So I’m gonna share something cool as ice. I actually found this 2 years ago but never posted it.

Let me introduce this nice thing

This sum is actually very nice. He wears a tie.

In fact if you input x=10^{-3}, and you would get this:

0.001002002003002004002004003004002006002004004005002006002006004004002008003004004006002008002006004004004009002004004008002008002006006004002010003006004006002008004008004004002012...

Ok so what’s dis? Notice that the decimals follow some pattern:

00, number, 00, number, 00, number….. And then we have some exceptions where “00” is replaced with “0” or “000”

What’s dis? There must be a pattern to the numbers right?

Let’s list down the number and see:

**1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12**

There doesn’t seem to be a pattern is there?

BUT YOU’RE WRONG THERE IS A PATTERN

Let’s start with the first number:

**1:**The number 1 has**1**divisor**2:**The number 2 has**2**divisor**2:**The number 3 has**3**divisor**3:**The number 4 has**3**divisor**2:**The number 5 has**2**divisor**4:**The number 6 has**4**divisor

Turns out that the **nth** number in the sequence is the number of divisors of **n**!

**Cool right?** Now why is dis so? Try to figure it out urselfgdby.

First of all, I’m sooooooooo sorry for breaking my promise and stuff. I know this is super terrible because yeah (breaking promises is bad). To make up for it, i will (probably) be writing more posts (so as long as Julian continues to pester me ceaselessly). I still want to aim for the 1 post every fortnight goal and i’ll still keep trying to achieve that someday.

Make sure you know what you’re getting yourself into before making a promise to a bunch of people

-Clyde Lhui 2016

Lately i have been getting into a lot of stuff. Stuff like learning Japanese, photography, music, cooking and a bunch of other stuff. I might be writing posts about those in future so yeah (i know i say this a lot and end up not writing but oh wells).

Well as you can probably tell by the title, this post is about science communication and scientific literacy (pretty self explanatory i guess). I feel very strongly about these 2 topics and that’s why I’m writing this post.

So first off, **DEFINITIONS**!

Science communicationgenerally refers to public communication presenting science-related topics to non-experts. This often involves professional scientists (called “outreach” or “popularization”), but has also evolved into a professional field in its own right. It includes scienceexhibitions, journalism, policy or media production.-Wikipedia

And…

Scientific literacy is the knowledge and understanding of scientific concepts and processes required for personal decision making, participation in civic and cultural affairs, and economic productivity. It also includes specific types of abilities.

-Literacynet.org

I know that you guys could have easily googled that but hey, 10 seconds saved is 10 seconds saved

So in short, science communication is talking to people who aren’t scientists about science and scientific literacy is knowing enough science to make logical and good decisions.

I think you are beginning to see how these 2 things are linked.

Perhaps i should further explain why i decided to write this post. In my daily life, i spend a lot of time with my friends (who are largely a group of nerds (who mostly take pride in their nerdhood)) and my family. Since i love science and i spend a lot of my time with my friends (who are nerds), we spend a lot of time talking about science and related topics. As such, when i talk to my family about a ‘science related topic’ (I’ll explain the apostrophe later), i notice whenever something is off.

My aunts and uncles like to send long WhatsApp messages about stuff they hear from their friends and 95% of those messages that i end up reading are wrong in one way or another. My mum once showed me this video:

Well I think most of you can see why this video is wrong.

(In case you didn’t figure it out, your stomach is part of your body which is at 37 degrees Celsius for the most part)

(Also ice water warming up is kind of a thing)

I know some of you must be thinking: “I’m not stupid, i wouldn’t believe things that don’t make sense.”

Well there are quite a lot of things that people misunderstand.

For starters, nuclear power. I am a strong advocate for nuclear power. It’s clean, reliable and fairly safe. Unlike solar panels or wind turbines that stop working once the sun stops shining or the wind stops blowing, nuclear power plants can work 24/7 and supply enough power to support entire power grids. Furthermore, new nuclear plant designs which have improved safety features are constantly being suggested, making future power plants safer than before. Despite this, many people have a very negative impression of nuclear power.

Does this look familiar?

