English Fun Math

Math Online Tom Circle

Different Maths:
1. English (UK/USA) Math is playfully tricky (Math Olympiad, Math Puzzles eg. this Numberpile video)

2. French Math is abstractly theoretical (Group theory explains Symmetry, Rubik’s cube game,… )

3. Chinese Math (Abacus, “Chicken-Rabbits Problem” 鸡兔问题, Singapore Modeling Math) is applied & procedural (明朝 13CE invented “Determinant” 行列式 before the 19th CE “Matrix” by UK Sylvester,Chinese Remainder Theorem 韩信点兵).

4. Indian Math (Vedic Math , Factorization, Ramanujian-Style Math) is mysteriously spiritual.

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AI with Advanced Math helps in discovering new drugs

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Why French excel in math ?

Math Online Tom Circle

Mathematics and quantitative finance, France

Since 1990, there have been 22 winners of the Fields Medal, widely regarded as the Nobel Prize of mathematics. Thirteen came from just two countries, Russia and France. Russia has more winners (seven), but more than twice the population, so the honours go to France, with six winners.

Cédric Villani, the 2010 Fields Medallist, cited national character. “Maths is an abstract way of looking at the world, which fits well with the French mentality. We apply algebra to everything.” Elite institutions help too. France’s brightest school leavers progress to the grandes écoles, which traditionally educate top scientists, administrators and presidents. For maths, you want Monsieur Villani’s alma mater, the École Normale Supérieure (ENS). All 10 French Fields Medallists learnt there. At ENS, no teacher can stay longer than 10 years. Instead of ancient dons, students get tutors at the forefront…

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Happy Birthday To Urbain Le Verrier, Who Discovered Neptune With Math Alone

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Percentage Mental Calculation

Math Online Tom Circle

4% of 75 = 75% of 4

Mental Steps:
75 x 4 – > 300 – > / 100 – > = 3 (ANS. )
Is it easier ?

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The Laws of Mathematics Zermelo-Fraenkel Axioms Set Theory

Math Online Tom Circle

The Foundation of Modern Math which is based on the “Set Theory”, hit a crisis due to Bertrand Russell “Barber’s Paradox”, until the Zermelo-Fraenkel Axioms came to its rescue.

Here is a humorous example of an ISP Internet Plan violating the 9 Z-F Axioms:

How ISPs Violate the Laws of Mathematics:


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Quora: Is Abstract Math a fluke (偶然) ?

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Math Online Tom Circle

Challenge: Put ’15’ and ’14’ in the right sequence (Impossible! )


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“Essential mathematical GIFs that will make mathematics finally make sense”

Math Online Tom Circle

“Essential mathematical GIFs that will make mathematics finally make sense” by Ali Kayaspor https://link.medium.com/z1nWgHm9GU

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RMM Olympiad 2019

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Best Machine Learning / Deep Learning Books

After some research, I find that the best book on Machine Learning / Deep Learning currently on the market is: Deep Learning with Python.

It is written by François Chollet, who is the creator of the famous Keras library (Google Tensorflow). The most popular language for machine learning is currently Python, which is the subject of this book. (Another language is R, but many people find R to be less user-friendly as compared to Python).

The book is suitable for those who are studying deep learning from scratch, with zero prior knowledge. Currently, machine learning is at the peak of its hype, with many employers seeking to utilize machine learning in one way or another. It is very useful for students to pick up this skill for potential usage in the future.

Deep Learning with Python

A key competitor to Keras/Tensorflow is Pytorch, developed by Facebook. It is supposedly even more user-friendly than Keras/Tensorflow. However, the drawback is that it is too new and few books/tutorials exist.

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Alexa Ranking Below 1 Million

Thanks to all readers, Mathtuition88 has reached Alexa Rank of below 1 Million!

Current Global Rank: 983,883

SG Rank: 11,701

This is the first time Mathtuition88.com has broken the 1000,000 Alexa Rank barrier.

For ranking, the lower the better. For instance Google is ranked number 1, and YouTube is number 2. Facebook is ranked number 3.

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Powerpac Ceramic Cooker Review

This should be the cheapest ceramic cooker on the market. PowerPac is a reliable Singaporean brand that has been on the market for quite some time.

At the price of $29.90, the Ceramic Cooker even comes with a free stainless steel pot with cover (CNY Promo). Unbeatable price! It is also free delivery.

It is ideal for home steamboat / hotpot cooking. By the way, ceramic cooker can cook all types of pots and pans (unlike induction cookers).

