We solve problems

Problem #PRU-100201

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

11–14 2.0

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

Problem #PRU-100202

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

Does there exist an irreducible tiling with \(1\times2\) rectangles of a \(4\times 6\) rectangle?

Problem #PRU-100203

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

Irreducibly tile a floor with \(1\times2\) tiles in a room that is a \(5\times8\) rectangle.

Problem #PRU-100204

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

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:

Problem #PRU-100205

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

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?

Problem #PRU-100206

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

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.

Problem #PRU-100367

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

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

Problem #PRU-100443

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers Tilings with ordinary and domino tiles

Can a \(5\times5\) square checkerboard be covered by \(1\times2\) dominoes?

Problem #PRU-100684

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

Can you cover a \(10 \times 10\) board using only \(T\)-shaped tetraminoes?

Problem #PRU-103885

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

11–12 2.0

At the disposal of a tile layer there are 10 identical tiles, each of which consists of 4 squares and has the shape of the letter L (all tiles are oriented the same way). Can he make a rectangle with a size of \(5 \times 8\)? (The tiles can be rotated, but you cannot turn them over). For example, the figure shows the wrong solution: the shaded tile is incorrectly oriented.