Chessboards, Hats, and Poetry :Chessboards, Hats, and Poetry :

Some Rigorous and Not-So-RigorousSome Rigorous and Not-So-Rigorous

Mathematical ResultsMathematical Results

C. L. Liu C. L. Liu 　劉炯朗　劉炯朗Tsing Hua, HsinchuTsing Hua, Hsinchu

數裡有詩？ 詩裡有數！

Hats

Poetry

0cbxax 2Mathematics

Chessboards

It all begins with a chessboard

Covering a Chessboard

88 chessboard

21 domino

Cover the 88 chessboard with thirty-two 21 dominoes

Enumeration

……

Number Theory, Probability, Statistics, Physics, Chemistry, …

Archimedes’ Stomachion Puzzle

17,152 ways

How do I love thee, Let me count the ways. － Elizabeth Barrett Browning

Enumeration

一蓑一笠一扁舟

一丈竿頭一只鈎

一水一拍似一唱

一翁獨釣一江秋

　　　一字詩∣康熙

一卷煙，一片山，幾點雲影，

一道水，一條橋，一支櫓聲，

一林松，一叢竹，紅葉紛紛；

　　　滬杭車中∣

徐志摩

Enumeration

Tyger! Tyger! Burning bright,

In the forests of the night.

What immortal hand or eye

Could frame thy fearful symmetry?

－ William Blake

Symmetry : Polya’s Theory of Counting

……

　天對地　雨對風

　爸爸對公公

　白板對紅中

　雷隱隱　雨濛濛

　小姑喚大嫂

　晚輩稱尊翁

　人間清暑殿

　天上廣寒宮

對 聯

又是春殘也

如何出翠幃

落花人獨立

微雨燕雙飛

寓目魂將斷

經年夢亦非

那堪愁向夕

蕭颯暮蟾輝

　

　　《春殘》　翁宏

落花人獨立

微雨燕雙飛

對 聯

無情對

一二三四五六

七忠孝仁愛禮義

廉　贈劉兆漢校

長

（忘

八）

（無

恥）

億萬千

百唐元明

清兆漢

Madam

Able was I ere I saw Elba

Palindrome

迴 文

潮迴暗浪雪山傾

遠浦漁舟釣月明

橋對寺門松徑小

檻當泉眼水波清

迢迢綠樹江天曉

靄靄紅霞海日晴

遙望四天雲接水

碧波千點數鷗輕

迴 文 詩

迴文 對聯

上海自來水來自海

上中山歸隱客隱歸山

中 斗六高材生材高六

斗

A Truncated Chessboard

21 domino

Cover the truncated 88 chessboard with thirty-one 21 dominoes

Truncated 88 chessboard

Proof of Impossibility

21 domino

Truncated 88 chessboard

Impossible to cover the truncated 88chessboard with thirty-one dominoes.

人有悲歡離合，月有陰晴圓缺，此事古難全。　　 水調歌頭－ 蘇軾

Truncated 88 chessboard

Proof of Impossibility

Impossible to cover the truncated 88 chessboard with thirty-one dominoes. There are thirty-two white squares and thirty black squares. A 2 1 domino always covers a white and a black square.

A Defective Chessboard

Triomino

Any 88 defective chessboard can be covered with twenty-one triominoes

Defective Chessboards

Any 2n2n defective chessboard can be covered with 1/3(2n2n -1) triominoes

Any 88 defective chessboard can be covered with twenty-one triominoes

Prove by mathematical induction

Mathematical Induction

The first domino falls.If a domino falls, so will the next

domino.

All dominoes will fall !

To see the world in a grain of sand, And heaven in a wild flower, Hold infinity in the palm of your hand, And eternity in an hour. 　　－ William Blake

Mathematical Induction

To see the world in a grain of sand,

And heaven in a wild flower,

Hold infinity in the palm of your hand,

And eternity in an hour.

　　－ William Blake

從一粒沙看世界，從一朵花看天堂，把永恒納進一個時辰，把無限握在自己手心。

一花一世界，一沙一天國，君掌盛無邊，剎那含永劫。

一砂窺塵世，一瓣證瑤台；隻手持無量，剎那悟如來。

Proof by Mathematical Induction

Basis : n = 1

Induction step :2 n+1

2 n+1

2 n 2 n

2 n

2 n

Any 2n2n defective chessboard can be covered with 1/3(2n2n -1) triominoes

If there are n wise men wearing white hats, then at the nth hour allthe n wise men will raise their hands.

