Ellipse Proof by Circle : Affine Transformation

Cumbersome Ellipse proof below :

By Affine Transformation the Ellipse to a Circle, the corresponding ratios preserved. Hence it is easier to prove in the circle below:

Similar triangles CQV ~ CTQ


圆幂 定理 II: The Affine Transformation FROM Circle to Ellipse. Also applicable to all other 2 Theorems.


Proof : Ellipse Area from Circle :



[7.1] 好书推荐

1. E. T. Bell: 《Men of Mathematics” 》 – [Comments:] Irish Hamilton & Quartenions – Bell said Halmiton later’s life was a tragedy in trying to make quartenions universally applied in Physics but failed. Prof Yang : Quartenions will be “universally useful” in future.

Note : Quartenions (1,i,j,k) & Special Relativity – Does it ring a bell to you the UNIVERSAL 4 FORCES (Gravity, Electric, Magnetic, Particles Weak force) :


2. E. Segrè (吴健雄的博士导师) & the story of the missing Periodic Table Element ’43’ (Tc) discovered by accident from unwanted trash.

[7.2] 重力势能和弹性势能




When Flutter does not save you – Level Up Coding

Flutter (or the other similar Framework “React-Native” ) is only doing its best at UI for single-code development in multi-platform (iOS, Android).

When you need to add on some non-UI features in your apps (eg. customized QR-code, customized foreign language keyboard, etc), you need to develop in native codes (C/C++… ) then called by Flutter / React-Native (Which Choice ? )


PISA 2018 China beat Singapore as Top

Chinoiseries 《汉瀚》[中/英/日/韩/法]


Analyse the Pisa 2018 :
1. Both China & SG are better in Math than Science by small gaps (1 & 8 resp.)

2. China beats SG in Math by VERY big gap (591-564 = 27), due to SG’s education mistake (no Algebra before PSLE) .
3. Reading big gap (555 – 535 = 20) : English is not SG’s mother tongue (mistake! ) , but 中文 is to China.

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新的数学教育链 Math Education Chain

21st century schools teach 20th century Math, not currently 19th century Math!


Key Points:

  1. “Push down” syllabus : Some University Math to Secondary & High Schools, the later’s Math to Primary schools.
  2. At each stage of education, no increase of workload to students.
  3. Example : Linear Algebra (2-3 dim Vector Space) in High-school Math.
  4. Separately teach Calculus (Differentiation & Integration) and Analysis (epsilon-delta) .
  5. Teach not solving more difficult Math problems but wider scope:
  6. 华罗庚: 学数学最好要读懂多种语文,eg. 中,英,& 法/德/俄。
  7. 倾中国的国力,开发中文的 电脑数学工具软件 (eg. Mathlab, Mathematica, Maple), 免费让中学/大学生用。

  • 英才数学教育:
    法国Prepas / Grandes Ecoles 数学

英国GCE H1/H2/H3

(HoTT) Homotopy Type Theory & Functional Programming




Relativistic Space Time, Fengshui, Isometry

风水 Fengshui (aka Geomancy) is Einstein Relativistic using Compass (东西南北上下 =Space) & Timing (吉凶日/时间 = auspicious date & time).

\Yu (Space) \Zhou (Time) = Space-Time
宇环 (universal surrounding)
宙时 (instant timing)


Axonometry = Isometry ( dengjiao toushi 等角透视,ie Angle Preserved Projection )

Chinese art is axonometric using isometry geometry, scrolling from right to left along time. Example : this digital animated 12-century painting

清明上河图 Qingming Festival River Scene(北宋)

“Why the world relies on a Chinese ‘perspective’” by Jan Krikke, China, AI and Quantum Physics


China Gaokao 2019 Interesting Math Question

Gaokao = Baccalaureate = GCE A Level

…[Solution scrolled below]

Let Ф = 0.618

… Golden Ratio Ф = 0.618

defined as:

“Head to Throat” vs “Throat to Navel” = a/b = Ф ,


“Head to Navel” vs “Navel to bottom of Legs” =(a+b) / c= Ф

Knowing :

Head to Neck = 26 cm &

Legs = 105 cm

Find the height (H) ?

