《数学与人类文明》数学与现代文明：

1. 哲学：希腊 Euclidean Geometry

2. 艺术 : Arabic 文艺复兴， 达芬奇, Golden Ratio

3. 工业革命：Descartes Analytic Geometry, Newton Calculus

4. 抽象: Gauss “Non-Euclidean Geometry” , Paradox in Set Theory, Godel “Incomplete Theorem” .

《数学与人类文明》数学与现代文明：

1. 哲学：希腊 Euclidean Geometry

2. 艺术 : Arabic 文艺复兴， 达芬奇, Golden Ratio

3. 工业革命：Descartes Analytic Geometry, Newton Calculus

4. 抽象: Gauss “Non-Euclidean Geometry” , Paradox in Set Theory, Godel “Incomplete Theorem” .

**IMO 2020** : China World Team Champion ( 5 Golds + 1 Silver), the only one in the World with Perfect Score.

This Chinese IMO coach comments :

IMO 2020 is the ** easiest** in the past 10 years, compared to 2015 (tougher) & 2018 (toughest).

王文湛 教授 (80) is a Math Prof in 清华，yet he can’t solve his 10-year-old grandson (Primary 4) Math Olympiad questions in Combinatorics.

The harms to kids math education:

1. Teach too early the higher math, only “acrobatic technique” , not genuine math education.

2. Waste parents’ money in unnecessary tuition for Olympiad math.

3. Pressure on kids.

In the 1988 IMO only 11 contestants solved this 6th problem, including 2 future Fields Medalists : Terrence Tao (12 years old) & G. PERLMAN.

The most elegant solution came from the 17 year-old Balgarian contestant using “Reductio Absurdum” Proof : Simple & “Violent” way.

https://m.toutiaoimg.cn/a6816353938204262927/?app=news_article_lite&is_hit_share_recommend=0

The USA IMO coach Prof Loh is the son of 2 Singaporean Math Prof (dad) & JC Math teacher (mom).

He uses the Chinese “drill” style + American “think” style to beat China IMO Team last few years.

Inequalities (for IMO Math) : Just remember this diagram (to reconstruct from memory each time)

邻边＜ 斜边

AM : Arithmetic Mean

GM : Geometric Mean

IMO questions could not be good Math of deep meaning, given that the contestants have to solve the tricky problems in a short time frame of 2 to 3 hours…IMO Prize is just an indication of Math capability, we can’t equate IMO winners as Mathematicians.

【专访美国奥数队总教练：奥数比赛对一个国家的数学水平有用吗？】复制这条信息€80avm€a56OR2€后打开👉今日头条极速版👈

2019 both China and USA co-win the IMO Team Champion, both teams consist of almost Chinese ethnic students (except 1 white american) & Chinese coaches.

**Key Points** :

**IMO** questions : exclude Calculus.

**IMO Boot Camp**: 3 month-training.

**Calculus** : In High Schools just learn formula & apply, in university learn the theory.

**France** is a Math power but weak in IMO, why?

Both Methods (I) & (II) below does not use

Algebra,which is not taught to primary school kids until secondary school (above 12 or 13 years old).

(I) **Ancient Chinese Arithmetic Method:**

例2：The sister is 13 years old this year, while the brother is 9 years old.

When the total of their ages is 40, how old will they be?

【这位妈妈太绝了！竟把小学6年奥数化为13句口诀】

*Trick: Age difference does not change over time, they + or – in tandem. When 2 ages change, the multiple between them also changes. *

**(II) Singapore Modeling Math**:

**［例2］**

Compare the Ancient Chinese Arithmetic (I) & Singapore Modeling Math (II):

- (II) is better for young kids to understand visually the logic behind,
- (I) is more a memorised trick but poor math education pedagogy. In [例 2] it is very hard for kids to understand why the 2 steps (40+4) & (40 – 4), although they lead to the correct answer.

Notes:

- Singapore Modeling Math (II) is invented by the Singapore Professor Lee Peng Yee 李秉彝 from Nanyang Technological University (National Institute of Education) in the 1980s, combining: Ancient Chinese Arithmetic [as shown in (I)], Polya Problem Solving Methodology, Visual Graphical Modeling
- French Fields Medalist Cédric Villani recommended the Singapore Modeling Math Pedagogy to all primary schools in France.

- The most famous Chinese Ancient Arithmetic “The Chicken-Rabbit Problem” (《孙子算经》，dated 3 AD – 5 AD）which had inspired Prof Lee Peng Yee. Watch video “鸡兔问题 :“.

https://tomcircle.wordpress.com/2019/03/14/chinese-math-olympiad-primary-trick/

- The Physics Noble Prize Laureate Paul Dirac’s “Coconuts & Monkeys Problem” : use Higher Math (Sequence & Linear Algebra, solved by Prof Richard Halmos) vs Singapore Modeling Math (3rd solution) .