The Wise Men and the Hats

Basis : n =1 At the 1st hour, the only wise man wearing a white hat will raise his hand.

Induction step : Suppose there are n+1 wise men wearing white hats.

At the nth hour, no wise man raises his hand.

At the n+1st hour, all n+1 wise men raise their hands.

……

The Wise Men and the Hats

One white hat

1st hour : hand raised

Two white hats

1st hour : silence

2nd hour : hands raised

Five white hats

1st hour : silence

2nd hour : silence

3rd hour : silence

4th hour : silence

5th hour : hands raised

以銅為鑑，可正衣冠；以古為鑑，可知興替；以人為鑑，可明得失。　　　　　 新唐書－魏徵傳

x • y x + y

I don’t know.I knew you would not know.However, neither do I.

Now, I know.Now, I know.

Now, I know.

I Don’t Know

Two Integers, 1 < x, y < 51

x = 4 , y = 13

Sound of Silence

水泉冷澀弦凝 絶，凝 絶不通聲漸歇，別有幽愁暗恨生，此時無聲勝有聲。　　　　　　　　　　 　琵琶行－白居易

Hello darkness, my old friend,I've come to talk with you again.           The Sound of Silence － Simon & Garfunkel 

To communicate through silence is a link between the thoughts of man. － Marcel Marceau 

Self Information : I (x) = - lg p (x)

Mutual Information : I (x, y) = - lg p (x) + lg p (x | y)

Information Theory

Measure of Information

Another Hat Problem

Design a strategy so that as few men will die as possible.

No strategy In the worst case, all men were shot.

Strategy 1 In the worst case, half of the men were shot.

Another Hat Problem

0 1 1 0 …….… 1

………..

1 1 0 　 .……… 1

1 0 .……… 1

1

= 1

= 0

Another Hat Problem

0 1 1 0 …….… 1

………..

1 1 0 　 .……… 1

0 ………. 1

1

= 1

= 1

1

Coding Theory

Representation of information in alternate forms for

efficiency reliability security

Algebraic Coding Theory

Cryptography

四　猜成語一句

欲罷不能

四　猜唐詩一句

山在虛無縹緲間

十千為萬

惜花春早起

愛月夜遲眠　

　贈胡萬川主

任斗酒十千同醉

月芳華廿九共

惜花

Yet, Another Hat Problem

Hats are returned to 10 people at random, what is the probability that no one gets his own hat back ?

張冠李戴

Recurrence Relations

bn = 0.92 bn-1

bn = 0.42 bn-1 + 0.5 bn-2

bn = 1.22 bn-1 + 0.3 bn-2 + 0.2 bn-3

b2004 = 0.92 b2003

1, 1, 2, 3, 5, 8, 13, …

fn = fn-1 + fn-2Fibonacci Numbers

Derangements

dn : number of derangements of n objects

dn = (n-1) dn-1 + (n-1) dn-2

d3 = 2 d2 + 2 d1 = 2 1 + 2 0 = 2

d1 = 0

d2 = 1

d4 = 3 d3 + 3 d2 = 3 2 + 3 1 = 9

d10 = 9 d9 + 9 d8 = 1,334,961

……

Derangement of 10 Objects

Number of derangements of n objects

]!

1)1(....

!3

1

!2

1

!1

11[!

nnd n

n

]!10

1)1(....

!3

1

!2

1

!1

11[!10 10

10 d

Probability !10

1)1(....

!3

1

!2

1

!1

11

!101010 d

36788.01 e

頂真格（鈎句，流水句）

雲鬢花顏金步搖 芙蓉帳暖度春宵

　春宵苦短日高起 從此君王不早

朝後宮佳麗三千人

三千寵愛在一身

君臣相顧盡霑衣

東望都門信馬歸

　歸來池苑皆依舊

太液芙蓉未央

柳　芙蓉如面柳如眉

對此如何不

淚垂

　　　　　　　　　

長恨歌∣

白居易

尋夢？撐一支長篙，

向青草更青處漫溯，

滿載一船星輝，

在星輝斑斕裏放歌，

但我不能放歌，

悄悄是別離的笙簫；

夏蟲也為我沉默，

沉默是今晚的康橋！

　　　　　　　