URL: https://qoo.tn/BrGOhR/Q100000595

[S$29.90](▼51%)[PowerPac Promo]PowerPac Ceramic Cooker (Any Pot) 2000 Watts (PPIC833) //Suitable for any use with any types of pots


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Free Philips Air Fryer (Enfagrow Promo)

Official URL: Enfagrow Homepage

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Group Theorems: Lagrange, Sylow, Cauchy

Math Online Tom Circle

1. Lagrange Theorem:
Order of subgroup H divides order of Group G

Converse false:
having h | g does not imply there exists a subgroup H of order h.
Example: Z3 = {0,1,2} is not subgroup of Z6
although o(Z3)= 3 which divides o(Z6)= 6

if h = p (prime number),
2. Cauchy Theorem: if p | g
then G contains an element x (so a subgroup) of order p.
$latex x^{p} = e $ ∀x∈ G

3. Sylow Theorem :
for p prime,
if p^n | g
=> G has a subgroup H of order p^n:
$latex h= p^{n}$

Conclusion: h | g
Lagrange (h) => Sylow (h=p^n) => Cauchy (h= p, n=1)

Trick to Remember:

g = kh (god =kind holy)
=> h | g
g : order of group G
h : order of subgroup H…

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陈省身 SS Chern – “The 2nd Gauss”

Math Online Tom Circle

陈省身香港科技大学 演讲经典 (Highly Recommended – spoken in English & Chinese)


他的女婿是香港科大校长。现任校长是 Prof 陈繁星 Tony F. Chan , the ‘Search Algorithm Thesis’ supervising Prof of the 2 Google founders in Stanford University, Applied Math Department)

Key Points:

  1. SS Chern won the “Wolf Prize” in the same year with the Hungarian “Vagabond” 流浪汉 Mathematician Erdos Paul.
  2. He was appointed the Scientific Advisor of President Reagan, during which Chern recommended to build the USA Center of Math in Berkeley University. After retirement, Chern built a similar center in China at his alma mater Nankai University 南开大学 where he graduated in 1930.
  3. SS Chern was an assistant lecturer of Prof Yang WuZhi 杨武之, who liked to invite Chern to his house for dinners, where he saw his 8-year old son Yang ZhenNing 杨振宁 (Nobel Prize Physics 1958).
  4. SS Chern has guided great PhD students eg. James Simons (Billionaire Financial Investor applying…

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70-million Bounded Gap Between Primes

Math Online Tom Circle

Since Ancient Greek :

  1. Euclid had proved there are infinite primes.
  2. Sieve of Eratosthenes to enumerate the primes.
  3. Recent time 3 Mathematicians GPY attempted another Sieve method to find the bounded gap(N) of primes in infinity, but stuck at one critical step.
  4. Dr. YiTang Zhang 张益唐 (1955 -) spent 7 years in solitude after failure in academic career, in 2013 during a 10-min walk at the deer backyard of his friend’s house, he found an Eureka solution for the GPY’s critical step: $latex boxed { epsilon = frac {1} {168}}&fg=aa0000&s=2 $ which gave the first historical bounded Gap (N) from an infinity large number to a limit of 70 million.


  • Chinese love the number “8” ba which sounds like the word prosperity 发 fa (in Cantonese) . He could have instead used 160, so long as $latex epsilon $ is small.
  • The UltimateGoal of the Bounded…

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Betta Fish Tricks

Betta fish are quite intelligent and curious. I trained my Betta fish to do some simple tricks.

I bought my 5 Gallon tank on Qoo10: Betta Fish Tank. The modern viewpoint is that Betta should not be kept in small bowls or vases (though they might still survive). A 2.5 gallon tank is considered the absolute minimum, with 5 gallon and above being a better choice.

Jumping Trick and Going Through Hoop Trick

Following Pen Trick

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小算盘 大乾坤 Abacus

Math Online Tom Circle

浙江临海市 算盘博物管主人 雷国华 收藏算盘20年,散尽储蓄,感慨涕下不已。

中国的算盘上面2珠 (2×5),下面5珠(5), 共15,加1 = 16 进位。古代是16位 (Hexadecimal) 制, 半斤 八两 (= 1/2 斤 x 16 两 = 8 两)。

日本改良成10位制: 上面1珠 (1×5),下面4珠(4), 共9,加1 = 10 进位。

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3D Rotation Matrices and Examples

The following rotation matrices rotate vectors by an angle \theta in an anticlockwise direction about the x-, y-, or z-axis respectively (the rotated axis points towards the observer).

\begin{aligned}  R_x(\theta)&=\begin{pmatrix}1 &0 &0\\  0 &\cos\theta &-\sin\theta\\  0 &\sin\theta &\cos\theta  \end{pmatrix}\\  R_y(\theta)&=\begin{pmatrix}\cos\theta &0 &\sin\theta\\  0 &1 &0\\  -\sin\theta &0 &\cos\theta  \end{pmatrix}\\  R_z(\theta)&=\begin{pmatrix}  \cos\theta &-\sin\theta &0\\  \sin\theta &\cos\theta &0\\  0 &0 &1  \end{pmatrix}  \end{aligned}

