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Title: C. L. Liu ???


1
Poetic Mathematics and Mathematical Poetry ?????
????!
  • C. L. Liu ???

2
Poetry
Chessboards
3
It all begins with a chessboard
4
Covering a Chessboard
2?1 domino
8?8 chessboard
Cover the 8?8 chessboard with thirty-two 2?1
dominoes
5
Enumeration

Number Theory, Probability, Statistics, Physics,
Chemistry,
6
Archimedes Stomachion Puzzle




































17,152 ways
How do I love thee, Let me count the ways.
- Elizabeth Barrett Browning
7
Enumeration
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?????????
8
Enumeration - Symmetry

Symmetry Polyas Theory of Counting
Tyger! Tyger! Burning bright, In the forests of
the night. What immortal hand or eye Could frame
thy fearful symmetry?
- William Blake
9
Archimedes Stomachion Puzzle












When rotational and reflexive symmetry are taken
into consideration 536 ways

10
Symmetry ??
river
11
Symmetry ??
12
Symmetry ??
13
The Torch Has Been Passed -John F.
Kennedy (20 January 1960)
Contrapuntalism??
We observe today not a victory of a party, but a
celebration of freedom-symbolizing an end, as
well as a beginning-signifying renewal, as well
as change. Let us never negotiate out of fear.
But let us never fear to negotiate. Now the
trumpet summons us again -not as a call to bear
arms, though arms we need not as a call to
battle, though embattled we are -but a call to
bear the burden of a long twilight struggle. And
so, my fellow Americans, ask not what your
country can do for you-ask what you can do for
your country. My fellow citizens of the world,
ask not what America will do for you, but what
together we can do for the freedom of man.

14
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15
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(??) (??)
16
? ?
Palindrome
madam Able was I ere I saw Elba
17
? ? ?
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????? ???????
18
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19
A Truncated Chessboard
Truncated 8?8 chessboard
Cover the truncated 8?8 chessboard with
thirty-one 2?1 dominoes
20
Proof of Impossibility
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??????, ??????, ?????? ?????-???
Truncated 8?8 chessboard
Truncated 8?8 chessboard
Impossible to cover the truncated 8?8 chessboard
with thirty-one dominoes.
21
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.
22
A Defective Chessboard
Any 8?8 defective chessboard can be covered with
twenty-one triominoes
23
Defective Chessboards
Any 8?8 defective chessboard can be covered with
twenty-one triominoes
Any 2n?2n defective chessboard can be covered
with 1/3(2n?2n -1) triominoes
Prove by mathematical induction
24
Mathematical Induction
The first domino falls. If a domino falls, so
will the next domino. All dominoes will fall !
To see a world in a grain of sand, And a heaven
in a wild flower, Hold infinity in the palm of
your hand, And eternity in an hour.              
          - William Blake
25
Mathematical Induction
To see a world in a grain of sand, And a heaven
in a wild flower, Hold infinity in the palm of
your hand, And eternity in an hour.              
        - William Blake
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26
Proof by Mathematical Induction
Any 2n?2n defective chessboard can be covered
with 1/3(2n?2n -1) triominoes
Basis n 1
Induction step
2 n1
2 n1
27
The Wise Men and the Hats
If there are n wise men wearing white hats, then
at the nth hour all the n wise men will raise
their hands.
Basis n 1 At the 1st hour, the
only wise man wearing a white hat will
raise his hand.
Induction step Suppose there are
n1 wise men wearing white hats. At
the nth hour, no wise man raises his hand.
At the n1st hour, all n1 wise men raise
their hands.
28
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
????,???? ????,???? ????,????? ????-????
29
I Dont Know
Two Integers, 1 lt x, y lt 51
x y
x y
I dont know.
I knew you would not know. However, neither do I.
Now, I know.
Now, I know.
Now, I know.
x 4 , y 13
30
Sound of Silence
???????,???????, ???????,????????
????-????
To communicate through silence is a link between
the thoughts of man.
                         - Marcel Marceau 
Hello darkness, my old friend, I've come to talk
with you again.           The Sound of
Silence - Simon Garfunkel 
31
Information Theory
Measure of Information
Self Information I (x) - lg p (x)
Mutual Information I (x, y) - lg p (x) lg p
(x y)
32
Another Hat Problem
No strategy In the worst case, all men were
shot.
Strategy 1 In the worst case, half of the men
were shot.
Design a strategy so that as few men will die as
possible.
33
Modulo 2 Addition
0 1
0 0 1
1 1 0
34
Another Hat Problem
..
0 1 1 0 .
1
1 1 0 . 1
1
1 0 . 1
0
1
35
Another Hat Problem
..
0 1 1 0 .
1
1 1 0 . 1
1
1
0 . 1
1
1
36
Coding Theory
  • Representation of information in alternate forms
    for
  • efficiency
  • reliability
  • security
  • Algebraic Coding Theory
  • Cryptography

