## 2008年3月28日 星期五

### The Mysterious Fish Number - 153

John 21:11 Simon Peter climbed aboard and dragged the net ashore. It was full of large fish, 153, but even with so many the net was not torn.

The above scripture is taken from the famous bible story "Jesus and the Miraculous Catch of Fish". Having read the scripture, you may wonder why the number of fish was counted and recorded with such a precision. Indeed, scholars throughout history have argued that 153 has a far deeper meaning!

Mathematically, 153 is a wonderful number.

(A) In the binary system, 153 is written as 10011001, which is a symmetric figure.

(B) 153 is a triangular number (1+2+3+...+16+17 = 153).

(C) 153 is the sum of the first 5 factorials (1!+2!+3!+4!+5! = 153).

(D) 153 is the sum of the cube of each of its digits (1^3+5^3+3^3 = 153).

Here comes the grand finale!

(E) Start with any multiple of 3. By repeatedly summing up the cube of each digit, you will always end up with 153 eventually.

Example 1: Take 66.

6^3+6^3 = 432
4^3+3^3+2^3 = 99
9^3+9^3 = 1458
1^3+4^3+5^3+8^3 = 702
7^3+0^3+2^3 = 351
3^3+5^3+1^3 = 153

Example 2: Take 2001.

2^3+0^3+0^3+1^3 = 9
9^3 = 729
7^3+2^3+9^3 = 1080
1^3+0^3+8^3+0^3 = 513
5^3+1^3+3^3 = 153

Do you know why?

### Grothendieck's 80th birthday

Today's the 80th birthday of Alexander Grothendieck. He is usually considered one of the greatest mathematicians of the 20th century. It's hardly an exaggeration that almost all modern algebraic geometers are influenced by his work (in particular, his famous EGA and SGA books). He is also quite a remarkable character: he withdrew from mathematics at the age of 42, and in 1991, he even left his home and disappeared. He is now said to live in southern France or Andorra and to entertain no visitors. Though he has been inactive in mathematics for many years, he remains one of the greatest and most influential mathematicians of modern times. Find out more about him in the following wiki article!

## 2008年3月27日 星期四

### 天秤找假幣（一）

1) 現在有 n 個金幣，已知其中一個是假的，而假的金幣重量與真的金幣不同，但我們不肯定是輕了還是重了。現在給你一個天秤，你每次可以在天秤的兩邊放一些金幣看看哪邊較重，還是兩邊重量一樣。問：

a) 若 n = 5，最少要秤多少次？
b) 若 n = 12，最少要秤多少次？

2) 情況跟第一題一樣，但今次有另外一個已被確定為真的金幣存在。當 n = 5 時，最少要秤多少次？

## 2008年3月24日 星期一

### Pi day

It has been a while now since March 14 - the pi day, but do you know how professors and students at Princeton University celebrates this math day? Here they go: they had a pi recitation competition and a pie eating contest on 3/14 1:59pm! (recall pi = 3.14159... and some graduate students there can remember up to a 100 digits!)

Find out also how the Indiana House of Representatives passed a bill proposing three formal - and inaccurate - definitions of pi, namely 3.2, 3.23 and four (!), and how a graduate student here made a skirt with the digits of pi sewn on it every year, just to celebrate this special day for mathematicians.

Have your pi and eat it too!

By Josephine Wolff
Senior Writer
Published: Friday, March 14th, 2008 on the Daily Princetonian

Applied math professor Ingrid Daubechies first learned about pi as a young child, when her father told her to go around the house and measure the circumference and diameter of every circle she could find.

“It made an incredibly strong impression on me,” Daubechies said. “I remember my father let me touch [the] circles I would otherwise never have been allowed to, like the precious china plates hanging on our wall. I got to take them down and measure them. My mother was very worried.”

Today is a day of celebration for Daubechies and her colleagues and students in Fine Hall, and not just because it’s the last day of midterms. March 14 is celebrated in math classes across the country to honor the number pi — 3/14 mirrors the first three digits of pi, 3.14.

Pi Day is a time of rejoicing — a time for pie-eating contests, pi-digit-recitation competitions, pi-themed clothing, pi jokes (What do you get when you divide the circumference of the sun by its diameter? Pi in the sky!) and other forms of unabashed nerdiness.

“What’s fun about pi is that everyone knows the number,” Daubechies said. “We all see it in elementary school. People feel they have an appreciation for what it means.”

Pi and popular culture

Popular television shows, like “The Simpsons” and “Star Trek,” have helped spread pi’s appeal to mainstream audiences.

In perhaps its most heroic role, pi saves the USS Enterprise starship in the “Star Trek” episode “Wolf in the Fold,” when the evil Redjac takes over the computer running the ship.