Example 1

Rotating \begin{pmatrix}1\\ 0\\ 0\end{pmatrix} 45^\circ anticlockwise about z-axis:

\begin{aligned}  R_z(45^\circ)\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}&=\begin{pmatrix}  \cos 45^\circ &-\sin 45^\circ &0\\  \sin 45^\circ &\cos 45^\circ &0\\  0 &0 &1  \end{pmatrix}  \begin{pmatrix}1\\ 0\\ 0\end{pmatrix}\\  &=\begin{pmatrix}\sqrt{2}/2\\\sqrt{2}/2\\0\end{pmatrix}.  \end{aligned}

Example 2

Rotating \begin{pmatrix}0\\ 1\\ 0\end{pmatrix} 45^\circ anticlockwise about z-axis:

\begin{aligned}  R_z(45^\circ)\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}&=\begin{pmatrix}  \cos 45^\circ &-\sin 45^\circ &0\\  \sin 45^\circ &\cos 45^\circ &0\\  0 &0 &1  \end{pmatrix}  \begin{pmatrix}0\\ 1\\ 0\end{pmatrix}\\  &=\begin{pmatrix}-\sqrt{2}/2\\\sqrt{2}/2\\0\end{pmatrix}.  \end{aligned}

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十二平均律 12-tone Equal Temperament

Math Online Tom Circle

《十二平均律》 是明朝 "布衣王子" 朱载堉 (1536年-1610年) 发明的,由当时在中国传教的意大利人 利马窦 (Matteo Ricci, 1552 – 1610), 传给欧洲的法国数学家 Marin Mersenne (Mersenne Prime ) 。现代音乐之父 巴哈 Bach 第一个采用,制作世界第一架钢琴有12黑白键,并作曲 《Bach 12-tone Equal Temperament》。

朱载堉把 “1” 到 “i” 的八度 (Octave) 分为等比 (ratio) 距离的12个半音 (half note), 每个音是前音的 $latex sqrt[12] {2 }$,第8高音频率 (frequency) “i” 巧好是第一音 “1”的2倍。他用81行的大算盘算出:

Side Note:

1977年法国大学数学教授在课堂好奇地问我,你们祖先如何解代数?是用算盘吗?当时计算机还不流行,复杂的算法只能用Log Table 或 计算尺 (Slide Rule) 。


Bach 12-tone Equal Temperament



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Richard Dedekind

Math Online Tom Circle

Julius Wilhelmina Richard Dedekind (6 Oct 1831 – 12 Feb 1916)

– Last student of Gauss at Göttingen
– Student and closed friend of Dirichlet who influenced his Mathematical education
– Introduced the word Field (Körper)
– Gave the first university course on Galois Theory
– Developed Real Number ‘Dedekind Cut‘ in 1872
– Accomplished musician
– Never married, lived with his unmarried sister until death
“Whatever provable should be proved.”
– By 1858: still yet established ?
$latex sqrt{2}.sqrt{3} = sqrt{2.3} $
– Gave strong support to Cantor on Infinite Set.


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HDB Low Water Pressure: Phone Hotline

For those living near the top floor of HDB blocks, you may experience low water pressure since the water tanks are typically built at the rooftop of HDB flats.

The pressure exerted by a column of liquid of height h and density ρ is given by the hydrostatic pressure equation \displaystyle\boxed{p=\rho gh}, where g is the gravitational acceleration.


By this equation, you can understand why those at the top floors experience low water pressure, due to the low height h (from the water tank). There is probably a need for a pump to increase the pressure. For low floors, it is typically ok, the natural pressure should be enough to make the water flow fast.

Water Pressure Low Hotline: 1800-CALL-PUB

The number you should call when experiencing low water pressure for HDB flats is:


My Experience with Low Water Supply

I experienced unbelievably quick action by PUB on Chinese New Year, around 9pm. I called the PUB hotline due to very low water pressure (insufficient for even bathing). Mentally, I was prepared that probably they could only fix it the next working day (which is 2 days later).

To my amazement, a repairman arrived 15-20 minutes later, and proceeded to fix the water tank. The water pressure was back to normal in just half an hour from the time I dialed the hotline. I was quite impressed by the high efficiency.

I am not sure what caused the problem in the first place though. This problem of low water pressure has been reported by Channel News Asia.

HDB Low Water Pressure What To Do

Firstly, you should check with neighbors on the same floor whether they experience the problem of low water pressure. This is to confirm whether the problem is with the water tank pump/pipes, or an isolated problem only pertaining to your unit.

