### Education

# Colletz Conjecture Finally Solved by Genius Miss Samiya Hussein

**Representations of the collatz conjecture****Final Proof of the Collatz Conjecture**

Lother Collatz was a German Mathematician, born 1910- 1990; he proposed the Collatz conjecture in 1937 in which he states that given any positive integer k, the sequence generated by iterations of the Collatz Function will eventually reach and remain in the cycle 4, 2, 1. Collatz states these hailstorm numbers will eventually fall to 1, for any positive interger. The Collatz Conjecture is the most famous unsolved math’s problem. The collatz conjecture is also known as the f(n) = (3n +1 ) /2n problem, the 3n + 1 is the conjecture. If the number is even divide it by 2 until you reach an odd number or 1. If it’s an odd number different from 1 then multiply by 3.

F(n) = { n/2 if n = 0

{ 3n+1 if n = 1

Example 3x+1= 3 +1 = 4 /2 = 2 /2 = 1

Here we have a loop sequence of 4, 2, and 1.

The Collatz Conjecture example below in which n = 10. Because number 10 is even, divide it by 2 and get 5. Now, 5 are odd. So, 3(5) + 1 = 16. 16 is even, so divide by 2 to get 8. Divide 8 by 2 gives 4. Then, 4/2 = 2. 2 divided by 2 is 1. 1 is odd, so ,3(1) +1 = 4. But we already got 4 in sequence, which goes down to 2, then 1. This 4–2–1 loop continues till infinity! So, our finite sequence of unique numbers observed, also called the hailstone sequence is 10, 5, 16, 8, 4, 2, and 1.

The Collatz Conjecture states take any positive integer that is the rule.

An integer is a number which is a whole number. A non-integer is not a whole number. A negative whole number and zero are classed as non-integers. Decimal numbers and fractions are also non integers.

**My Proof of the Collatz Conjecture**

For two weeks I have been researching the Collatz conjecture, and like most Mathematicians and Scientists have spent many hours trying to prove this unsolved problem. I did many calculations and still I found it came back down to this loop sequence and ending at 1.**My Proof of the Collatz Conjecture**

After many hours of study; I began to realise that Mathematicians and Scientists had missed the whole concept of this equation. The Collatz Conjecture is true up until a certain point. After a certain point, it becomes untrue.

**Proof:**

To prove the collatz conjecture I will consider an example which is

3* 1000000 = 3000000

Apply the Collatz conjecture: Here we have:

3000000/2 = 1,500,000

1,500,000 /2 = 750,000

750,000 /2 = 375,000

375,000 /2 = 187,500

187,500/2 = 93,750

93,750 /2= 46,875

46,875 /2= 23,437.5

This point is where the Collatz conjecture is not true. Because 23,437.5 have converted into a decimal, and the Collatz Conjecture states to use only positive integers, so this is where it ends. It is no longer an integer; it is no longer a whole number.

Let’s just ignore the rule and continue to apply the conjecture with the decimal, as follows:

23,437.5 /2 = 11,718.75

At this point as I see it’s odd add 1 which gives 11,719.75

You see the + 1 does not add onto the .75 leaving it as odd again.

**My Proof and Conclusion:**

The Collatz Conjecture is true when every Collatz sequence eventually produces 4, 2, 1, 4, 2, and 1. In our case the Collatz Conjecture is true up to the number 46,875, and after these numbers is: 23,437.5, which means we found a, b, c different from 4, 2, and 1 that would disprove the Collatz Conjecture. More generally, if there exists an n-tuple, a_1, a_2, . . . , a_n different from 4, 2, 1, which is eventually infinitely repeated in the sequence of residues, then this would constitute a disproof. Moreover, if we could prove that the only possible n-tuple is 4, 2, and 1 that would only add to the body of evidence supporting the conjecture. The 23,437.5 is a non-Integers and therefore cannot carry on with the Collatz Conjecture. The key to our proof of the Collatz conjecture was that a number must end up in an integer number but this is not true in the Collatz case due to which the Collatz conjecture is false. If the number keeps showing values in the decimal then how can it get down to 1?

The rule is to use only positive integer in the Collatz conjecture. Here you can see there is a decimal point in the 23,437.5 so you cannot possibly carry on the sequence, because decimal point numbers are not allowed to be used in the Collatz conjecture. It seems that Mathematicians and Scientists have gone against the rules of the Collatz Conjecture. As a mathematician I agree that this theory is true because the problem with the Collatz conjecture that makes it so intimately difficult is the unpredictable nature of the cycles. That reason alone is why this problem is so difficult compared to the n+1 problem, which is an incredibly simple inductive proof. Any existing method to prove the Collatz Conjecture using the 3x+1 function directly will fail because the operations are incomplete. But, I hear you say, that the conjecture and the function are the same thing”. I understand this, and I would not contradict that. What I am saying is you need a superset function that is provable and when you have that proof, the smaller case of 3x+1 will have already been proved.