Well it should. I have seen countless news reports about climate change showing images or videos these pumping out massive white clouds.

The only thing is these are the **cooling towers** of nuclear power plants.

Most people see these images and go “OH NO! We are pumping all that carbon dioxide into the atmosphere!?”. However, the white clouds coming out of the cooling towers are literal clouds: clouds of water droplets.

(Hopefully) by now you should understand the fact that we are prone to having a lot of misconceptions. It could be due to the way the media presents facts, the way social media promotes controversial content or any other reason out there. Regardless of the reason, i hope that you understand that this is pretty bad and it’s something that is extremely hard to avoid. I myself cannot claim that all the knowledge i possess is 100% accurate (in fact i do get things wrong pretty often).

What i hope you get out of reading this post is that **knowledge is never absolute **and that life and learning is all about constantly renewing our knowledge by being sceptical and challenging our own beliefs. We all need to keep reading and keep discussing so as to improve the accuracy of our knowledge. I think it’s also important for us to keep an open mind and not to immediately say “No that’s wrong.” when someone has contrasting beliefs (well you could but remember to provide your reasons and explain your views).

Never stop questioning your beliefs and perhaps one day there won’t be Geography teachers believing that the Earth goes around the sun in a day and rotates once around its axis in a year.

Thanks for reading!

Clyde Lhui

]]>What do I mean by this? Well, supposed your algorithm is to turn the top side once. After repeating this algorithm 4 times, you would return back to the solved state right? In the problem above, “any algorithm” would refer to literally ANY ALGORITHM. For instance, your algorithm might be: R’ D D R D R’ D’ R, and repeating this algorithm a sufficient number of times on a solved cube would make it eventually return to its solved state.

This problem seems impossible right? At the very least, it probably involves some uni graph theory concept and stuff right? Before you close off (pls dont), the solution to this problem requires no prerequisites, just a grasp of logic.

Ok, here comes the solution:

To tackle this problem, I will be using this technique called “Proof by contradiction”.

For instance, if I want to proof that statement A is false, I assume otherwise; that A is true. Then I show that if A is true, absurd shit that doesn’t make sense would happen (ie. we would see a contradiction). Since A being true causes a contradiction, A must be false. Hence proved that A is false.

Got the general gist of this technique? Cause this would be fundamental in our proof.

Alright, the actual proof:

Let’s first assume that there exists an algorithm (Let’s call it “f”) that would not cycle. In other words, repeatedly applying “f” on a cube would cause it to always land in a state that it has not been before:

However, this would imply that a rubix cube has an INFINITE number of different states! This is a contradiction, since it is well known that a rubix cube has a finite number of states (Left as exercise of reader). Since there is a contradiction, we can safely conclude that after applying “f” sufficiently number of times, it would cause the cube to return to a state it was before.

We are not done however, as this conclusion results in 2 possible cases:

Case 1 is where it returns to the solve state (which we want), and Case 2 is where it returns not to the solved state but to some random state within the cycle (which we want to disprove).

To disprove case 2, we employ prove by contradiction again:

Let’s define “f^{-1}” as the inverse algorithm of “f”. In other words:

Basically, “f^{-1}” is an algorithm that REVERSES what “f” does.

So now, what happens if we replace all “f” with “f^{-1}” in the picture for Case 2? Well, it becomes this:

Now we will show how absurd this would be. Notice the box for “State n”. Notice how this diagram implies that applying the algorithm “f^{-1}” on state n would result in 2 possible states (State n-1 and State k). This is absurd because we know that applying an algorithm on a rubix cube will result in only 1 possible state (ie. each state should only point to 1 other state). Hence we have reached a conclusion that Case 2 is IMPOSSIBLE. Which leaves us with Case 1 as the only case that is possible.

Hence, proved, that repeatedly applying any algorithm on a solved Rubixs Cube would cause it to eventually return back to its solved state.

An update to the previous post: I have played with the simulator and recorded some of the animations

Hoped you enjoyed this post!

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Lagged my computer a lot just trying to render it. In fact I wasn’t even able to extract the animation from the school’s computer.