If your neighbors also experience low water pressure, then most likely it is a problem pertaining to the water tank pump/pipes. You should dial the PUB hotline (1800-2255-782) and inform them of the low water pressure. They will liaise with the Town Council and send a technician to investigate. (The service is free of charge.)

If the situation still does not improve, you may wish to purchase this “Wind Storm” High Pressure Shower Head. Basically, it turns a low pressure stream of water into a high pressure spray of water, to the extent that you can feel a “wind” blowing. It also saves water so it will pay for itself in the long run (by reducing your water bills).

URL: https://qoo.tn/BpGvgR/Q100000595

[S$11.90](▼77%)[Restock] Wind Storm Shower Head ★Powerful High Pressure Shower heads ★ For Bath


Also see: Free Water Saving Kit (Singapore)

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Dedekind Cut : “Prove 0.9999… = 1”

Math Online Tom Circle

Key Notes:

  1. Rational Numbers Q eg. 1/2, … Complete but non Continuous.
  2. Irrational numbers eg. $latex sqrt {2}$
  3. Real number is Complete and Continuous
  4. Dedekind Cut

The Proof of “0.999… = 1” by using Set Theory :

$latex A subset B$
$latex B subset C$
$latex implies A = B $

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Bit Coin and Block Chain

Math Online Tom Circle

Bit Coin is only one application of Block Chain.

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Happy Chinese New Year

Happy Chinese New Year to all readers of Mathtuition88.com!

Do check out and follow my Facebook page, where I post interesting news on Math/Education.

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Machine Learning is Just Mathematics! Free Machine Learning Resources

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Learn NUMPY in 5 minutes – BEST Python Library!

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Formula for 3×3 Matrix Inverse

Let \displaystyle A=\begin{pmatrix}a &b &c\\  d& e& f\\  g& h& i  \end{pmatrix}.

\displaystyle A^{-1}=\frac{1}{\det(A)}\begin{pmatrix}  ei-fh &ch-bi &bf-ce\\  fg-di &ai-cg &cd-af\\  dh-eg &bg-ah &ae-bd  \end{pmatrix}.

This method is about the same speed as the Gaussian-Elimination method. However, if you have already calculated det(A), using the formula may be slightly faster.

The formula provides an insight on why A is singular (not invertible) if det(A)=0. (Because, if det(A)=0, in the formula we would be dividing by 0 which is not allowed.)

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Men’s wallet with coin compartment Singapore

Despite many cashless initiatives like PayLah, Singapore is not yet a truly cashless country. Coins are still needed to buy coffee or tea at the “Kopitiam”, or to put a $1 coin for pushing a trolley at supermarkets. Even if you carry paper notes, when you receive coins as change you will need a place to put them.

Many wallets do not have coin compartments. On Qoo10, this brand “Arete First” is a Singaporean brand that sells Leather Wallets at a reasonable price, with coin compartment. The quality is good, and it is genuine leather. It can also put a decent number of cards (nine card slots).

Do check it out if you are interested in a wallet with coin compartment.

URL: Arete First Genuine Leather Wallet

[S$19.90](▼88%)[Arete First] Genuine Leather Wallet for Men | RFID blocking | Bifold | Coin compartment


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Horned Nerite Snails

Some video footage of my new Horned Nerite Snails in the Betta Tank. Really amazing and bright colors.

Taken using Canon PowerShot: Canon Camera and Camera Case Singapore

The fish tank is 5 Gallon: Betta Fish Tank

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Imaginary Numbers

Math Online Tom Circle

FINAL EPISODE – Riemann Complex Plane : 4 dimension but viewed in 3 dimensions

Episode 1 – 9 : History

Episode 10: Complex function

Function : 1 input mapped to 1 output

Multi-function : 1 input mapped to (n > 1 ) output

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Math for AI : Gradient Descent

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My most exciting Bishop’s Opening ever

Game URL: https://lichess.org/pKHiu5ey/white#1

I did make a few serious blunders, hence it is not my best game. But it is certainly quite exciting. My heart pounded furiously as I played 22. f6! sacrificing my queen… If my opponent takes the queen, it will be checkmate in 3. (He didn’t.)

White has just played 22. f6!

My opponent, being 2000 rated, played quite well. He missed punishing my blunder at around move 14.

Currently, my white opening repertoire involves the Bishop’s opening (1. e4 e5 2. Bc4), with a view towards playing f4 whenever possible to transpose into Vienna/King’s Gambit.

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Forest Man of India

Previously, I blogged about the Mountain Man of India:

Mountain Man Motivational Story

Today, I learnt that there is also a Forest Man of India.

By planting one tree a day (for many years), Jadav Payeng (a.k.a Forest Man of India) has planted a forest bigger than Central Park in New York!

This principle of doing small things everyday also applies to students and learning. By doing one Math question per day, at the end of one year, a student would have completed 365 Math questions! This would be a considerable amount that would certainly help to boost the student’s Math grade and also Math understanding.