Note: The tricky part is to know “the neck is below the throat” & “the legs start some distance below the navel”, hence 2 inequalities :

a < 26, c > 105

H = (a + b) + c = Фc + c = (1+Ф) c = 1.618c
H > 1.618 x105
H > 169.9

H = a + b + (c)
= a + b + (a+b) / Ф
= (a+b) (1+1/Ф)
= (a +a/Ф) (1+1/Ф)
= a(1+1/Ф)(1+1/Ф)
< 26.(1+1/Ф)(1+1/Ф)

H < 178.2

ANS = H ~ 175 (B)

苏联数学三巨头之 Gelfand 盖尔范德


(想看更多合你口味的内容,马上下载 今日头条)

5G Concert by Players Worldwide

这就是5G的厉害:The Power of 5G

六国音乐家在六地通过5G网络,合奏巴赫C小调前奏曲与赋格。6 countries play over 5G simultaneously with no transmission latency.

第一小提琴德国,Violin 1 (Germany)
第二小提琴西班牙,Violin 2 (Spain)
第三中提琴蒙古,Viola (Mongolia)
第四大提琴瑞典,Cello (Sweden)
第五钢琴美国, Piano (USA)
第六定音鼓日本, Timpani (Japan)


Different Views of Category, Type & Set

Different views of objects 对象 by:

1. Category 范畴 “Cat” (morphism* between Objects, Functors ‘F‘ between Cats);

2. Set 集合 (a “smaller Cat”, only objects);

3. Type 类型 (deal only with same kind of objects: Int, String, Boulean…).

Note : Category can be a Set (SET) , Group, Ring, Vector Space (Vect) , “Topo” (Topology) … any algebraic structure with Associative Morphism (Map or ‘Arrow’ ) between them.

Note (*) : A morphism 态 in layman’s term is best illustrated by geometry:

2 triangle objects are similar 相似 = homomorphism 同态

2 triangle objects are congruent = isomorphism 同构


Note: Analogy –
Category : School
Type : Boy Class, Girl Class
Set : Students mixed of Boys, Girls

RIP Sir Michael Atiyah

Singapore Maths Tuition

Rest in peace, Sir Michael Atiyah. Many scientists have called Atiyah the best mathematician in Britain since Isaac Newton.

Read also our previous posts:

Source: New York Times

Michael Atiyah, a British mathematician who united mathematics and physics during the 1960s in a way not seen since the days of Isaac Newton, died on Friday. He was 89.

The Royal Society in London, of which he was president in the 1990s, confirmed the death but gave no details. Dr. Atiyah, who was retired, had been an honorary professor in the School of Mathematics at the University of Edinburgh.

Dr. Atiyah, who spent many years at Oxford and Cambridge universities, revealed an unforeseen connection between mathematics and physics through a theorem he proved in collaboration with Isadore Singer, one of the most…

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Logic & Math (Set Theory)

Cambridge Prof Peter Smith:

You can download the book at the bottom link from the below web site.


Philosophy using Math – that is cool!

Set Theory is Philosophy, see this “Set Proofing Technique” taught in French Baccalaureate high school but Cambridge GCE A level ignores :

Prove : A = B
You need to prove 2 ways:
A ⊂ B
B ⊂ A
=> A = B

In the Bible 《John 14:11》
Jesus said to his disciples:
“Believe me when I say that I am in the Father and the Father is in me”.

Proof: Father (God) = Jesus

“Father in Me” :
Father ⊂ Jesus
“I am in the Father” :
Jesus ⊂ Father
Jesus = Father (God)
[QED. ]

Merry Xmas! 圣诞快乐!

The amazing power of word vectors

the morning paper

For today’s post, I’ve drawn material not just from one paper, but from five! The subject matter is ‘word2vec’ – the work of Mikolov et al. at Google on efficient vector representations of words (and what you can do with them). The papers are:

From the first of these papers (‘Efficient estimation…’) we get a description of the Continuous Bag-of-Words and Continuous Skip-gram models for learning word vectors (we’ll talk about what a word vector is in a moment…). From the second paper we get more illustrations of the…

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Maths is a potential life changer — from wages to dating


When you are stuck with too many choices in life, whether buying houses recommended by agents, finding schools for children, hiring staff among the hundreds of CVs…apply ‘e’ the number from nature (logarithm) to make smart and efficient decision under the constraints of time and resources.

Theory of Optimal Stopping” = 1/e ~ 37%


e = 2.7 18281828 45 90 45…

If you do a house search with 20 properties, then by the Theory of Optimal Stopping, pick the first property which is better than the first 7 properties you see (7. 4 = 20 x 37%).

[Note] Ponder over the hidden Philosophy behind – “Brilliant Limit” :

\displaystyle \boxed{ \lim_{n \to \infty}\left({\frac{n!}{n^n}}\right)^{\frac{1} {n}} = \frac{1} {e}}

The Hardest H3 Math Question (Combinatorics)

Singapore Maths Tuition

I think this may be one of the hardest H3 Math Questions in history. It is taken from RI H3 Prelim 2018. It seems that even in top schools like RI, there are less than 50 people taking H3 Maths in any given year. Part (d) is extremely hard to get the formula for general r. In fact during the exam it is probably wise to skip such questions or give partial answers (e.g. the formula for r=3) as it is not worth the time for 3 marks.

