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 pieces? 8 8 2 / 9

Domino tiling — a) a) Is it possible to tile this board by 2 × 1 domino pieces? 8 8 2 / 9

Domino tiling — a) a) Is it possible to tile this board by 2 × 1 domino pieces? 8 8 2 / 9

Domino tiling — a) a) Is it possible to tile this board by 2 × 1 domino pieces? 8 8 2 / 9

Domino tiling — a) a) Is it possible to tile this board by 2 × 1 domino pieces? 8 No, because the number of squares (63) is odd . � 8 2 / 9

Domino tiling — b) 8 8 3 / 9

Domino tiling — b) b) Is it possible to tile this board by 2 × 1 domino pieces? 8 8 3 / 9

Domino tiling — b) b) Is it possible to tile this board by 2 × 1 domino pieces? Yes. 8 8 3 / 9

Domino tiling — b) b) Is it possible to tile this board by 2 × 1 domino pieces? Yes. � 8 8 3 / 9

Domino tiling – c) 8 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? 8 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ... 8 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ... 8 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ... 8 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ... 8 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ... 8 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ... 8 We failed. 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? We can try ... 8 We failed. So what? 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? No. 8 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? No. 8 Color the board in a chess pattern. 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? No. 8 Color the board in a chess pattern. 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? No. 8 Color the board in a chess pattern. Each domino takes one white and one black square. 8 4 / 9

Domino tiling – c) c) Is it possible to tile this board by 2 × 1 domino pieces? No. 8 Color the board in a chess pattern. Each domino takes one white and one black square. However, there are 8 32 white and 30 black squares. � 4 / 9

Bugs – a) a) There is a bug at each square of a 5 × 5 grid. 5 / 9

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. 5 / 9

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. 5 / 9

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. 5 / 9

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. 5 / 9

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. 5 / 9

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. 5 / 9

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. � 5 / 9

Bugs – b) b) There is a bug at each square of a 5 × 5 grid. 6 / 9

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. 6 / 9

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. 6 / 9

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. 6 / 9

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 ........................ 6 / 9

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. 6 / 9

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. 6 / 9

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. 6 / 9

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. � 6 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? 10 10 7 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? We can try ... 10 10 7 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? We can try ... 10 10 7 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? We can try ... 10 10 7 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? We can try ... 10 We failed. 10 7 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? We can try ... 10 We failed. So what? 10 7 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? No. 10 10 7 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? No. 10 Color the board . . . . . . . . . . . . 10 7 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? No. 10 Color the board as shown. 10 7 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? No. 10 Color the board as shown. 10 7 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? No. 10 Color the board as shown. Each tetramino takes 2 black and 2 white squares. 10 7 / 9

Tetraminos Is it possible to tile a 10 × 10 board by tetraminos? No. 10 Color the board as shown. Each tetramino takes 2 black and 2 white squares. However, there are 13 ∗ 4 = 52 black and 10 12 ∗ 4 = 48 white squares. � 7 / 9

A bathroom floor A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. 8 / 9

A bathroom floor A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. 8 / 9

A bathroom floor A rectangular bathroom floor was tiled by tiles of two kinds: 2 × 2, and 1 × 4. 8 / 9

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. 8 / 9

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. 8 / 9