Problems

(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?

Can you fully cover the surface of a cube with 16 identical colourful rectangular paper strips? The paper strips may wrap around the sides of the cube, and no overlapping is allowed. For example, two of the paper strips may look like this:

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.

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

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?

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

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

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

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