## Task 4 - 330714

A puzzle game consists of four puzzle pieces of the shapes $A, B, C, D$, and $E$ shown in the first image. Each of these 20 puzzle pieces is composed of four squares with a side length of $1 \mathrm{~cm}$.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0544.jpg?height=71&width=257&top_left_y=433&top_left_x=480)

A

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0544.jpg?height=140&width=139&top_left_y=433&top_left_x=770)

B

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0544.jpg?height=131&width=188&top_left_y=434&top_left_x=937)

C
![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0544.jpg?height=172&width=410&top_left_y=424&top_left_x=1164)

$\mathrm{E}$

a) The area of a square with a side length of $4 \mathrm{~cm}$ is to be completely covered by puzzle pieces of a uniform shape, without any pieces overlapping or extending beyond the square.

Investigate which of the shapes $A, B, C, D, E$ can achieve this, and with which of these shapes there are even multiple possibilities!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0544.jpg?height=271&width=274&top_left_y=864&top_left_x=891)

b) The image shows how the area of a square with a side length of $5 \mathrm{~cm}$, with the shaded central square removed, can be covered by six puzzle pieces. The additional requirement is that three different types of pieces are used, and exactly two pieces of each type.

Give at least four more possibilities that meet this requirement!

c) The area of a rectangle with side lengths of $8 \mathrm{~cm}$ and $4 \mathrm{~cm}$ is to be covered by eight puzzle pieces. Four different types of pieces should be used, and exactly two pieces of each type.

Give two possibilities that differ in the types of pieces used!

Hint: Two coverings are considered different if and only if there is no reflection or rotation that transforms one into the other. The puzzle pieces do not distinguish between "top" and "bottom"; each piece can therefore also be "flipped".
a) For the shapes $A, B, C, D$, there are, for example, the possibilities shown in the figure:
![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0544.jpg?height=222&width=1196&top_left_y=1830&top_left_x=430)

For $A$, there are no further possibilities (in the sense of the "Hint"), as each tile $A$ already occupies a length of $4 \mathrm{~cm}$.

For $B$, there are also no further possibilities, as each tile $B$ has a length and width of $2 \mathrm{~cm}$, so the tiles can only be placed side by side and below each other.

For $D$, there are also no further possibilities; because at a corner (e.g., bottom left) of the square to be covered, a tile $D$ (apart from rotation and reflection) can only be placed as shown by the double-shaded area; the missing part of the field at the bottom right requires the placement of the single-shaded area, which then determines the positions of the last two tiles $D$.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0544.jpg?height=217&width=226&top_left_y=2427&top_left_x=915)

For the shape $E$, there are no possibilities; because at a corner, a tile $E$ can only be placed as shown in the upper figure, and with any way to cover the shaded field with another tile $E$ (without overlapping the previous one), this tile would extend beyond the square.

b) The figure shows four examples of the required type.
![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0545.jpg?height=270&width=1152&top_left_y=454&top_left_x=454)

c) The figure shows (with $A, B, C, D$ and $A, B, C, E$ as the used types) two examples of the required type. (Other arrangements of the tiles are also possible.)
![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0545.jpg?height=210&width=854&top_left_y=861&top_left_x=604)

Solutions of the 1st Round 1993 taken from [5]

### 4.35.2 2nd Round 1993, Class 7