through a spiritual and/or cultural lens. In terms of spirituality, the moon generally represents cyclic changes as well as divine feminine energy, intuitive wisdom and guidance. Each different phase is meant to represent a different meaning in someone' s life. For example, the full moon is the time of heightened power and during the last quarter moon it is believed that it is the time for personal inquiry - when you should dig deep, seek answers and integrate.