Spock tells the computer to “compute to the last digit the value of pi,” destroying the computer with this impossible demand and thereby freeing the ship.

In an episode of “The Simpsons,” “Lisa’s Sax,” two school girls play a patty-cake-style hand game while chanting: “Cross my heart and hope to die, Here’s the digits that make pi: 3.1415926535897932384...”

In another “Simpsons” episode, “Marge in Chains,” Kwik-E-Mart proprietor Apu testifies that he can recite pi to 40,000 decimal places. “The last digit is one!” Apu says, correctly.

“Mmmm, pie,” Homer says.

In preparation for the episode, “Simpsons” writers wrote to NASA asking for the 40,000th digit of pi, and NASA responded by sending back a printout of the first 40,000 digits.

One trillion digits!

Forty thousand may seem like a lot of digits, and in fact, 40 decimal digits are all that we really need for practical calculations, Daubechies said.

She referenced a 1996 article about pi, in which lead author David Bailey, the chief technologist at the Lawrence Berkeley National Laboratory, wrote that the “value of pi to 40 digits would be more than enough to compute the circumference of the Milky Way galaxy to an error less than the size of a proton.”

The first 40,000 digits, however, are barely the tip of the computational iceberg that has helped researchers calculate more than one trillion digits of pi in the past few decades. Yasumasa Kanada at the University of Tokyo has computed more than 6.4 billion digits and currently holds the world record.

Not everyone, however, has been eager to see pi grow so long.

In 1897, the Indiana House of Representatives passed a bill proposing three formal — and inaccurate — definitions of pi. The bill was based on the work of a Dr. Goodwin, who fancied himself an amateur mathematician and claimed he had definitively computed three official values for pi: 3.2, 3.23 and four. Goodwin also copyrighted his ideas and announced that he would allow only schools in Indiana to teach them for free and that everyone else in the country would have to pay him a royalty if they wished to teach or use these “facts.”

The bill was approved by the Senate Education Committee, but Purdue mathematician C.A. Waldo put an end to it before it could be passed into law.

Perhaps the most famous truncation of pi is in the “Second Book of Chronicles.” King Solomon is constructing the Temple of Jerusalem, and he builds a tub of “ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about” (1 Kings 7:23). In other words, the circular tub has a diameter of 10 cubits and a circumference of 30 cubits, in which case pi would equal exactly three.

The rabbi Nehemiah, in 150 AD, pointed out that while the diameter of the tub had been measured from the outer rim of the thick stone, the circumference measure was taken along the inner circle, and it was this discrepancy that accounted for the inaccurate portrayal of pi.

A brief history of a lengthy number

The first of pi’s trillion calculated digits date back to the ancient Egyptians and Babylonians, who used 3.16 and 3.125, respectively, as rough estimates of pi.

Mathematicians across the globe have been computing more and more accurate estimates of pi for centuries since, but it was a set of mid-20th-century advances in computing that allowed for the discovery of billions and billions of digits.

John von Neumann, one of the first faculty members at the Institute for Advanced Study in Princeton, played a crucial role in designing ENIAC, the first electronic computer, which in 1949 more than doubled the number of known digits by correctly calculating 2,037 of them.

Today, calculating long values of pi is a common method for testing new computer chips, Daubechies said. A great deal also remains unknown about pi, she added, most notably the question of whether its decimal expansion is normal, that is, whether each digit occurs with relatively equal random frequency in pi.

Irrational holiday

Lillian Pierce ’02 GS said she usually celebrates Pi Day at 3:14 p.m. in 314 Fine Hall or the common room next to it. In the annual pi-reciting and pie-eating contests held in Fine, Pierce added, the physics team usually beats the math team at eating.

“Pi itself is one of the fundamental constants of mathematics. So you could say that it is very important. But it’s also very beautiful,” Pierce said. “Pi Day is a great way for mathematicians to poke fun at themselves. If everyone else is always poking fun at us, it’s only fair we should have a turn too.”

The first Pi Day celebration at Princeton took place in 1988, said Yang Mou ’10, president of the undergraduate math club. This year’s celebration is today at 1:59 p.m. in the Fine Hall common room.

Pierce also makes a pi dress every year for the occasion. In the past, she has embroidered roughly 50 digits of pi around the hem of a dress. This year, however, she is going for “sheer quantity of digits.” If she uses small enough numbers, she estimates, she may be able to fit as many as 1,000 digits around a skirt.

Pierce, like Daubechies, first discovered the joys of pi as a girl when she read a 1992 New Yorker article about pi by Richard Preston.

“I’d always liked numbers, but reading about this many numbers all strung out in an infinitely long, beautiful sequence was incredibly inspiring,” Pierce said.