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Goby and Shrimp Video (GEP Science)

Amazing “teamwork” between a goby fish and a shrimp! Basically, the shrimp is in charge of digging the burrow, but it is almost blind. The goby fish acts as the sentry to alert the shrimp in case of danger.

This video may be useful for GEP science, it may be tested as an example of “symbiotic relationships” in animals. GEP Science syllabus includes a brief introduction to different types of symbiosis (Mutualism, Commensalism and parasitism).

It is also very entertaining and can promote interest in science for children.

Also check out:

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Python Online Courses for Teenagers/Adults

If your child is interested in a Computer Science/Data Science career in the future, do consider learning Python beforehand. Computer Science is getting very popular in Singapore again. To see how popular it is, just check out the latest cut-off point for NUS computer science, it is close to perfect score (AAA/B) for A-levels.

According to many sources, the Singapore job market (including government sector) is very interested in skills like Machine Learning/ Deep Learning/Data Science. It seems that Machine Learning can be used to do almost anything and everything, from playing chess to data analytics. Majors such as accountancy and even law are in danger of being replaced by Machine Learning. Python is the key language for such applications.

I just completed a short course on Python: Python A-Z™: Python For Data Science With Real Exercises! The course fee is payable via Skillsfuture for Singaporeans, i.e. you don’t have to pay a single cent. (You have to purchase it first, then get a reimbursement from Skillsfuture.) At the end, you will get a Udemy certificate which you can put in your LinkedIn profile.

The course includes many things from the basic syntax to advanced visualization of data. It teaches at quite a basic level, I am sure most JC students (or even talented secondary students) with some very basic programming background can understand it.

The best programming language for data science is currently Python. Try not to learn “old” languages like C++ as it can become obsolete soon. Anyway the focus is on the programming structure, it is more or less universal across different languages.icon

Udemy URL: Python A-Z™: Python For Data Science With Real Exercises!

Related posts on Python:

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Superposition Principle of Linear Algebra

Superposition Principle

If \mathbf{u} is a solution to \bf{A}\mathbf{x}=\mathbf{b}, and \mathbf{v} is a solution to \bf{A}\mathbf{x}=\mathbf{c}, then \mathbf{u}+\mathbf{v} is a solution to \bf{A}\mathbf{x}=\mathbf{b}+\mathbf{c}.


Properties of Matrix Transpose

1) (A^T)^T=A
2) (A+B)^T=A^T+B^T
3) (kA)^T=kA^T
4) (AB)^T=B^T A^T

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Hangout @ Chinatown

Includes a review on “The Coconut Club”, which serves one of the best Nasi Lemak in Singapore.

Cooking For Fun

Well, today I have one day off and my husband also has time. We planned to go to Chinatown and have lunch there. He wanted to try a nasi lemak restaurant there. It’s the first time for us to go to Chinatown and hunt for non-Chinese food.

Chinatown of Singapore is really close to to the CBD. You can actually walk from Chinatown to CBD and also see blocks of tall buildings from Chinatown.

There are many places to have nice food in Chinatown, such as Amoy Street and Chinatown Food Street.

Chinatown has many traditional buildings of Nanyang style.

The restaurant we went is at Ann Siang Rd.

There are many bars, restaurants and cafes there.

The blue restaurant is the one we were going.

The Coconut Club.

It is not a very big restaurant. Their signature dish is nasi lemak, which can be told from the menu.


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To those starting JC life in today’s Singapore

Excellent post by top GP tutor Mr Steven Ooi. This post sums up the current competitiveness of the Singaporean education system. Similarly for Math, there is a knowledge “arms-race” to the extent that a typical Primary 3 student should know how to calculate 1+2+3+…+100 in case this question pops up in the exam. The textbook syllabus remains just as easy as in the past, but what is tested has become 10 times more difficult.

GP & English Tutor with First Class Honours, NUS

jc singapore

[Writer’s note: While this article is addressed specifically to Singapore-based students, it may hold some truth for students in many other places as well.]

Dear young friend,

I’m writing this to share some thoughts that I used to convey to every one of my students when I was still a full-time tutor in Singapore (I retired in 2016). It is just my personal opinion, formed through years of experience as a student in the 1980s and 1990s, then as a tutor in the 2000s and 2010s.

Observing student life in the 90s and then in the 2010s, I can only conclude that the school system in Singapore has become 10 times more competitive than during my time. At least 10 times. During the 90s, most of my junior college classmates didn’t even attend private tuition. We could hang out at the bowling alley, the arcade or the hawker centre most…

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Malayan Aqua Fern

This “Malayan Aqua Fern” is a very mysterious plant, there is very little information on Google about it.

There are two views on it, based on online writings.

