Chessboards, Hats, and Poetry : Some Rigorous and Not-So-Rigorous Mathematical Results

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Chessboards, Hats, and Poetry Some Rigorous and
Not-So-RigorousMathematical Results
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• C. L. Liu ???
• Tsing Hua, Hsinchu

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

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




































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

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
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Palindrome
Madam Able was I ere I saw Elba
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A Truncated Chessboard
Truncated 8?8 chessboard
Cover the truncated 8?8 chessboard with
thirty-one 2?1 dominoes
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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.
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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.
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Proof of Impossibility
1
0
0
0
0
1
1
1
0
1
1
0
1
1
0
0
1
0
0
0
0
1
1
0
1
1
0
0
0
0
1
1
1
1
1
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
1
0
1
0
1
0
0
1
1
1
1
0
0
0
0
1
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.
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Modulo-2 Arithmetic
0 1 2 3 4 5 6 ..
even odd even odd even odd even ..
white black white black white black white ..
on off on off on off on ..
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Coloring the Vertices of a Graph
vertex
edge
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2 - Colorability
A necessary and sufficient condition No
circuit of odd length
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2 - Colorability
Necessity If there is a circuit of odd length,
Sufficiency If there is no circuit of odd
length, use the contagious
coloring algorithm.
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3 - Colorability
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3 - Colorability
The problem of determining whether a graph is
3-colorable is NP-complete. ( At the present
time, there is no known efficient algorithm for
determining whether a graph is 3-colorable.)
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Fermats Last Theorem
2-colorability is tractable, 3-colorability is
unknown.
2-satisfiability is tractable, 3-satisfiability
is unknown.
Fermats Last Theorem
x n y n z n
n 2 there are integer solutions 3
2 4 2 5 2
n ? 3 no integer solution
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4 - Colorability Planar Graphs
Kuratowskis subgraphs
All planar graphs are 4-colorable. How to
characterize non-planar graphs ? Genus,
Thickness,
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A Defective Chessboard
Any 8?8 defective chessboard can be covered with
twenty-one triominoes
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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
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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
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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
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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
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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.
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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
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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
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Sound of Silence
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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 
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Information Theory
Measure of Information
Self Information I (x) - lg p (x)
Mutual Information I (x, y) - lg p (x) lg p
(x y)
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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.
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Another Hat Problem
..
0 1 1 0 .
1
1 1 0 . 1
1
1 0 . 1
0
1
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Another Hat Problem
..
0 1 1 0 .
1
1 1 0 . 1
1
1
0 . 1
1
1
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Coding Theory

• Representation of information in alternate forms
  for
• efficiency
• reliability
• security

• Algebraic Coding Theory

• Cryptography

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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 says
P. Third person guesses
Probability of winning 1/2
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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
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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.
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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 and Non-adaptive Algorithms

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Yet, Another Hat Problem
Hats are returned to 10 people at random, what is
the probability that no one gets his own hat
back ?
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Recurrence Relations
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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
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Derangement of 10 Objects
Number of derangements of n objects
Probability
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Permutation

1 2 3 4
a
b
c
d
Positions
Objects
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Placement of Non-taking Rooks

1 2 3 4
a
b
c
d
Positions
Objects
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Permutation with Forbidden Positions

1 2 3 4
a
b
c
d

1 2 3 4
a
b
c
d
Positions
Positions
Objects
Objects
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Placement of Non-taking Rooks

1 2 3 4
a
b
c
d

1 2 3 4
a
b
c
d
Positions
Positions
Objects
Objects
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At Least One Way to Place Non-taking Rooks

1 2 3 4
a
b
c
d

Eva Faye Gigi Helen
Adam
Bob
Carl
Dan
Positions
Objects
Theory of Matching/Marriage Stable
Marriage Monogamic/Polygamic Marriage
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Stable Marriages
Eva Faye Gigi Helen
Adam 3 1 2 4
Bob 4 3 1 2
Carl 4 2 1 3
Dan 3 4 1 2
Eva Faye Gigi Helen
Adam 1 2 1 2
Bob 2 3 2 3
Carl 3 4 3 1
Dan 4 1 4 4
Bob
Gigi
Adam
Eva
Helen
Carl
Bob
Faye
unstable
stable
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Stable Marriages
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Stable Marriages
Eva Faye Gigi Helen
Adam 3 1 2 4
Bob 4 3 1 2
Carl 4 2 1 3
Dan 3 4 1 2
Eva Faye Gigi Helen
Adam 1 2 1 2
Bob 2 3 2 3
Carl 3 4 3 1
Dan 4 1 4 4
(1)
(4)
(2)
(2)
Adam
Eva
Adam
Eva
(1)
(2)
(4)
(1)
Bob
Faye
Bob
Faye
(3)
(2)
(3)
(1)
Carl
Gigi
Carl
Gigi
(3)
(1)
(4)
(1)
Dan
Helen
Dan
Helen
stable
stable
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Stable Marriages
Eva Faye Gigi Helen
Adam 3 1 2 4
Bob 4 3 1 2
Carl 4 2 1 3
Dan 3 4 1 2
Eva Faye Gigi Helen
Adam 1 2 1 2
Bob 2 3 2 3
Carl 3 4 3 1
Dan 4 1 4 4

Adam Faye Faye Faye Faye
Bob Gigi Gigi Gigi Gigi
Carl Gigi Faye Helen Helen
Dan Gigi Helen Helen Eva

Eva Adam Bob
Faye Dan Dan
Gigi Adam Adam
Helen Carl Carl
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Stable Marriages
Theorem There is always a set of stable
marriages.
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Theorem Monogamy is optimal.
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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.
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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.
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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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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.
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Step 1
Balance
Step 2
Balance
Imbalance
Step 3
Step 3
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(No Transcript)
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Step 1
Imbalance
Step 2
Imbalance
Step 3