See also our related blog posts:

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H3 Mathematics Resource Page

Singapore Maths Tuition

H3 Mathematics is the pinnacle of the Junior College Mathematics syllabus in Singapore. It contains a glimpse of actual Math that Mathematicians do, and it requires true mathematical understanding and technique to do well. (H1/H2 math requires a lot of practice, but not true understanding. It is quite common for students to “apply the method” and get the correct answer without having any idea of what they are actually doing.)

Topics in H3 Mathematics include Functions, Sequence and Series, Combinatorics, and even Number Theory. Certain schools also include topics like Linear Algebra and Differential Equations. Certainly, the H3 Math questions have a Math Olympiad style to them.

Here are some practice questions for H3 Math (more will be added in the future), with some hints. Questions are adapted from actual H3 prelim papers.


Q1) The function $latex f$ is such that $latex f(x+2)=af(x+1)-f(x)$, for all real $latex x$ and…

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Solution to HP A4 Printer Paper Mysterious Question

Singapore Maths Tuition

A while ago, I posted the HP A4 Paper Mysterious Question which goes like this:

Problem of the Week

Suppose $latex f$ is a function from positive integers to positive integers satisfying $latex f(1)=1$, $latex f(2n)=f(n)$, and $latex f(2n+1)=f(2n)+1$, for all positive integers $latex n$.

Find the maximum of $latex f(n)$ when $latex n$ is greater than or equal to 1 and less than or equal to 1994.

So far no one seems to have solved the question on the internet yet!

I have given it a try, and will post the solution below!

If you are interested in Math Olympiad, it is a good idea to invest in a good book to learn more tips and tricks about Math Olympiad. One excellent Math Olympiad author is Titu Andreescu, trainer of the USA IMO team. His book 104 Number Theory Problems: From the Training of the USA IMO Team is…

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How to type Chinese characters in LaTex (on Mac, using TeXShop)

Singapore Maths Tuition

This is one of the easiest ways to type Chinese characters in LaTeX on Mac, using the default TeXShop editor. (If you know of an easier way, please let me know in the comments below!)

I have tried for hours, experimenting with different packages, before “discovering” the following steps. Hope it helps!

Step 1) Add “usepackage{ctex}” to the beginning of the document. This will load the main package ctex.

Step 2) It is very important to save the LaTeX file in UTF-8 format, otherwise all Chinese characters will appear as question marks. The preferred way to do this is via:

TeXShop > Preferences > Encoding = Unicode (UTF-8). (see image below)

This will “permanently” set the format as UTF-8 by default. If you don’t do this, an annoying thing that can happen is that your TeX file reverts to “non-UTF8” upon saving. That means, the Chinese characters may appear correctly…

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WiFi Password = Integral Answer

China 南京航空航天大学 Nanjing University of Aeronautics and Astronautics set the WiFi password as the answer of this integral (first 6 digits).

Can you solve it?

(If can’t, please revise GCE “A-level” / Baccalaureate / 高考 Calculus 微积分)

Answer : Break the integral (I) into 2 parts:

I = A(x) + B(x)

\displaystyle A(x) = \int_{-2 }^ {2} x^{3}. \cos \frac{x}{2}.\sqrt{4-x^2}dx

\displaystyle B(x) = \int_{-2 }^ {2} \frac{1}{2}\sqrt{4-x^2}dx

A(x) = – A(-x) => Odd function
=> A(x) = 0 since its area canceled out over [-2, 2]

B(x) = B(-x) => Even function
\displaystyle\implies B(x) = 2\int_{0 }^ {2} \frac{1}{2}\sqrt{4 - x^2}dx
\displaystyle\implies B(x) = \int_{0 }^ {2} \sqrt{4 - x^2}dx

Let x = 2 sin t => dx = 2 cos t. dt

x = 2 = 2 sin t => sin t = 1 => t = π / 2

x = 0 = 2 sin t => sin t = 0 => t = 0

\displaystyle B(x) = \int_{0 }^ {\pi/2} \sqrt{4 - 4.\sin^{2} {t} }. (2 \cos t. dt)

\displaystyle \implies B(x) = \int_{0 }^ {\pi/2} 2.\cos t. (2 \cos t. dt)

