SLIDE 1
Mathematical colorings
Kaloyan Slavov Department of Mathematics ETH Z¨ urich kaloyan.slavov@math.ethz.ch February 9, 2017
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Domino tiling — a)
8 8
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Mathematical colorings Kaloyan Slavov Department of Mathematics - - PowerPoint PPT Presentation
Mathematical colorings Kaloyan Slavov Department of Mathematics - - PowerPoint PPT Presentation
Mathematical colorings Kaloyan Slavov Department of Mathematics ETH Z urich kaloyan.slavov@math.ethz.ch February 9, 2017 1 / 9 Domino tiling a) 8 8 2 / 9 Domino tiling a) a) Is it possible to tile this board by 2 1 domino
SLIDE 2
SLIDE 3
Domino tiling — a)
8 8 a) Is it possible to tile this board by 2 × 1 domino pieces?
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SLIDE 4
Domino tiling — a)
8 8 a) Is it possible to tile this board by 2 × 1 domino pieces?
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SLIDE 5
Domino tiling — a)
8 8 a) Is it possible to tile this board by 2 × 1 domino pieces?
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SLIDE 6
Domino tiling — a)
8 8 a) Is it possible to tile this board by 2 × 1 domino pieces?
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SLIDE 7
Domino tiling — a)
8 8 a) Is it possible to tile this board by 2 × 1 domino pieces? No, because the number of squares (63) is odd.
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SLIDE 8
Domino tiling — b)
8 8
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SLIDE 9
Domino tiling — b)
8 8 b) Is it possible to tile this board by 2 × 1 domino pieces?
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SLIDE 10
Domino tiling — b)
8 8 b) Is it possible to tile this board by 2 × 1 domino pieces? Yes.
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SLIDE 11
Domino tiling — b)
8 8 b) Is it possible to tile this board by 2 × 1 domino pieces? Yes.
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SLIDE 12
Domino tiling – c)
8 8
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SLIDE 13
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces?
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SLIDE 14
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ...
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SLIDE 15
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ...
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SLIDE 16
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ...
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SLIDE 17
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ...
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SLIDE 18
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ...
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SLIDE 19
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ... We failed.
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SLIDE 20
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ... We failed. So what?
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SLIDE 21
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? No.
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SLIDE 22
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? No. Color the board in a chess pattern.
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SLIDE 23
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? No. Color the board in a chess pattern.
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SLIDE 24
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? No. Color the board in a chess pattern. Each domino takes
- ne white and
- ne black square.
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SLIDE 25
Domino tiling – c)
8 8 c) Is it possible to tile this board by 2 × 1 domino pieces? No. Color the board in a chess pattern. Each domino takes
- ne white and
- ne black square.
However, there are 32 white and 30 black squares.
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SLIDE 26
Bugs – a)
a) There is a bug at each square of a 5 × 5 grid.
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SLIDE 27
Bugs – a)
a) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws horizontally or vertically to a neighboring square.
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SLIDE 28
Bugs – a)
a) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws horizontally or vertically to a neighboring square. Prove that some square will remain empty.
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SLIDE 29
Bugs – a)
a) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws horizontally or vertically to a neighboring square. Prove that some square will remain empty. Proof.
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SLIDE 30
Bugs – a)
a) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws horizontally or vertically to a neighboring square. Prove that some square will remain empty. Proof. Color the board in a chess pattern.
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SLIDE 31
Bugs – a)
a) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws horizontally or vertically to a neighboring square. Prove that some square will remain empty. Proof. Color the board in a chess pattern. Each bug changes the color of its square.
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SLIDE 32
Bugs – a)
a) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws horizontally or vertically to a neighboring square. Prove that some square will remain empty. Proof. Color the board in a chess pattern. Each bug changes the color of its square. There are 13 black and 12 white squares.
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SLIDE 33
Bugs – a)
a) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws horizontally or vertically to a neighboring square. Prove that some square will remain empty. Proof. Color the board in a chess pattern. Each bug changes the color of its square. There are 13 black and 12 white squares. = ⇒ some black square will remain empty.
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SLIDE 34
Bugs – b)
b) There is a bug at each square of a 5 × 5 grid.
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SLIDE 35
Bugs – b)
b) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws diagonally to a neighboring square.
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SLIDE 36
Bugs – b)
b) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws diagonally to a neighboring square. Prove that at least 5 squares will remain empty.
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SLIDE 37
Bugs – b)
b) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws diagonally to a neighboring square. Prove that at least 5 squares will remain empty. Proof.