First viewpoint on Malayan Aqua Fern:

The “Malayan Aqua Fern” is none other than Peacock Fern, or Selaginella willdenowii. It is NOT an aquatic plant, and will die if fully submerged under water. This is quite reasonable, based on the pictures of Peacock Fern online, it does look like the pictures of the “Malayan Aqua Fern”. Nonetheless, it can be planted in a crab/terrapin tank where it is not fully submerged under water.

The “Malayan Aqua Fern” may give the illusion of surviving under water when it may be slowly dying (takes around a month to die).

Websites that support this view:

Second view:

The “Malayan Aqua Fern” is an amphibious plant, able to live both submerged or in moist conditions. If this is true, then “Malayan Aqua Fern” would be an incredible aquarium plant as it looks very good in a fish tank.

Websites that support this view:

Is Malayan Aqua Fern Aquatic or Not?

The main question is whether the Malayan Aqua Fern is aquatic or not? Can it grow fully submerged under water? If you have experience with the Malayan Aqua Fern, please leave your comments below!

Malayan Aqua Fern and Betta

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Chromatic Fingering for Moonlight Sonata 3rd Movement

In Beethoven’s Moonlight Sonata 3rd movement, there is a very fast stretch of chromatic notes in bar 187.

I was wondering if the most basic fingering is good, or a more advanced fingering involving the 4th or even 5th finger would be better. (Read more on advanced chromatic fingering here: http://www.pianisttopianist.com/?p=486)

Some people on Reddit suggest some very fanciful fingerings, including “using LH (Left Hand!!) (faster playing up)4321,4312,4321,4321. So, groups of 4321 with the exception starting on the C. That part goes 4312. Just play all the way up with the LH and smack that top A (was it?) with yer Right Hand”! This fingering is apparently attributed to Alfred Brendel.

It turns out that at least 2 professional pianists seem to use the basic chromatic fingering: 2 13 123 13 123 … etc. They do not use the fourth finger at all. The starting fingering of the first note (G#) may vary.

The first pianist is Valentina Lisitsa. Watch at 0.25 speed to see the fingering at around 6:07 time. Lisitsa seems to use a “special trick” by using her left hand to play the lowest G# in the chromatic range.

The second pianist is Rousseau. You can see his fingering clearly, it is the basic chromatic fingering at around 6:34 time.

Hence, it seems that for Moonlight sonata it is not needed to use the advanced chromatic fingering. Using the pedal during the chromatic (both pianists use it to some extent) gives the illusion of a fast chromatic.

For other pieces like Fantasie Impromptu, the advanced chromatic fingering works well for greater speed.

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BTS Fan Products

BTS is one of the most popular Korean bands. Previously, we wrote on how Scientists Use Math To Discover Why BTS Jin Is The Most Handsome. Their dance steps are also amazing, hence it is no question why they are so popular worldwide. Basically, nowadays top artistes need to be good at dance + singing in order to be popular.

BTS seems very popular locally in Singapore, their concert tickets are often sold out within an hour. By the way, if a youth says that they are in “the Army”, it may not mean that they are soldiers/National Cadet Corps. BTS fans are also known as “the Army”. I once conversed with a taxi driver who told me that his daughter is in RGS, and most of her classmates are fans of K-pop including BTS. Korea is also now a top tourist destination for Singaporeans.

Both Qoo10 and Amazon have BTS fan paraphernalia, with Amazon having a more extensive collection. Most of the items ship to Singapore, with the delivery charge being quite affordable considering the huge distance from America to Singapore.

An example of a BTS cosmetic set from Qoo10.

[S$64.90](▼36%)[VT Cosmetic]【VT X BTS】★THE SWEET SPECIAL EDITION SET(Cushion(Choice 21 or 23 Tint 2 pcs included) /TTBEAUTY


From Amazon, there are collections of BTS jackets, sweaters, and more. We only showcase those that are at least 4 stars out of 5 for the Amazon rating. Basically, any product with substantial ratings of above 4/5 stars on Amazon are quality products. Especially, if they have the “Amazon’s choice” label, they are personally curated and approved by Amazon’s team.

babyHealthy Kpop BTS Love Yourself Hoodie Suga Jimin V Rap-Monster Sweater Jacket Pullover (4.1/5 stars)

BTS Kpop Socks 7 Pairs (Jungkuk, J-Hope, Suga, V, Jin, Jimin, Rap Mon) (4.8/5 stars, Amazon’s choice)

babyHealthy BTS Baseball Jacket Uniform Bangtan Boys Suga Jin Jimin Jung Kook Sweater Coat (4.3/5 stars)

KPOP BTS New Logo Hoodie Bangtan Boys Unisex Sweatershirt JIN SUGA JIMIN V (4.1/5 stars)

BTS – 2018 SEASON’S GREETINGS Set (4.6/5 stars, Amazon’s choice)

BTS 1st Album [Dark and Wild] CD + PhotoCard + PhotoBook K-POP BANGTAN (4.5/5 stars, Amazon’s choice)

Most of BTS videos have 100 million views or more:

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Free American Tourister 25” Luggage (Dumex)

Very good deal here, free 25 inch luggage (American Tourister brand) with purchase of Dumex milk powder!

