mathematical puzzles and not so puzzling mathematics
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Mathematical Puzzles and Not So Puzzling Mathematics. C. L. Liu National Tsing Hua University. It all begins with a chessboard. Covering a Chessboard. 21 domino. 8 8 chessboard. Cover the 8 8 chessboard with thirty-two 21 dominoes. 21 domino. A Truncated Chessboard. - PowerPoint PPT PresentationTRANSCRIPT
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Mathematical Puzzles Mathematical Puzzles and and
Not So Puzzling MathematicsNot So Puzzling Mathematics
C. L. LiuC. L. LiuNational Tsing Hua UniversityNational Tsing Hua University
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It all begins with a chessboard
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Covering a Chessboard
88 chessboard
21 domino
Cover the 88 chessboard with thirty-two 21 dominoes
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A Truncated Chessboard
21 domino
Cover the truncated 88 chessboard with thirty-one 21 dominoes
Truncated 88 chessboard
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Proof of Impossibility
21 domino
Truncated 88 chessboard
Impossible to cover the truncated 88 chessboard with thirty-one dominoes.
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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.
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An Algebraic Proof
1 x x2 . . . . . . . . . . . . . . . . . . . . . . . x7
y7
............y2
y
1
(1+x) xi y j (1+y) x i y j
(1+x+x2+. . . x7) (1+y+y2+. . . y7) – 1 - x7y7
= (1+x) xi y j + (1+y) x i y j xi yj
Impossible !Let x = -1 y = -1 -2 = 0
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Modulo-2 Arithmetic
1 2 3 4 5 6 …..
odd even odd even odd even…..
odd even
odd even odd
even odd even
0 1
0 0 1
1 1 0
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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
vertex
edge
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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
The problem of determining whether a graph is 3-colorable is NP-complete. ( At the present time, there is no knownefficient algorithm for determining whether a graph is 3-colorable.)
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4 - Colorability : Planar Graphs
All planar graphs are 4-colorable.
How to characterize non-planar graphs ? Genus, Thickness, …
Kuratowski’s subgraphs
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A Defective Chessboard
Triomino
Any 88 defective chessboard can be covered with twenty-one triominoes
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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
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Principle of Mathematical Induction
To show that a statement p (n) is true
1. Basis : Show the statement is true for n = n0
2. Induction step : Assuming the statement is true for
n = k , ( k n0 ) , show the statement is true for n = k + 1
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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
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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+1th hour, all n+1 wise men raise their hands.
……
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Principle of Strong Mathematical Induction
To show that a statement p (n) is true1. Basis : Show the statement is true for n = n0
2. Induction step : Assuming the statement is true for n = k , ( k n0 ) , show the statement is true for n = k + 1 n0 n k,
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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.
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Another Hat Problem
x n x n-1 x n-2 x n-3 ……………… x1
………..
x n-1 x n-2 x n-3 ……… x1
x n-2 x n-3 ……… x1
x n-1 x n-3 ……… x1
x n-2
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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 rightLosing : 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
00011011
1PP0
pattern call
000001010011100101110111
111PP1P1P0PP1PPP0PPP0000
Probability of winning = 3/4
More people ?
Best possible ?Generalization : 7 people, Probability of winning = 7/8
Application of Algebraic Coding Theory
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A Coin Weighing Problem
Twelve coins, possibly one of them is defective ( too heavyor too light ). Use a balance three times to pick out thedefective coin.
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1 2 3 4 5 6 7 8
G 9 10GG 11
12G 109
Step 1
Step 3
Step 2
Balance
Step 3Balance Imbalance
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7G
1 2 3 4 5 6 7 8
1 3 452 6
Step 1
Step 2
Imbalance
Step 3Balance
21
Step 3Imbalance
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1 2 3 4 5 6 7 8
1 3 452 6
Step 1
Step 2
Imbalance
43
Step 3Imbalance
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Another Coin Weighing Problem
Application of Algebraic Coding Theory
• Adaptive Algorithms• Non-adaptive Algorithms
Thirteen coins, possibly one of them is defective ( too heavyor too light ). Use a balance three times to pick out thedefective coin. However, an additional good coin is availablefor use as reference.
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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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Apples and Oranges
ApplesApples OrangesOrangesOrangesOrangesApplesApples
Take out one fruit from one box to determine the contentsof all three boxes.
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Derangements
AA BB CCa b c
a c b
b a c
b c a
c a b
c b a
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Derangement of 10 Objects
Number of derangements of n objects
]!
1)1(....!3
1!2
1!1
11[!n
nd nn
]!10
1)1(....!3
1!2
1!1
11[!10 1010 d
Probability !101)1(....
!31
!21
!111
!101010 d
36788.01 e
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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
Positions
Objects1 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
Objects1 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
Rook Polynomial :
R (C) = r0 + r1 x + r2 x2 + …
ri = number of ways to place i non-taking rooks on chessboard C
R (C) = 1 + 6x + 10x2 + 4x3
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At Least One Way to Place Non-taking Rooks
1 2 3 4
a
b
c
d
Positions
Objects1 2 3 4
a
b
c
d
Positions
Objects
Theory of Matching !
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Conclusion
Mathematics is about finding connections, betweenspecific problems and more general results, and between one concept and another seemingly unrelatedconcept that really is related.