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SLIDE 38
Bugs – b)
b) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws diagonally to a neighboring square. Prove that at least 5 squares will remain empty. Proof. Color the board ........................
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SLIDE 39
Bugs – b)
b) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws diagonally to a neighboring square. Prove that at least 5 squares will remain empty. Proof. Color the board ........................ Each bug changes the color of its square.
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SLIDE 40
Bugs – b)
b) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws diagonally to a neighboring square. Prove that at least 5 squares will remain empty. Proof. Color the board as shown. Each bug changes the color of its square.
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SLIDE 41
Bugs – b)
b) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws diagonally to a neighboring square. Prove that at least 5 squares will remain empty. Proof. Color the board as shown. Each bug changes the color of its square. There are 15 black and 10 white squares.
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SLIDE 42
Bugs – b)
b) There is a bug at each square of a 5 × 5 grid. At a given instant, each bug craws diagonally to a neighboring square. Prove that at least 5 squares will remain empty. Proof. Color the board as shown. Each bug changes the color of its square. There are 15 black and 10 white squares. = ⇒ at least 5 black squares will remain empty.
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SLIDE 43
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos?
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SLIDE 44
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos? We can try ...
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SLIDE 45
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos? We can try ...
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SLIDE 46
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos? We can try ...
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SLIDE 47
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos? We can try ... We failed.
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SLIDE 48
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos? We can try ... We failed. So what?
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SLIDE 49
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos? No.
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SLIDE 50
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos? No. Color the board . . . . . . . . . . . .
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SLIDE 51
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos? No. Color the board as shown.
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SLIDE 52
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos? No. Color the board as shown.
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SLIDE 53
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos? No. Color the board as shown. Each tetramino takes 2 black and 2 white squares.
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SLIDE 54
Tetraminos
10 10 Is it possible to tile a 10 × 10 board by tetraminos? No. Color the board as shown. Each tetramino takes 2 black and 2 white squares. However, there are 13 ∗ 4 = 52 black and 12 ∗ 4 = 48 white squares.
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SLIDE 55
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4.
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SLIDE 56
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4.
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SLIDE 57
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4.
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SLIDE 58
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost.
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SLIDE 59
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available.
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SLIDE 60
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor?
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SLIDE 61
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No.
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SLIDE 62
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No. Color the board . . . . . . . . . . .
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SLIDE 63
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No. Color the board as shown.
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SLIDE 64
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No. Color the board as shown.
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SLIDE 65
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No. Color the board as shown. Each 2 × 2 tile covers exactly 1 black square.
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SLIDE 66
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No. Color the board as shown. Each 2 × 2 tile covers exactly 1 black square.
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SLIDE 67
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No. Color the board as shown. Each 2 × 2 tile covers exactly 1 black square. Each 4 × 1 tile covers 0 or 2 black squares.
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SLIDE 68
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No. Color the board as shown. Each 2 × 2 tile covers exactly 1 black square. odd! Each 4 × 1 tile covers 0 or 2 black squares.
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SLIDE 69
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No. Color the board as shown. Each 2 × 2 tile covers exactly 1 black square. odd! Each 4 × 1 tile covers 0 or 2 black squares. even!
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SLIDE 70
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No. Color the board as shown. Each 2 × 2 tile covers exactly 1 black square. odd! Each 4 × 1 tile covers 0 or 2 black squares. even!
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SLIDE 71
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No. Color the board as shown. Each 2 × 2 tile covers exactly 1 black square. odd! Each 4 × 1 tile covers 0 or 2 black squares. even! The number of black squares (30) is even (in the picture).
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SLIDE 72
A bathroom floor
A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. Before gluing the pieces, one of the 2 × 2 tiles got lost. A spare 4 × 1 tile is available. Is it still possible to tile the floor? No. Color the board as shown. Each 2 × 2 tile covers exactly 1 black square. odd! Each 4 × 1 tile covers 0 or 2 black squares. even! The number of black squares (30) is even (in the picture). = ⇒ the number of 2 × 2 tiles in any tiling must be even!
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SLIDE 73
Tetraminos revisited
10 10 Is it possible to tile a 10 × 10 board by 4 × 1 tetraminos?
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SLIDE 74
Tetraminos revisited
10 10 Is it possible to tile a 10 × 10 board by 4 × 1 tetraminos? No.
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SLIDE 75
Tetraminos revisited
10 10 Is it possible to tile a 10 × 10 board by 4 × 1 tetraminos? No.
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SLIDE 76
Tetraminos revisited
10 10 Is it possible to tile a 10 × 10 board by 4 × 1 tetraminos? No. Apply the “Bathroom floor” problem.
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