This is the official Dumex Qoo-10 store, by the way.

[S$325.90](▼23%)[Dumex]Free 25 luggage(worth $220)【Bundle of 6】Dumex Mamil Gold Stage 2/ 3 / 4 / 5 Milk Formula 1.6kg


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Mathematician Couple’s Wedding Math Questions

Math Online Tom Circle


It is fun for the guests who are also mathematicians like the wedding couple…

Why don’t the IT couples do the same with computer questions (eg. private cloud, serverless, containers, DevOps,… ) , lawyers with legal cases, doctors with medical questions …

Example 1 is obvious to this guest whose PhD research topic is in Differential Geometry “Chern Class” invented by Prof S. S. Chern (陈省身) :

Example 2 is obvious to this guest unless he/she can’t count the number of sides (8):

Example 3 is to the guest with some background in secondary school organic chemistry but forgotten, definitely not linear…

Example 4 to the hobbyist of Rubik’s cube, but the answer can be guessed easily by elimination: definitely not 2019 (this year), or π.

Answer in Multiple choices(Only 1answer is correct, the other 2 answers are obviously wrong) :

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The Best Way to Take Online Exams

Today the technology has changed a lot and the way the exams are being taken is changing too with the changing times. Gone are the days when people used to take the paper and pen test. This was time consuming. But today people go for the online tests that not only help them to save their money but also help them as these tests are convenient. There are many advantages of this type of exam and we have mentioned a few of them for your understanding.

Advantages of the online exams

  • Can be taken as per the convenience

These exams can be taken as per the convenience of the higher authority and the higher authority and the candidates can decide a suitable time. This is the way both can save their time. This is a suitable method for all. You will not have to worry as this can be taken as per the ease of both.

  • Helps to save money as well as time

This exam is taken online and hence money will be saved. This is very accurate way of test as it does not involve paper and pen. This will also help to save the money for transportation. As the candidates will write the exam from their place, the transportation money will be saved. This is the most accurate way to take an exam as the results will be there in no time. There is no need to take the print out of results and this is the way we can also say that this is an eco-friendly approach to the examination. Many people go for this as this does not need paper and this helps save paper.

  • The convenient way to take a test

This is the easy and simple way to take an exam. There will be auto grade and there will be multiple answers pattern. The candidate will be given four options for each of the question asked on this online exam platform and there will be negative marking for the wrong answers given. The candidates need to be very careful while they are giving answers. This is a completely automated way of exam and hence this is more perfect in nature. There will be no wasting of time for the distribution of paper and the paper will be uploaded in no time. There is also no need to have a moderator to keep a watch on the candidates as there will be online monitoring. The tests are set up in such a manner that no student can get the answers in a book or anywhere else.

  • It will be an economical way, just go for it

This is the economical manner to take an exam. There is no paper involved, and everything will be done online only. The candidates will only need to have an internet connection and a device. They can also give the exam on their phone and this is a simple and easy manner. There is no need to rent a class and this is the way the higher authority as well as candidates can save their money. Also, there is no need to appoint a moderator to keep a watch on the students.

  • Safe, secure and accurate way that is

This is the safe and accurate way to take an exam. You only need to create a big question bank that has many questions. The student will have to select the question sets. This is the way there is a variety and safety too at the same time.

Though there are many advantages, there are a few things that you need to keep in mind while the exam is going on. We have mentioned a few points for you to keep in mind.

  • It is free way of exam

There will be no check on the students and to see if they are copying from some book. This is the reason there is a need to alter the questions from time to time. The questions should not be very simple, and they must be asked in such a manner that the candidates will not be able to find them from books easily.

  • The results will be out instantly and there will be no time waste

The results will be there immediately after the exam is over. There will be no time waste and waiting over.

Why to go for an online examination and how does that work

There are many advantages as well as disadvantages of the online exam system. There are many reasons for which you need to go for this option. One needs to create a test with the help of generator. The candidates can easily get started with the exam. They need to search through the various sites and see which test they are fine with. As you set exam you need to write a description in which you need to mention what you expect from the candidates at the time of examination. You can add some questions there. You can also add some audios or videos for a better effect.

Easy to share

These tests are easy to share, and the candidates will be able to share the tests with ease. You need to create your own timetable and set a date and time for the exam. After the exam is created you can put that online. You can upload thee exam and the students can find that. You will be able to have a look at the progress of the students. You can also make sue of the statistics and graphs so that you will be able to measure the growth of the candidates.