　再別康橋∣徐志摩

頂真格（鈎句，流水句）

風雨送春歸，

飛雪迎春到，

已是懸崖百丈冰，

猶有花枝俏。

俏也不爭春，

只把春來報。

待到春花爛漫時，

她在叢中笑。

　　　　　　　

卜算子詠梅∣毛澤東

頂真格（鈎句，流水句）

柳色青

柳色青青

青滿城

滿城風雨煙光送

風雨煙光送遠行

遠行君向歸山路

君向歸山路前去

前去離亭芳草青

離亭芳草青無數

無數山

山彎水潺湲

彎水潺湲行路難

行路難時時往還

往還多是名場客

多是名場客行急

行急無論多少程

無論多少程千百

千百人　戀芳春

人戀芳春不似君

不似君家有老親

家有老親常倚閭

常倚閭　望君馬

望君馬到金台下

到金台下桂花香

桂花香報報高堂

高堂正屆稀齡壽

正屆稀齡壽春酒

春酒遲君衣錦傾

遲君衣錦傾金斗

金斗酌

酌春風

春風人共醉

人共醉融融

頂真格（鈎句，流水句）

數裡有詩

詩裡有數

？

！？

！

怎一個愁字 了得

尋尋覓覓，冷冷清清，

淒淒慘慘戚戚。

乍暖還寒時候，最難將息。

三杯兩盞淡酒，怎敵他，

晚來風急。

雁過也，正傷心，卻是舊時相識。

滿地黃花堆積，憔悴損，

如今有誰堪摘。

守著窗兒，獨自怎生得黑，

梧桐更兼細雨，到黃昏，點點滴滴。

這次第，怎一個愁字了得。

　　　　　　　　　　　　聲聲慢∣

李清照

怎一個愁字 了得

愁

只恐雙溪舴艋舟，

載不動許多愁。

　　　武陵春∣

李清照

一斛明珠萬斛愁，

關山漂泊腰肢細。

　　　圓圓曲∣

吳梅林

舟載斛量

新妝宜面下朱樓，

深鎖春光一院愁。

　　　　　　春詞∣

劉錫

禹白髮三千丈，

離愁似個長。

　　　　　　秋浦歌∣

李白

問君能有幾多愁，

恰似一江春水向東流。

　　　　　　虞美人∣

李後主

寂寞深閨，柔腸一寸愁千縷。

　　　　　點絳唇∣

李清照

一懷愁緒，幾年離索。

　　　　　釵頭鳳∣

陸游

一點芭蕉一點愁。

　　　　　雙調水仙子夜雨∣

徐再

思凝眸處，從今又添，

一段新愁。

　　　　　鳳凰台上憶吹簫∣

李清

照

與爾同消萬古愁。

　　　　　　將進酒∣

李

白 一種相思，兩處閒愁。

　　　　　　一剪梅∣

李清照

怎一個愁字 了得

尋尋覓覓，冷冷清清，

淒淒慘慘戚戚。

乍暖還寒時候，最難將息。

三杯兩盞淡酒，怎敵他，

晚來風急。

雁過也，正傷心，卻是舊時相識。

滿地黃花堆積，憔悴損，

如今有誰堪摘。

守著窗兒，獨自怎生得黑，

梧桐更兼細雨，到黃昏，點點滴滴。

這次第，怎一個愁字了得。

　　　　　　　　　　　　聲聲慢∣

李清照

2 > 1

Concluding Remarks

Mathematics is about finding connections, betweenspecific problems and more general results, and between one concept and another seemingly unrelatedconcept that really are related.

Fermat’s Last Theorem Taniyama-Shimura Conjecture

Gerhard FreyKen Ribet

EllipticEquations

ModularForms

AndrewWiles

Concluding Remarks

Poetry finds connections between moon and flowers,spring and autumn, orders and chaos, and happiness and sorrow, and weaves them into a fabric of many splendors.

春花秋月何時 了，往事知多少 。小樓昨夜 又東風 ，故國不堪回首月明中。雕欄玉砌應猶在，只是 朱顏改。問君能有幾多愁， 恰似一江春水向 東流 。　　　　　　　　　　　　　　　　　　　　 虞美 人－ 李煜

Concluding Remarks

In the eyes of a mathematician,

In the eyes of a poet,

And through their eyes,

In our eyes,

The world is a beautiful world,

And life a beautiful life.