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Found: 79

Tile a \(5\times6\) rectangle in an irreducible way by laying \(1\times2\) rectangles.

Does there exist an irreducible tiling with \(1\times2\) rectangles of

(a) \(4\times 6\) rectangle;

(b) \(6\times 6\) rectangle?

Irreducibly tile a floor with \(1\times2\) tiles in a room that is

(a) \(5\times8\); (b) \(6\times8\).

Having mastered tiling small rooms, Robinson wondered if he could tile big spaces, and possibly very big spaces. He wondered if he could tile the whole plane. He started to study the tiling, which can be continued infinitely in any direction. Can you help him with it?

Tile the whole plane with the following shapes:

Robinson Crusoe was taking seriously the education of Friday, his friend. Friday was very good at maths, and one day he cut 12 nets out of hardened goat skins. He claimed that it was possible to make a cube out of each net. Robinson looked at the patterns, and after some considerable thought decided that he was able to make cubes from all the nets except one. Can you figure out which net cannot make a cube?

It is known that it is possible to cover the plane with any cube’s net. (You will see it in the film that will be shown at the end of this session). But Robinson, unfortunately, lived on an uninhabited island in the 19th century, and did not know about the film. Try to help him to figure out how to cover the plane with nets \(\#2\), \(\#6\), and \(\#8\) from the previous exercise.

A strange wonderland creature is called a painting chameleon. If the queen puts the painting chameleon on a chess-like board then he moves one square at a time along the board either horizontally or vertically. When he moves, he either changes his colour to the colour of the square he moves to, or he paints the square he moves to into his own colour. The queen puts a white painting chameleon on an all-black board \(8\times8\) and orders the chameleon to paint the board into a chessboard. Can he succeed?

Remove a \(1 \times 1\) square from the corner of a \(4 \times 4\) square. Can this shape be dissected into \(3\) congruent parts?

Ten little circles are drawn on a squared board \(4\times4\).

Cut the board into identical parts in such a way that each part contains 1, 2, 3, and 4 drawn circles correspondingly.

Philip and Denis cut a watermelon into four parts. When they finished eating watermelon (they ate the whole thing), they discovered that there were five watermelon rinds left. How is it possible, if no rind was cut after the initial cutting?