# Quod Erat Demonstrandum

## 2012/01/30

### 由 3 出發

Filed under: Fun — johnmayhk @ 3:52 下午

e.g. 1

$3$

$=\sqrt{1+8}$

$=\sqrt{1+2\times 4}$

$=\sqrt{1+2\sqrt{16}}$ (more…)

## 2012/01/29

### [TED] Robert Lang全新型態的摺紙

Filed under: Fun — johnmayhk @ 7:53 下午

TED Filmed Feb 2008

## 2012/01/28

### 標誌裡的數學

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 7:54 下午

## 2012/01/20

### arctan,pi,complex numbers

Filed under: mathematics,Pure Mathematics — johnmayhk @ 12:02 下午
Tags:

$\pi=\tan^{-1}1+\tan^{-1}2+\tan^{-1}3$

$\tan^{-1}(\frac{1}{2})+\tan^{-1}(\frac{1}{3})=\frac{\pi}{4}$

（易知 $\tan^{-1}(\frac{1}{n})+\tan^{-1}(n)=\frac{\pi}{2}$，故上述兩式等價。）

## 2012/01/18

### [FW] NOVA scienceNOW | Twin Prime Conjecture | PBS

Filed under: Fun — johnmayhk @ 9:51 下午
Tags:

### [News] BSD猜想有望破解 華數學家轟動學界

Filed under: Report,University Mathematics — johnmayhk @ 11:06 上午

BSD猜想有望破解 華數學家轟動學界 (more…)

## 2012/01/17

### Core Math 某題-counting

Filed under: mathematics,NSS — johnmayhk @ 12:50 下午

10 人（當中包括 A 和 B）排 10 人隊，若 A 不能排第一，B 不能排第尾，問排法多少？

## 2012/01/15

### Ｄ數字

Filed under: Fun — johnmayhk @ 12:04 上午

$3'=?$

1. 若 $a$ 是質數，則 $a'=1$
2. $(ab)'=ab'+a'b$

## 2012/01/14

### 續 Core Math 某題

Filed under: mathematics,NSS — johnmayhk @ 9:32 上午

I’m asked to generalize the solution of the previous post, okay, do it.

Let $n$ be a positive even integer.

Set 1: {$a_1,a_2,\dots,a_{n/2},a_{n/2+1},\dots,a_n$}
Set 2: {$a_1,a_2,\dots,a_{n/2},0,a_{n/2+1},\dots,a_n$}

where $a_1 < a_2 < \dots < a_{n/2} < 0 < a_{n/2+1} < \dots < a_n$.

Let $\sigma_1, \sigma_2$ be the standard deviations of Set 1 and Set 2 respectively.

Show that $\sigma_1 > \sigma_2$.

Solution (by ugly brute force) (more…)

## 2012/01/12

### Core Math 某題

Filed under: mathematics,NSS — johnmayhk @ 10:25 下午

Set 1: {$a,b,d,e$}
Set 2: {$a,b,c,d,e$}

（其中 $a < b < c < d < e$

$\sigma_1, \sigma_2$ 分別是 Set 1，Set 2 的標準差（standard deviation），即

$\sigma_1^2=\frac{a^2+b^2+d^2+e^2}{4}-(\frac{a+b+d+e}{4})^2$

$\sigma_2^2=\frac{a^2+b^2+c^2+d^2+e^2}{5}-(\frac{a+b+c+d+e}{5})^2$

$\sigma_1^2 > \sigma_2^2$

## 2012/01/11

### [FW] When learning maths, abstract symbols work better than real-world examples

Filed under: mathematics,Report — johnmayhk @ 10:40 上午

## 2012/01/09

### [YouTube] Dangerous Knowledge (Philosophy, Physics, Mathematics) -BBC

Filed under: Fun,Report — johnmayhk @ 4:00 下午

## 2012/01/07

### 某題-級數

Filed under: Fun,HKALE,Pure Mathematics — johnmayhk @ 12:04 上午

1 , 12 , 123 , 1234 , … , 12345678910 , 1234567891011 , …

1 , 21 , 321 , 4321 , … , 10987654321 , 1110987654321 , …

$1,\frac{12}{21},\frac{123}{321},\frac{1234}{4321},\dots$

## 2012/01/06

### 變形金剛原型？

Filed under: Fun,Report — johnmayhk @ 7:38 上午

http://www.scientificamerican.com/article.cfm?id=computational-origami-robot

(OT) Old stupid jokes…

Who is transformer’s sister?
Tran (more…)

## 2012/01/05

### 斐波那契數與二項係數

Filed under: HKALE,Pure Mathematics — johnmayhk @ 9:51 上午

$(a+b)^n=\displaystyle\sum_{r=0}^nC^n_ra^{(n-r)}b^{r}$

$(fg)^{(n)}=\displaystyle\sum_{r=0}^nC^n_rf^{(n-r)}g^{(r)}$