2. Chessboards, Hats, and Poetry : Some Rigorous and Not-So-Rigorous Mathematical Results C. L. Liu 劉炯朗 Tsing Hua, Hsinchu 數裡有詩？ 詩裡有數！

3. Hats Poetry Mathematics Chessboards

4. It all begins with a chessboard

5. Covering a Chessboard 8  8 chessboard 2  1 domino Cover the 8  8 chessboard with thirty-two 2  1 dominoes

6. Enumeration …… Number Theory, Probability, Statistics, Physics, Chemistry, …

7. Archimedes’ Stomachion Puzzle 17,152 ways How do I love thee, Let me count the ways. － Elizabeth Barrett Browning

8. Enumeration

9. 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 ……

10. 對 聯對 聯

11. 對 聯對 聯

12. 無情對

13. Madam Able was I ere I saw Elba Palindrome 迴 文迴 文

14. 迴 文 詩迴 文 詩

15. 迴文 對聯

16. A Truncated Chessboard 2  1 domino Cover the truncated 8  8 chessboard with thirty-one 2  1 dominoes Truncated 8  8 chessboard

17. Proof of Impossibility 2  1 domino Truncated 8  8 chessboard Impossible to cover the truncated 8  8 chessboard 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 8  8 chessboard

18. Proof of Impossibility Impossible to cover the truncated 8  8 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.

19. A Defective Chessboard Triomino Any 8  8 defective chessboard can be covered with twenty-one triominoes

20. Defective Chessboards Any 2 n  2 n defective chessboard can be covered with 1/3(2 n  2 n -1) triominoes Any 8  8 defective chessboard can be covered with twenty-one triominoes Prove by mathematical induction

21. 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

22. 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。

23. Proof by Mathematical Induction Basis : n = 1 Induction step : 2 n+1 2 n Any 2 n  2 n defective chessboard can be covered with 1/3(2 n  2 n -1) triominoes

24. If there are n wise men wearing white hats, then at the n th hour all the n wise men will raise their hands. The Wise Men and the Hats Basis : n =1 At the 1 st 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 n th hour, no wise man raises his hand. At the n+1 st hour, all n+1 wise men raise their hands. ……

25. The Wise Men and the Hats One white hat 1 st hour : hand raised Two white hats 1 st hour : silence 2 nd hour : hands raised Five white hats 1 st hour : silence 2 nd hour : silence 3 rd hour : silence 4 th hour : silence 5 th 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 

26. x yx + y I don ’ t know. I knew you would not know. However, neither do I. Now, I know. I Don’t Know Two Integers, 1 < x, y < 51 x = 4, y = 13

27. 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

28. Self Information : I ( x ) = - lg p ( x ) Mutual Information : I ( x, y ) = - lg p ( x ) + lg p ( x | y ) Information Theory Measure of Information

29. 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.

30. Another Hat Problem 0 1 1 0 …….… 1 ……….. 1 1 0.……… 1 1 0.……… 1 1 = 1 = 0

31. Another Hat Problem 0 1 1 0 …….… 1 ……….. 1 1 0.……… 1 0 ………. 1 1 = 1 1

32. Coding Theory Representation of information in alternate forms for efficiency reliability security Algebraic Coding Theory Cryptography

35. 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

36. Recurrence Relations b n = 0.92 b n-1 b n = 0.42 b n-1 + 0.5 b n-2 b n = 1.22 b n-1 + 0.3 b n-2 + 0.2 b n-3 b 2004 = 0.92 b 2003 1, 1, 2, 3, 5, 8, 13, … f n = f n-1 + f n-2 Fibonacci Numbers

37. Derangements d n : number of derangements of n objects d n = (n-1) d n-1 + (n-1) d n-2 d 3 = 2  d 2 + 2  d 1 = 2  1 + 2  0 = 2 d 1 = 0 d 2 = 1 d 4 = 3  d 3 + 3  d 2 = 3  2 + 3  1 = 9 d 10 = 9  d 9 + 9  d 8 = 1,334,961 ……

38. Derangement of 10 Objects Number of derangements of n objects Probability

39. 頂真格（鈎句，流水句）

43. 數裡有詩 詩裡有數 ？ ！ ？ ！

44. 怎一個愁字了得

45. 愁

50. 2 > 1

51. Concluding Remarks Mathematics is about finding connections, between specific problems and more general results, and between one concept and another seemingly unrelated concept that really are related. Fermat’s Last TheoremTaniyama-Shimura Conjecture Gerhard Frey Ken Ribet Elliptic Equations Modular Forms Andrew Wiles

52. 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 

53. 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.