\displaystyle\implies B(x) = \int_{0 }^ {\pi/2} 4 \cos^{2} t. dt

\displaystyle \cos ^{2} t = \frac {1 + \cos 2t} {2 }

\displaystyle\implies B(x) = \int_{0 }^ {\pi/2} (2 + 2\cos 2t) . dt

\displaystyle\implies B(x) = (2 t) \Bigr|_{0 }^ {\pi/2} + (2. \frac{1}{2} \sin 2t) \Bigr|_{0 }^ {\pi/2}

\displaystyle\implies B(x) = (\pi ) + \sin \pi = \pi

\boxed{ I = \pi = 3.14159}

A smarter method using Analytic Geometry: A circle of radius 2 is

x^2 + y^2 = 4 \implies y = \sqrt {4 -x^2}

Programming is Math Proof: Structured Programming


  • Dijkstra, Edge Wyber (born 1930 Rotterdam)
  • Goto is harmful
  • Structures: sequence, selection, iteration

Three Programming Paradigms:

1. Structured Programming (1968 Dijkstra)

  • Impose discipline on direct transfer of control aka “Goto“.
  • “If/ then /else, do/while” control structures are structured.
  • Language: Pascal, etc

2. Object-Oriented ‘OO’ (1966 Ole Johan Dahl & Kristen Nygaard)

  • Impose discipline on Indirect transfer of control (Polymorphism, Inheritance, Encapsulation:constructor‘ function of class, it’s local variables = instance variables).
  • Language: C++/ C# / Objective-C, Java / JavaScript, Go, Python, etc.

OO = Data + Function
Class = Object + Method

3. Functional Programming ‘FP’ (1958 John McCarthy’s LISP language, based on Math “Lambda Calculus” from Alonzo Church 1936).

In LISP: Data = Function

  • Impose discipline upon assignment (side effect, immutability of data, Referential Transparency *).
  • Category Theory = Program : Monad, etc
  • Language: Pure FP: {Lisp, Clojure, Haskell}, Hybrid OO+FP: {Scala, Kotlin}, etc.

4. Any more ?

All Programs can be constructed from just 3 structures (Böhm and Jacopini, 1966):

Sequence / Selection / Iteration.

Dijkstra’s Math Proofs for:

1. Sequence – by simple enumeration.

  • Math Technique: trace the inputs of the sequence to the outputs of the sequence.

2. Selection – by reapplication of enumeration.

  • Each path thru the selection was enumerated. If both paths eventually produced appropriate Math results, then the proof was solid.

3. Iteration – by induction.

  • Proved the case for 1 by enumeration.
  • Assume if N case was correct.
  • Proved N+1 case correct by enumeration.
  • Also proved the starting and ending criteria of the iteration by enumeration.

Note (*): Referential Transparency means – a function (f) with a given parameter always returns the same result.

Eg. Trigonometric function (f) = sin 30 = 0.5 (always! )

In FP, a program is many layers of composition of functions of function, with each function guaranteed (math proven) always returning the same result for given parameters (aka arguments). This is software safety with no surprising unexpected result due to side effects (like database search / Web search / IO output errors).


Clean Architecture – A Craftsman’s Guide to Software Structure and Design (by Robert C. Martin)

[Singapore National Library NLB #004.22]

IT Application’s 5 Evolution Stages

From Mainframe/Mini Server-based Monolithic Application since 80s, to
Browser-based thin-client-thick-Server Application from mid-90s, to
MicroServices-based Applications in 2018’s…

Small is beautiful !!

“Dinausaur” Monolithic Applications give way to Microservice-based Application.

Microservice is a realisation of “mini”-SAAS (Software As A Service).

Component-based Software a la hardware components VLSI is becoming a reality.


Pure to Applied Math: Self-driving Cars & “Sum of 2 Squares” Polynomial

Key Points:

  • 1900 Hilbert’s 17th Conjecture: Non-negative Polynomial <=> sum of 2 squares (Proved by Emile Artin in 1927)
  • Computing Math : approximate by optimisation with “Linear Programs” which are faster to compute.
  • Princeton Mathematicians applied it to self-driving cars.



Sum of 2 Squares <=> always non-negative ( 0)

13 = 4 + 9 = 2^{2} + 3^{2}

P (x) = 5x^2+16x+13 = (x+2)^{2} + (2x+3)^{2} \geq 0

Self-driving Car: Trajectory = P (x)

P(x) < 0 where the car’s position in the trajectory;

Obstacles are positions where P (x) 0.

This is one of the many cases of Pure Math turned to be Applied Math in last few decades. Other examples:

Is Applied Math => Pure Math ?