37
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38
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39
Yet, Another Hat Problem
A person may say, 0, 1, or P(Pass) Winning No
body is wrong, at least one person is
right Losing One or more is wrong
Example 010 010 win
000 lose 0P0
win 0P1 lose
40
Yet, Another Hat Problem
A person may say, 0, 1, or P(Pass) Winning No
body is wrong, at least one person is
right Losing One or more is wrong
Strategy 1 Everybody guesses
Probability of winning 1/8
Strategy 2 First and second person always say
P Third person guesses
Probability of winning 1/2
41
Strategy 3
observe call
00 01 10 11 1 P P 0
pattern call
000 001 010 011 100 101 110 111 111 PP1 P1P 0PP 1PP P0P PP0 000
Probability of winning 3/4
42
A Coin Weighing Problem
Twelve coins, possibly one of them is defective (
too heavy or too light ). Use a balance three
times to pick out the defective coin.
43
Another Coin Weighing Problem
Thirteen coins, possibly one of them is defective
( too heavy or too light ). Use a balance three
times to pick out the defective coin. However, an
additional good coin is available for use as
reference.
  • Application of Algebraic Coding Theory
  • Adaptive Algorithms
  • Non-adaptive Algorithms

44
The Tea Kettle Principle - Problem 1
45
Problem 1 solution
46
Problem 2
47
Problem 2 solution-engineer
48
Problem 2 solution-mathematician
49
Problem 1
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50
Problem 1 solution
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?? ??
51
Problem 2
?? ??
?? 101
52
Problem 1 solution
?? ???
?? ??
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?? 101
53
On the Tea Kettle Principle - - - - -
- - - - - - - - - - - - - - - - - - - -
- - -
The author wishes to thank Dr. Frank Fang for
translating this article into Chinese. The
author wishes to thank Dr. Frank Fang for
translating the preceding footnote into
Chinese. The author wishes to thank Dr. Frank
Fang for translating the preceding footnote
into Chinese.
54
Fibonacci sequence of numbers
55
Recurrence Relation
Homogeneous solution Characteristic equation
Boundary condition Particular solution
56
Yet, Another Hat Problem
Hats are returned to 10 people at random, what is
the probability that no one gets his own hat
back ?
????
57
Derangements
dn number of derangements of n objects
dn (n-1) dn-1 (n-1) dn-2
d1 0
d2 1
d3 2? d2 2 ? d1 2 ? 1 2 ? 0 2
d4 3? d3 3 ? d2 3 ? 2 3 ? 1 9

d10 9 ? d9 9 ? d8 1,334,961
58
Recurrence Relation
Exponential generating function
59
Derangement of 10 Objects
Number of derangements of n objects
Probability
60
The Josephus Problem
1
2
10
3
9
4
8
5
7
6
61
Recurrence Relation
J (1) 1
J (2n) 2 J (n) - 1
J (2n1) 2 J (n1)
J (2kl ) 2 l 1
J (10) J (232) 2 2 1 5
J (14) J (236) 2 6 1 13
J (100) J (2636) 2 36 1 73
62
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63
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64
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65
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66
!
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67
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68
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69
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70
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71
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72
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73
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74
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75
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.
76
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.
???????,?????? ???????,?????????? ???????,?????? ?
??????,?????????? ????-???
77
Concluding Remarks
The mathematicians patterns, like the painters
or the poets, must be beautiful the ideas, like
the colours or the words, must fit together in a
harmonious way. Beauty is the first test there
is no permanent place in the world for ugly
mathematics.
G.H. Hardy
78
Concluding Remarks
Poetry is a mirror which makes beautiful that
which is distorted.
Percy Bysshe Shelley
79
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.
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