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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

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

David Smith cut out 12 nets. He claimed that it was possible to make a cube out of each net. Roger Penrosae 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. Show how you can cover the plane with nets below:

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-100377

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Features of dissection pieces

A big square was cut into smaller squares. Sebastian used all the pieces and constructed two squares with different side lengths by glueing the pieces together. Show an example of how he could do that.

Problem #PRU-100380

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Features of dissection pieces

It was Sebastian’s younger brother who cut the big square in Example 2. Now you need to help him to cut one of the squares (which Sebastian obtained after glueing the pieces) into smaller congruent triangles. But please make sure the elder brother can do the same thing as before: to divide the resulting congruent triangles into two groups and to glue the pieces of each group together to make two squares with different side lengths.

Problem #PRU-100396

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers

(a) A picnic spot has a form of a 100 m\({}\times {}\)100 m square. Is it possible to partially cover it with non-intersecting square picnic blankets so that the total sum of their perimeters will be greater than 10,000 m?

(b) One sunny day almost every citizen came to the picnic spot from point (a). All of them brought square picnic blankets. In a local newspaper there was mentioned that the total area of grass covered with picnic blankets was greater than 20,000 m\(^2\). Do you think it was possible or did they make a mistake in their computations?