I used Desmos for the animation: https://www.desmos.com/calculator/ycn9inng2a

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**Explore here:** https://www.desmos.com/calculator/n3iwydn0sa

**Explore here:** https://www.desmos.com/calculator/ukty5kjfbq

**Context here:** https://www.youtube.com/watch?v=0z1Rp3g8U7c&t=67s

**Explore here:** https://www.desmos.com/calculator/3wjeuxlwuc

**Context here: **https://www.youtube.com/watch?v=w-JSWtu1oMU

**Explore here: **https://www.desmos.com/calculator/u6gfbffvmz

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Seems rather uninteresting right now but when you plot it, you’ll get INFINITE HEARTS!

Check it out here: https://www.desmos.com/calculator/ntqotaqyp2

]]>Anyway, this is a post regarding a class of functions that are fun (subjective), of which I might make an elaborated post about after I’m done with my Analytic Number Theory post.

I’ve yet to look into this thoroughly since I only started doing so today 1150pm so here is a special case:

Yeah, does not look so impressive… YET.

Here’s the cool part: Take out your calculator (preferably a scientific one), and ensure that it is set to Radians.

Now, find “F(1)”. Did you get “F(1) = 1”?

How about “F(2)”? Did you get “F(2)=2”?

Continue to do this until “F(4)”, do you see the pattern?

What about “F(5)”? Oh? It loops back to “F(5)=1″…

How about “F(6)”? How about “F(2015)”?

So here’s it: “F(n)” outputs the remainder when “n” is divided by “4”!

So “F(2015)=3” since “2015” divided by “4” gives a remainder of “3”!

Now how did the godly Julia derive such an amazing (actually not so) function?

Sorry not going through how goodnite.

How the function looks like when graphed out:

]]>As mentioned before, i will be writing about black holes this week.

The previous post from this series was done by my co-writer, Jackson, who mysteriously disappeared after posting it. Don’t worry, he’s still alive and well helping me solve my darn math and physics problems.

Okay so on to black holes.

In the previous post, Jackson covered the formation and structure of black holes.

To make up for the previous post’s lack of graphics, here’s a helpful diagram of how a black hole looks like:

Yes. It is black. Or rather, dark. As mentioned before, the gravity from the region beyond the event horizon is so strong that even light cannot escape. As such, it is well, dark.

Some of you might be thinking: *“So as time passes, all the matter in the universe will end up in black holes. Since nothing can escape a black hole, whatever goes in, doesn’t come out. So we’ll all end up as black holes one day!”*

If you seriously thought that, then you were thinking what i thought when i first learned about black holes.

However, that (in reality) is incorrect.

That brings me to what this post is about. This post is about the thing that kills black holes. How all the black holes will meet their end.It’s a story about the weirdness of the universe, and also how cool it can be.

This post is about Hawking radiation.

So **what is Hawking radiation?**

Simply put, Hawking radiation is radiation that is emitted by black holes. Over time, the black hole loses mass as it emits more Hawking radiation.

According to Einstein’s famous equation, E=mc^2, the energy from the radiation emitted is related to the mass loss by a factor of c^2 as such, more energy lost, more mass lost.

And what happens after the black hole loses all its mass? It basically disappears. It evaporates.

That’s also why scientists can make micro black holes in particle accelerators without getting the entire earth sucked into oblivion. The micro black holes evaporate in an extremely short span of time, thus we are all safe!

Also, it gets its name from the famous physicist Stephen Hawking (the cool guy who helped us understand black holes a lot more and is also famous from the awesome move *Theory of Everything*)

At this point you’re probably wondering **how can Hawking radiation escape from black holes when basically nothing can escape from black holes?**

Well, to answer that, we need to look into quantum physics, where things coming from nothing and teleportation make it possible for the Hawking radiation escape.

The first theory of how Hawking radiation escapes from black holes is by the separation of particle-antiparticle pairs that form in the vacuum of space.

So some of you might be familiar with Heisenberg’s Uncertainty Principle. Most know it for its ‘position-momentum’ uncertainty version. The theory states that the more certain an observer is of an object’s position, the less certain he will be about the object’s momentum. However, this uncertainty is extremely minute and undetectable by most of us in our daily lives. But it is indeed experimentally proven and consistent with many other theories and experimental results.

