Open Cubic Root

Mental Trick

It was discovered by the Martial Art writer Liang Yusheng 武侠小说家 梁羽生 (《白发魔女传》作者), who met Hua Luogeng (华罗庚) @1979 in England:
2³= [8]
8³= 51[2]
3³= 2[7]
7³= 34[3]

The last digit pairs :
[2 <->8] , [3 <-> 7]
Others unchanged.

Example:

$latex \sqrt[3]{658503} = N$
Last three digits 503 <-> ...[7]
First three digits 658:
 (8³ =512)< 658 < (729 = 9³)
=>  8...
Answer : $latex \sqrt[3]{658503} = N$= 87
Note: Similar trick for opening $latex \sqrt[23] {200 digits}$ by an indian lady Ms Shakuntala (83) dubbed "Human computer".

[caption id="attachment_2408" align="alignnone" width="560"] Human computer[/caption]

数学大师: 无法

我现正在上2010全国华乐比赛-大师班。
大师音乐评分标准：
1. 大法（基本功）
2. 小法（特色）
3. 无法（风格）

数学大师Math Masters 也是如此，
1. 大法（基本功 Classical Math）
2. 小法（特色 Math Olympiad techniques）
3. 无法（风格 Abstract Math -> French "Math Composition"）

Probability: Dice

Throw 2 dies, the sum of points i, j and probability P (i+j):

2(1/36) 3(1/18) 4(1/12) 5(1/9) 6(5/36) 7(1/6)
8(5/36) 9(1/9) 10(1/12) 11(1/18) 12(1/36)
=> Bell curve symmetric both sides, peak 1/6 (sum 7).

Vector Algebra

Vector changes Geometry to Algebra

1. No complexity of Analytical Geometry
2. Remove the astute dotted (helping) line in Geometry
3. No need diagram: Use only 2 vector properties:
Head- to-Tail:
$latex \vec{AC}=\vec{AB}+\vec {BC}$
Closed Loop:
$latex \vec{DE}+\vec{EF}+\vec{FD}=0$
4. Enable Computer automated proof of Geometry via Algebra.

Example: 任意四边形 Quadrilateral ABCD with M,N midpoints of AB, CD, resp.
Prove： MN=1/2(BC+AD)
Proof: (by vector):

Consider MBCN:
MN=MB+ BC+ CN..(1)

Consider MADN:
MN=MA+ AD+ DN..(2)

(1) +(2):
2MN=(MB +MA) +
(BC +AD) +(CN +DN)

but (MB +MA) =0,
(CN +DN) =0 [same magnitude but different direction cancelled out ]

=> MN=1/2 (BC +AD)

Special cases:
1. A = B (=M)
=> triangle ACD
AN = 1/2 (AC +AD)
2. BC // AD
=> Trapezium ABCD
MN=1/2 (BC +AD)
=> MN // BC // AD

Chicken & Egg Problem

Chicken or Egg First?

The Chinese Qing Dynasty Emperor QianLong (1711 – 1799) asked the French Jesuit P. Michael Benoist:

"Chicken or Egg, which one appeared first ?"

Benoist answered,

"Your Majesty, the Bible 'Genesis' said God created all birds (include chicken) and fishes on the 5th day, so chicken first !"

 

Fun Identity

1+2+3+...+n = n(n+1)/2

both sides * α
α + 2α + 3α +...+ nα = nα(nα+1)/2

both sides * sin
 sin α + sin2α +...+ sin nα = sin nα.sin (nα+1)/2

(Discovered by Hua Luogeng )

http://en.wikipedia.org/wiki/Hua_Luogeng

De Moivre

De Moivre (1722)

$latex (Cos A+iSin B)^n = Cos nA + i.Sin \:nA$
- French Protestant, jailed by the Catholic King Louis XIV (14th), exiled to UK.
- Gave Tuition in the Duke home where he met Newton with the newly published book “Principia” and the "New Math" Calculus, he self-paced study to become expert in Calculus.
- He never got a Professor job despite Newton and Leibniz’s help.
- He observed gambling in coffee shops, invented the Game Theory.
- He liked to sleep. One day he declared he would sleep 20 mins more per day. On 73rd day when he accumulated 24 hrs more, he died at 87 yrs old !