Picture Order (From Left) :1. 2. 3. 4.

**1. ****Caucher Birkar**** ****(UK / Kurdish – Iran, 40**)

https://www.bbc.com/news/science-environment-45032422

**2. Alessio Figalli (Italy, 34**)

https://www.quantamagazine.org/alessio-figalli-a-mathematician-on-the-move-wins-fields-medal-20180801/

**3. Akshay Venkatesh (Australia / India, 36**)

Studies number theory and representation theory.

https://www.quantamagazine.org/fields-medalist-akshay-venkatesh-bridges-math-and-time-20180801/

**4. Peter Scholze (Germany, 30**)

Intersection between number theory and geometry

How likely is it that a mathematics student can’t solve IMO problems?

Is there a fear of embarrassment in being a math Ph.D. who can’t solve problems that high-school students can? by Cornelius Goh

**Problem A3**

A function f is defined on the positive integers by:

for all positive integers n,

Determine the number of positive integers n less than or equal to 1988 for which f(n) = n.

What is the explanation of the solution of problem 3 from IMO 1988? by Alon Amit

One basket of eggs.

1粒1粒拿，正好拿完。

Remove 1 by 1, nothing left in basket.

2粒2粒拿，还剩1粒。

Remove 2 by 2, one left in basket.

3粒3粒拿，正好拿完。

Remove 3 by 3, nothing left in basket.

4粒4粒拿，还剩1粒。

Remove 4 by 4, one left in basket.

5粒5粒拿，还差1粒才能拿完。

Remove 5 by 5, short of one to complete.

6粒6粒拿，还剩3粒。

Remove 6 by 6, 3 left in basket.

7粒7粒拿，正好拿完。

Remove 7 by 7, **nothing** left in basket.

8粒8粒拿，还剩1粒。

Remove 8 by 8, one left in basket.

9粒9粒拿，正好拿完。

Remove 9 by 9, nothing left in basket.

请问筐里最少有几粒鸡蛋 ？

At least how many eggs are there in the basket?

[Hint] This is a Chinese Remainder Problem (” 韩信点兵”)

—— [Solution] —–

Let there be minimum X eggs in the basket.

Remove 1 by 1, nothing left in basket:

X = 0 mod (1) …[1]

=> trivial & useless !

Remove 2 by 2, one left in basket:

X = 1 mod (2) … [2]

Remove 3 by 3, nothing left in basket:

X = 0 mod (3) … [3]

Remove 4 by 4, one left in basket:

X = 1 mod (4) … [4]

Remove 5 by 5, short of one to complete:

X = -1 mod (5) … [5]

Remove 6 by 6, 3 left in basket:

X = 3 mod (6) … [6]

Remove 7 by 7, nothing left in basket:

X = 0 mod (7) … [7]

Remove 8 by 8, one left in basket:

X = 1 mod (8) … [8]

Remove 9 by 9, nothing left in basket:

X = 0 mod (9) … [9]

Simplify [6] = [3]

X = 3 mod (6)

X = 3 mod (3×2)

X = 3 mod (3)

X = 0 mod (3) … [3]

Notice [3], [7],[9]: X is multiple of 3, 7, 9

=> X is mulitple of LCM (3,7,9) = 63

**X = 0 mod (63) … [3,7,9]**

Similarly,

X = 1 mod (2)

X = 1 mod (4)

X = 1 mod (8)

=> X = 1 mod (LCM {2, 4, 8}) [**Note @**]

=>** X = 1 mod (8) … [8]**

**[Note @]:**

X = 1 mod (2)

=> X – 1 = 2n

Similarly,

X = 1 mod (4) <=> X – 1 = 4m

X = 1 mod (8) <=> X – 1 = 8t = 4.(2n)t’ = 2.(4m)t”

These 3 equations <=> X = 1 mod (8)

Simplify the 9 equations by considering only the remaining 3 equations:

**X = 0 mod (63) … [3,7,9]**

**X = -1 mod (5) … [5]**

**X = 1 mod (8) … [8]**

Since (8, 5, 63 ) are **co-primes pair-wise –**** ***ie (8, 5), (8, 63), (5, 63) are relative primes to each other in each pair* – we can apply the Chinese Remainder Theorem to the last 3 equations:

…[#]

By sequentially “mod” the equation [#] by 8, 5, 63, we get:

(63×5).u1 = 1 mod (8) **… [8]**

315.u1 = (39×8+3).u1 = 1 mod (8)

3.u1 = 1 mod (8)

=> **u1 = 3 **[as 9 = 1 mod (8)]

(63×8).u2 = -1 mod (5)**… [5]**

504.u2 = (50×11 – 1).u2 = -1 mod (5)