Apart from ‘position-momentum’ uncertainty, there is also ‘energy-time’ uncertainty. And due to this, in an extremely short span of time, there will be an extremely huge energy uncertainty, causing the energy to possibly go high enough that random particles may be generated. Once again, Einstein’s equation: E=mc^2 comes into play here as mass in the particles is a result of the energy uncertainty. However, as we all know, energy has to be conserved (or else evil scientists would have created an infinite energy producing machine and taken over the world with crazy weapons or maybe just selling us power at ridiculous prices) as such, the particles soon disappear as they are annihilated by the antiparticle that forms with it. As such, the universe remains in equilibrium and evil scientists don’t destroy the Earth.

This also occurs at the edges of the black holes. At the event horizon.

Sometimes, a pair might form with one particle forming within the event horizon with the other particle forming outside.

As such the antiparticle will fall into the black hole while the particle goes out into the unexplored wilderness that is the universe. Since the antiparticle must fall in as it can’t escape, all pairs that form this way will probably end up releasing a particle outside the black hole. Thus, we observe it as Hawking radiation.

The second way Hawking radiation could possibly form is by quantum tunnelling.

This effect occurs when a particle bypasses an energy barrier without reaching the required energy. For example, certain chemical reactions require a certain temperature (which is proportionate to the average energy of the particles) for the reaction to occur. As such, quantum tunnelling would be when a particle in the solution reacts without reaching the required energy.

This is possible in black holes too. The energy barrier is the event horizon, the particle is a photon.

Regarding how this effect actually works, it relies on the fact that particles can be described as waves (something that is pretty complex and something that I’m leaving out in this post).

The idea of Hawking radiation is extremely important as we can learn a lot from observing Hawking radiation. Since it’s basically the only thing that we can see coming out of a black hole (actually not exactly but hey who cares) it serves an important role in our understanding of these immensely powerful balls of uncertainty.

There still is much more to cover on black holes. Being such an extreme object, it provides insight into what happens under extreme conditions.

Well, that’s it for this fortnight’s post, look forward to the next one!

Thanks for reading!

Clyde Lhui

]]>These symbols (∑ and ∏) might look very daunting at first but I assure you, even a toddler can understand what it means.

“Hey Fredrick, wanna try a new game?”

Let’s first start with the summation: ∑ | Sigma notation

The above symbol basically means “To sum” or “Add everything together”. You guys know what is addition right? 1+1=3 ( ͡° ͜ʖ ͡°)? No? Ok nvm.

But sum what with what? Well, let me introduce the dummy variable:

The dummy variable is basically an instruction of how to sum. For the example on the left, “n” is the dummy variable.

“a” is the starting point

“b” is the end point

and “f(n)” is the function you want to sum over.

What does this all mean?

Well, basically:

That’s it! Notice how “a” is the starting point and “b” is the end point?

Let’s take a look at some examples:

For the third example, note that “x” is the dummy variable and not y.

Now, what happens if you are walking along a beautiful path on a bright sunny day when suddenly a burly man pulls out a butterfly knife with a pearl-embedded, elephant tusk handle and put it against your throat. Adrenaline rushes through your veins as your limbs go numb with fear while your eyes watch the empty space ahead, futile in making sense of your awaiting doom. With a deep croak, your adversary spat

“We meet again. Now let’s talk.”

He pulls out a slip of paper. The paper cried as it flapped against the wind.

“Take a look. Take a close look at this. Explain.”

The slip of paper makes it’s way to inches from your face. Close up, you piece together the various symbols on the paper and break out in cold sweat. On the paper, largely printed, you saw:

Now what the hell is that “infinity” symbol doing there? Julian your master failed to cover this in his lectures, and you know you are doomed, for you don’t know what it means.

You don’t want to end up in the situation above do you? So listen up!

The infinity symbol up there means that you sum to infinity. You don’t stop summing, you just add and add and add and add…….. You are adding an infinite number of terms.

You don’t stop!

You might be asking now, wouldn’t the sum diverge to infinity? In other words:

Well the answer is: Yes, most of the time. **MOST **of the time.