2005 年：3 月 27 日
2006 年：4 月 16 日
2007 年：4 月 8 日
2008 年：3 月 23 日
2009 年：4 月 12 日
2010 年：4 月 4 日
2011 年：4 月 24 日

## 2008年3月21日 星期五

### 數學資料庫五歲了!!!

MD是一個以推廣數學為宗旨的網站，慶祝生日也當然要與數學拉上關係。我們很榮幸邀得香港中文大學的張婉琳小姐主持於三月十五日主持了一個學術講座，向我們的會員為「An introduction to Strong Conical Hull Intersection Property」這個題目作介紹，大家都獲益不少。

MD人才濟濟! 1,2,3... 笑~

## 2008年3月20日 星期四

### 選擇公理

1.設P(n)為命題“在n個非空集合中可以在每個集合中抽一個元素出來”；
2.考慮P(1)，我們可以在那一個非空集合中任意抽一個元素出來，故P(1)成立；
3.假設P(k)對於一些自然數k成立，考慮k+1個非空集合，由歸納假設知我們可以在頭k個集合中抽一個元素出來，而我們又可以在第k+1個中任意抽一個元素出來，故P(k+1)成立；
4.由數學歸納法原理知P(n)對一切自然數成立；
5.證明完畢。

1. 巴拿赫-塔斯基悖論：存在一個方法，可以將一個三維實心球分成有限個部分，然後通過旋轉和平移，重新組合為兩個半徑和原來相同的完整的球；
2. 塔斯基分割圓問題：存在一個方法，可以將平面上的一個圓分割成有限多塊，然後通過平移，重新拼合成面積相同的正方形；
3. 在以下情況中，存在一個策略令只有有限個犯人不被釋放:

## 2008年3月17日 星期一

### 循環論證 (I) --- 圓周公式

X : 給出一個圓形，圓周為 $r$，圓面積的公式 $\pi r^2$能夠運用積分的方法來證明。

X : 那麼，請問圓形的圓周 $2\pi r$ 可否運用類近的積分算式去「證明」呢？

X 提問時，特別強調證明一詞。

$\pi$定義熟悉的讀者，應該明白背後的原因吧。

$L$ 為給出圓形的圓周，設圓的圓心為 $\left(0,0\right)$，所以圓的方程為$x^2+y^2=r^2$

Let $L$ be the circumference of the given circle, suppose the circle center at origin, so the equation of circle is $x^2+y^2=r^2$.

$y=\sqrt{r^2-x^2}$

By arc length integral and symmetry:

$L=4\int_{0}^{r}\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$

After simplification :

$L=4r\int_{0}^{r}\frac{1}{\sqrt{r^2-x^2}}\,dx$

Therefore :

$L=4r\left[\sin^{-1}\frac{x}{r}\right]_{0}^{r}=4r\left[\frac{\pi}{2}-0\right]=2\pi r$

For reference:
Definition of Pi
Arc length forumla

## 2008年3月16日 星期日

### 造馬

1 x 3 x 5 x ... x (2n-1) = (2n-1)!! 種

C(2,2) x C(4,2) x ... x C(2n,2) = n! (2n-1)!!

## 2008年3月10日 星期一

### MD Academic Seminar Mar 2008

Strong Conical Hull Intersection Property (strong CHIP in short) is a geometric concept introduced in 1990’s for studying constrained convex optimization problems. In the seminar, a short account of its historical background, its mathematical formulation and its significance in recent researches on optimization will be given.

No specific prerequisites will be needed for the seminar.

REFERENCES:
1.
Deutsch, F. (1998): The role of the strong conical hull intersection property in convex optimization and approximation. In: Chui, Schumaker, eds., Approximation Theory IX, Vanderbilt University Press, Nashville, TN.
2.
Heinz H. Bauschke, Jonathan M. Borwein, Wu Li (1999): Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization. Mathematical Programming 86, 135-160
3.
Deutsch, F., Li, W., Ward, J.D. (1997): A dual approach to constrained interpolation from a convex subset of Hilbert space. J. Approx. Theory 90, 385-414

## 2008年3月6日 星期四

### 猜包揼

[這是網友singmay向數學資料庫手記提供的文章。]

[1] 這點可以用Stirling’s formula證明。Stirling’s formula是 n! 的近似公式，有興趣的朋友可以在這裡找到一篇很好的介紹：

[2] 這裡隨機的嚴格定義是所有可能組合出現的概率也相等。

[3] 另外一個有趣的解釋是預先被選的那人出「包」或「揼」已沒有所謂，所以成功機會剛好是兩倍。

[以下是 marco_dick 的補充。]