-1.u2= -1 mod (5)

**u2 = 1 **[as -1 = -1 mod (5)]

(5×8).u3 **=** 0 mod (63)**… [3,7,9]**

40.u3 = 0 mod (63)

**u3 = 0** [as 0 = 0 mod (63) ]

[#]:

X = (63×5).u1 + (63×8).u2 +(5×8).u3 mod (8x5x63)

X = (63×5).**3** + (63×8).**1** + (5×8).**0** mod (2520)

X =** 63 x 23** = 1449 mod (2550)

Prof ST Yau **丘成桐** , Chinese/HK Harvard Math Dean, is the only 2 Mathematicians in history (the other person is Prof Pierre Deligne of Belgium) who won ALL 3 top math prizes: Fields Medal 1982 (at 27, proving Calabi Conjecture), Crafoord Prize (1994) , Wolf Prize (2010).

**Key Takeaways **:

1. **On Math Education**:

◇ Compulsary Math training for reasoning skill applicable in Economy, Law, Medicine, etc.

◇ Study Math Tip: read the new topic notes 1 day before the lecture, then after lecture do the problems to enhance understanding.

◇ Read Math topics even though you do not understand in first round, re-read few more times, then few days / months / years / decades later you will digest them. (做学问的程序).

◇ Do not consult students in WHAT to teach, because they don’t know what to learn.

◇ Love of Math beauty is the “pull-factor” for motivating students’ interest in Math.

◇ Parental Pressure.

2. “3D” facial photo using Math

3. Pi-Music: 1 = “do”, 2 = “re”, 3 =”me”…

Pi =3.1415926…

4. **Math Olympiad**: Prof ST Yau had criticised publicly it as a bad Math training, not the “real” Math.

An audience tested Prof ST Yau on a Math (Accounting) Puzzle which he couldn’t solve on the spot. He said Mathematicians are poor in +-×÷ arithmetic.

5. **Chinese students in USA**: China sends over 200,000 students to USA universities. They are good in secondary / high school Math with *known* solutions, but poor in graduate PhD Math which requires “**out-of-the-box**” independent thinking skill for finding *unknown* solutions. Recent few years Chinese students (eg. Stanford Prof 李骏 : 1989 Harvard PhD) in USA have improved standard in PhD research.

6. Research is not for fame. It takes many years to think through an interesting topic.

**Reference**:

1. Prof ST Yau’s **Best Seller** Book 《The Shape of Inner Space》avail @ NLB (Ref #530.1) 11 copies in most NLB branches@ AMK, Bishan etc.

2. Interview Prof ST Yau by HK TV (**Cantonese**)

3. 丘成桐 (2008) 评中国 和 美 国的教育 : 中国学生不爱看课外书, 因为考试太重, 课余时间花在玩电脑游戏。

4. 丘成桐 (2016): 中国大学本科要注重基础教育, 才能培养世界级一流人才

**Calculate**:

Let

Or:

Quite messy to expand out:

This 14-year-old vienamese student in Berlin – Huyen Nguyen Thi Minh discovered a smart trick using the identity:

or more general,

He multiplies x by (3-1):

.

.

.

**When n = 16,**

The result is not surprising to China but to USA:

♢Recently China government bans IMO training in schools.

♢Obama was surprised that the USA IMO team consists of predominantly Chinese American students.

IMO Math is like ‘Acrobatics’ to real ‘Kung-fu’, it is not real Math education, but special ‘cute’ techniques to solve tough ‘known’ solution problems. Real Math is long R&D solving problems with UNKNOWN solution (eg. Fermat’s Last Theorem, Riemann Conjecture,…)

2 types of Math: **Algorithmic ** or **Deductive (演绎)**. Chinese long traditional ‘abacus’ mindset, procedural computational Math is Algorithmic, applied to special cases (eg. astronomy, calendar, agriculture, architecture, commerce,…). European Greek’s Euclid deductive, step-by-step axiom-based proofing, is theoretical, generalized in all cases (Geometry, Abstract Algebra,…)

Look at the Fields Medal (aka ‘Nobel Prize’ of Math) super-power – France – which has produced 1/3 of the Fields Medalists, but performing so-so in IMO. In contrast, China has **ZERO** Fields Medalists, albeit dominating IMO championship for more than 2 decades!

**IMO 2015:**

https://www.imo-official.org/results.aspx

**USA 1st,**

**China 2nd,**

South Korea 3th,

North Korea 4th,

Vietnam 5th,

Australia 6th

Iran 7th

Russia 8th

Canada 9th

**Singapore 10th ** [2012 Individual World’s Champion ]

Ukraine 11th

Thailand 12th

Romania 13th

**France 14th**

…

United Kingdom 22th