You see, for some sums, it doesn’t approach infinity. For instance, you can proof that

Try it on your calculator! Or… research on Infinite Geometric Progressions… or not.

When it comes to infinite sums, there can be some quite unexpected results. For instance:

And yes, the π is indeed the one we are familiar with ( π=3.1415…)

To learn more about it, you can research on the Basel Problem, or the Riemann Zeta Function.

There are other types of summations, such as cyclic sum, symmetric sum and sum-over-n-that-satisfies-some-conditions (I’m don’t know the actual name for this type of sum, hopefully somebody can help fill me up on that). Anyway, I’m not going to go through those sums cause I’m lazy. Let’s move on to ∏ (product):

∏ is essentially the same as ∑, just that instead of summing, you multiply!

Now let’s take a look at random examples:

For the second example, note that “n” is the dummy variable and not “x”.

Similar to infinite sums, there are also infinite products, where you multiply FOREVER! HAHAHA!

Now, what happens if you are walking along a beautiful path on a bright sunny day when suddenly a burly man pulls out a butterfly knife with a pearl-embedded, elephant tusk handle and put it against your throat. Adrenaline rushes through your veins as your limbs go numb with fear while your eyes watch the empty space ahead, futile in making sense of your awaiting doom. With a deep croak, your adversary spat

“We meet again. Now let’s talk.”

He pulls out a slip of paper. The paper cried as it flapped against the wind.

“Take a look. Take a close look at this. Tell me. Tell me the exact value.”

The slip of paper makes it’s way to inches from your face. Close up, you piece together the various symbols on the paper and break out in cold sweat. On the paper, largely printed, you saw:

You faint like a Victorian lady, for you do not know.

Well, for this case, I’ll probably let you die, because I’m too lazy to give you a solution to the above problem. (Also because I’m not sure (Also because I don’t know)). However, I can give you the answer though:

Basically, my point is, just like an infinite sum, an infinite product can also yield interesting results.

Challenge: Using the above result regarding the infinite product, proof that:

If the LHS and the RHS exists. Hence, proof that:

First part of the challenge question can be solved with Sec 4 syllabus while the second part involves A-level syllabus. However, the second part is easier as compared to the first.

I hope that now you know what ∏ and ∑ means! Just don’t go around doing stuff like this:

Or this:

**Thank you for your kind attention**

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So I just ended my mid-year exams (or MYEs as we like to call them) today and I have decided to start blogging once more! Yes, from now on I’ll be posting (or rather will be trying my best to post) regularly. Well in intervals of once a fortnight! (for those who have absolutely no idea as to what a fortnight is, it’s basically 2 weeks).

*“Why is this post called Accountability?”*

Yes I hear you dear reader.

So recently (while I was on my mega blogging hiatus) I have been doing a bunch of stuff. One of these things is, you guessed it, WATCHING YOUTUBE VIDEOS!!!

So while I was watching YouTube videos, I stumbled upon this channel called College Info Geek . He’s a dude called Thomas Frank and he makes these videos about productivity and study tips. There’s a video on his channel about how he was able to wake up at 6am every day:

So what I’m doing is basically what he’s doing just that I’m using the system with a couple of modifications.

So what am I going to actually do?

First, I’m posting this post which is basically a commitment. Every person who reads my blog will now expect me to post something 2 weeks later. And if I do end up not posting anything for 2 weeks, people gonna be angry.

So what if people are like:

Then won’t I just end up slacking off?

Welcome to part 2 of the system.

I made a deal with my friend. So if he sees that I don’t post anything for 2 week, I’ll buy him a drink and after 3 weeks I buy him a meal. It basically escalates to the point where I owe him a Starbucks drink if I don’t post for 5 weeks.

So yeah, accountability.

Well I actually have quite a lot of topics to discuss. Like I said I have been doing *a lot *of stuff while I wasn’t blogging.

From meeting Nobel laureates to playing piano and even watching anime.

Lots of anime.

Expect a lot more in the future. The next post will probably be something about black holes since it has been quite a while since I last did something on that topic.

Well, look out for the next post within the next 2 weeks!

Thanks for reading!

Clyde Lhui

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