We solve problems

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?

Problem #PRU-100400

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

Can you cover the surface of a cube with 16 identical colourful rectangles? No overlappings are allowed and the cube has to be fully covered.

Problem #PRU-100406

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

The cube from Example 3 is a present and one layer of a gift-wrap is totally not enough. Can you cover it with another 15 identical rectangles? You can assume the covering from Example 3 was thin and it did not affect the shape of a cube. As before no overlappings are allowed and the surface has to be fully covered by rectangles.

Problem #PRU-100443

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

Can a $5 \times 5$ square checkerboard be covered by $1 \times 2$ dominoes?

Problem #PRU-100476

Problems Methods Colouring Colouring (other) Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers

a) What is the answer in case we are asked to split the figure below into $1 \times 4$ rectangles instead of $1 \times 5$ rectangles?

(b) In the context of Example 1 what is the answer in case we are asked to split the figure into $1 \times 7$ rectangles instead of $1 \times 5$ rectangles?

Problem #PRU-100686

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

Can you cover a $10 \times 10$ square with $1 \times 4$ rectangles?

Problem #PRU-100687

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

Two opposite corners were removed from an $8 \times 8$ chessboard. Can you cover this chessboard with $1 \times 2$ rectangular blocks?

Problem #PRU-100690

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

One small square of a $10 \times 10$ square was removed. Can you cover the rest of it with 3-square $L$ -shaped blocks?

Problem #PRU-100694

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

Can you cover a $13 \times 13$ square using two types of blocks: $2 \times 2$ squares and $3 \times 3$ squares?

Problem #PRU-109018

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

12–14 3.0

A circle is covered with several arcs. These arcs can overlap one another, but none of them cover the entire circumference. Prove that it is always possible to select several of these arcs so that together they cover the entire circumference and add up to no more than $720^{\circ}$ .