School System Video (Do not make a fish climb trees)

Singapore Maths Tuition

Singapore is being mentioned around 4:54. Very nice video. The truth is that the classroom of today is still nearly the same as the classroom of 150 years ago. There needs to be a “Educational Revolution” parallel to that of the Industrial Revolution. Many children cannot fit into the single classroom model, leading to growth in diagnosis of behavioral “problems” such as ADHD in developed nations.

Americans who are tired of “Common Core” may want to check out Singapore Math for their kids, which is highly acclaimed in the educational realm.

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Population Differential Equations and Laplace Transform

Singapore Maths Tuition

Malthus Model
$latex displaystyle frac{dN}{dt}=BN-DN=kN$

$latex N$: Total population

$latex B$: Birth-rate per capita

$latex D$: Death-rate per capita

$latex k=B-D$

Solution to D.E.:
$latex displaystyle boxed{N(t)=widehat{N}e^{kt}},$

where $latex widehat{N}=N(0)$.

Logistic Equation
$latex begin{aligned}

Logistic Case 1: Increasing population ($latex widehat{N}<N_infty$)
$latex begin{aligned}

The second expression can be derived from the first: divide by $latex s$ in both the numerator and denominator.

Logistic Case 2: Decreasing population ($latex widehat{N}>N_infty$)
$latex begin{aligned}

Logistic Case 3: Constant population ($latex widehat{N}=N_infty$)
$latex displaystyle N(t)=N_infty$

Basic Harvesting Model: $latex displaystyle boxed{frac{dN}{dt}=(B-sN)N-E}.$

$latex E$: Harvest rate (Amount harvested per unit time)

Maximum harvest rate without causing extinction: $latex boxed{dfrac{B^2}{4s}}$.

$latex displaystyle boxed{beta_1,beta_2=frac{Bmpsqrt{B^2-4Es}}{2s}}.$

$latex beta_1$: Unstable equilibrium population

$latex beta_2$: Stable equilibrium population

Extinction Time: $latex displaystyle boxed{T=int_{widehat{N}}^0frac{dN}{N(B-sN)-E}}.$

Laplace transform of $latex f$
$latex displaystyle F(s)=L(f)=int_0^infty e^{-st}f(t),dt$

Tip: Use this equation when the questions…

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11-year old math and chess prodigy in Singapore

Singapore Maths Tuition

Source: Channel News Asia

Aarushi Maheshwari solved the famous “Cheryl’s Birthday Problem” when she was only 9. She is also a chess champion and can play blindfold chess.

Watch the video below to learn more!

Also read our previous post on The Most Accomplished 10-Year-Old (Gifted pupil).

For those who want to learn more about Olympiad Math and International Chess, check out the previous two links. Math and Chess are two of the most intellectually challenging activities that can develop the intelligence of kids.

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The Inventors of the 10 Computer languages

  1. Python (Dutch Guido van Rossum, 1956)
  2. Java (Canadian James Gosling 1955)
  3. Javascript (USA Brendan Eich, 1961)
  4. C (USA Dennis Ritchie, 1941 – 2011 )
  5. C++ (Denmark Bjarne Stroustrup, 1950)
  6. Ruby (JAPAN Yukihiro “Matz” Matsumoto, 1965)
  7. Perl (USA Larry Wall, 1954)
  8. Pascal (Switzerland Niklaus Wirth, 1934)
  9. Lisp (USA John McCarthy, 1927 – 2011)
  10. PHP (Denmark Rasmus Lerdorf, 1968)


Below the 3 hotest Functional Programming language influenced by Lisp:

11. Kotlin(Russia Andrey Breslav)

12. Scala (USA Martin Odersky)

13. Haskell (USA)

14. Clojure (USA Rich Hickey)

How to help gifted children deal with hyper-sensitivity

Singapore Maths Tuition

This should be useful for parents of gifted kids.

Also read:

Books for Gifted Children

Book by Truly Gifted Kid (GEP Book)

Source: Aleteia

Gifted children can be challenging and exhausting: unable to handle frustration, finding it difficult to accept boundaries, talking endlessly, negotiating the slightest rule or order … However, according to Jeanne Siaud-Facchin, a psychologist who is a specialist in gifted children (see our previous article in which she explains what makes a child gifted), creating boundaries and setting limits is a vital necessity for their emotional development, and the only way to avoid the creation of permanent and escalating conflicts. How should we respond to these tantrums due to a child’s heightened emotional state? We’ve spoken to some experienced moms who share what to do to calm and reassure our sensitive children.

Why are many gifted children challenging and exhausting?

Of course, not all gifted children…

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