The easy and simple it works

This is the easy way to take the exam. You need to create an exam and upload that. This is the way you save your energy as well as time. Just go for this type of test and have a good time.

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Cheap Portable Aircon

How to stay Cool in Singapore

Singapore’s heat and humidity makes it quite unbearable to live without an air-conditioner. Especially for students who study, a hot and humid environment makes it harder to concentrate.

For long term purposes, most families will install a split aircon system like Mitsubishi Starmex.

[S$1,898.00](▼58%)[MITSUBISHI ELECTRIC]Mitsubishi Inverter Multi Split AirCon FN Series – SYS. 2 – 4 [5 Ticks]


However, for short term purposes (e.g. rental, or do not want the hassle of installing a permanent air-con), a portable aircon is the solution. A portable aircon will be enough to cool a small room (i.e. most HDB rooms and small Condo rooms).

Whatever you do, DO NOT buy those mist fans, they are close to useless since Singapore’s humidity is already close to 90%. Those mist fans work well in hot and dry desert-like environments, possible like Saudi Arabia or Australia. They work by making the mist evaporate, which does not work well in Singapore due to the high humidity. Not to mention, the constant mist may lead to some mold problems.

You might as well buy a powerful fan with vortex action:



Cheap Portable Aircon Singapore

A good brand for cheap portable aircon is Europace. The last Europace portable aircon I bought lasted close to 10 years, and counting.

[S$399.00](▼51%)[Europace][LAST 5 SETS] EuropAce Portable AirCon (10KBTU / 12KBTU) (EPAC 12T8/10Q/10P/12P) – 5 YEARS WARRANTY*


This is another newer model of Europace:

[S$399.00](▼64%)[Europace]EuropAce Portable Air Conditioner with HepaCarbon filter 10K/12K/14K BTU- (EPAC 10T6/12T6/14T6)


A cheaper brand would be Trentios. This price is probably the cheapest for a brand new portable aircon.

[S$288.00](▼68%)[TRENTIOS]Trentios/Avolta Portable Aircon 10000/12000/14000 BTU / (Include Air Freshener/Dehumidifying)


Bonus Cooling Tip

Another good way of staying cool at home is to wear Uniqlo DRY-EX T-Shirts. They are much more cooling than cotton shirts. Wearing one of these can be equivalent to a drop of 1-2 degrees Celcius. Sometimes the online Uniqlo stores do have 50% discount, do grab them during the discount if possible.

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Cute Graph Cheatsheet

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Sir Michael Atiyah obituary

Math Online Tom Circle

Atiyah is so-called the “2nd Newton” in being both a Mathematics & a Physics Grandmaster, his life achievement extends the ancient Greek Geometry to Topology and Algebra since Euclid till now, providing useful application to Quantum Physics.


Vector Bundle 向量丛

K-Theory : Vector Bundle invariant

In Sept 2018, Atiyah claimed in a seminar of his astonishingly simplest proof of the Riemann Hypothesis (one of the 6 outstanding millenian problems with US$1 million Clay Prize) – but sadly nobody believes him even after his death 3 months later !


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China Education Documentary

Similar to Singapore, China has different tiers of universities. The top tier in China is Tier 1 (一本). Within Tier 1, there is a further differentiation of “Project 985” and “Project 211” universities, the former being more selective. That is, Project 985 universities is a subset of Project 211 universities.  “Project 985” universities consists of 39 top universities, including Peking University and Tsinghua University.

The above documentary exposes some fraudulent private universities that are out to con money from simple, rural folk. It also shows the hardship experienced by the young people in China, where salaries are low but living expenses can be quite high.

Summary from IMDB:

In ancient times in China, education was the only way out of poverty; in recent times it has been the best way. China’s economic boom and talk of the merits of hard work have created an expectation that to study is to escape poverty. Many Chinese parents see getting a degree as a way of ensuring their children have better, less impoverished lives than their own. But in 1997, the Chinese government privatized universities and education became a commodity. It’s a money-making business where there’s a profit to be made – The University Entrance Exams Day – the day students receive their results – often determines the future of one’s whole life. If you get good results, you get into the state-subsidized universities. If you don’t, there are other universities you can get into but they are more costly, often substandard, and run as lucrative businesses. Some unscrupulous colleges employ marketing techniques to target poor, rural families who are often less streetwise to their tactics. More often than not, these families can’t afford the tuition fees and some end up selling livestock and even their homes to cover the cost of their child’s education. But many colleges can’t live up to their promises and each year, more than two million graduates in China do not find work. For students and families who incurred debt while studying, education is no longer a way out of poverty but a cause of it along with unemployment and despair.

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