Understanding Mathematical language is the key to understanding the Collatz Conjecture. It is important to observe what the Collatz Conjecture in mathematical terms is asking one to do.

Integers play an important role in the equation. It is clear that Mathematicians have ignored this rule and carried on with calculations up to trillions, whilst including non-integers; which defeat the whole concept of what the Conjecture is asking one to do.

**Author:** *Miss Samiya Hussein is from Lancaster in the UK, and studied Engineering, and she has a very high iQ,*

Kenneth is a proud native of sydney, born and raised there. However, he pursued his education abroad and studied in Australia. Kenneth has worked as a journalist for almost a decade, making valuable contributions to prominent publications such as Yahoo News and The Verge. Currently, he serves as a journalist for The Hear Up, where he focuses on covering climate and science news. You can reach Kenneth at [email protected].

### Education

# 10 Key Principles for Effective eLearning Content Development

The rise in popularity of eLearning has been unprecedented over the past few years. It’s a powerful tool that’s being used to facilitate learning in various fields, from healthcare to finance, and more.

However, creating effective eLearning content can be a challenging task. That’s why in this blog, we’ll look at the ten key principles for effective eLearning content development.

## 1. Keep the Content Engaging

Effective eLearning content should be engaging, interactive, and immersive. Students tend to learn better when the content is visually appealing, interactive, and able to capture their attention. Find out at octivo.io these kinds of content when you try one of their eLearning courses.

## 2. Ensure the Course Content Is Relevant

The course content that you create needs to be relevant to the learners. Ensure that it addresses their needs, interests, and pain points. This relevance will keep them motivated to learn and retain the information.

## 3. Use Multimedia Content

Multimedia content is a crucial element in online education. It includes images, videos, simulations, and animations. It helps to enhance learners’ engagement, increase retention, and make learning more interactive.

## 4. Make It Accessible

Ensure that the eLearning content you create is accessible to everyone, including those with disabilities. This principle considers web accessibility standards to make it accessible to all learners, including those with visual and auditory difficulties.

## 5. Use a Variety of Assessment Methods

To evaluate the learning outcomes of eLearning content, use a variety of assessment methods. These methods can include quizzes, case studies, simulations, and role-playing exercises. A variety of assessment methods will ensure that you cater to all the learners’ needs.

## 6. Create User-Friendly Content

User experience is critical when it comes to eLearning. Ensure that the content is easy to navigate, and learners can find what they are looking for quickly. Design an interface that is user-friendly and intuitive.

## 7. Use Instructional Design Principles

Instructional design is the science and art of creating instructional material that facilitates learning. Ensure that you use instructional design principles to guide your content development. These principles include the ADDIE model, Bloom’s Taxonomy, and the ARCS model.

## 8. Make It Interactive

Interactivity engages learners and enhances learning outcomes. Effective eLearning has interactive elements that provide opportunities for learners to engage with the content directly. These elements include simulations, quizzes, and assessments.

## 9. Make It Mobile-Friendly

The world is moving towards mobile devices, and so should eLearning. Ensure that your content is mobile-friendly, meaning it’s viewable on any mobile device. This feature will provide learners with increased flexibility, allowing them to learn wherever and whenever they want.

## 10. Develop Measurable Learning Objectives

Learning objectives help learners understand what they will achieve at the end of the course. Ensure that the learning objectives are specific, measurable, attainable, relevant, and time-bound (SMART).

Developing measurable objectives will help you track the progress of the learners. You will also measure the impact of the eLearning content.

## Use These Key Principles for Effective eLearning Content Development

Effective eLearning content development is a continuous process. It requires a lot of effort, creativity, and a variety of design and development principles. In summary, learners benefit from eLearning content when it’s engaging, relevant, and accessible.

Learning is easier when the program is interactive and easy to use on mobile devices. It should also have clear learning goals. Students learn better when instructional design principles and various assessment methods are used.

If you enjoyed this article and would like to read more like this, check out the rest of our blog page right now.

*Having completed my education in English, I’ve cultivated a successful career as a content writer. My tenure includes valued collaborations with distinguished professional organizations, reflecting my commitment to producing high-quality content.*

**Contact me on this mail:** [email protected]