2. Baron Munchausen claims that a rectangle $15 \mathrm{~cm} \times 9$ cm can be cut into three rectangles of equal perimeter in four different ways. Is the baron correct? Ways are considered different if the rectangles into which they are cut have different perimeters.
Answer: correct.

Solution. 1) It is possible

to cut

a rectangle into three equal parts with two straight lines parallel to the shorter side (see fig.). In this case, the perimeter will be equal to $2 \times 9 + 2 \times 15 / 3 = 18 + 10 = 28$.

2) It is possible to cut
the rectangle into three
equal parts with two straight lines,
parallel to the longer
side (see fig.). In this
case, the perimeter will be equal to
$2 \times 15 + 2 \times 9 / 3 = 30 + 6 = 36$.
3) It is possible to cut off a rectangle with a cut along the longer side, and then divide the remaining rectangle into two equal parts with a perpendicular cut.

Using an equation, we find the shorter side of the long rectangle:

$2(9 - x) + 2 \times 15 / 2 = 15 \times 2 + 2x;$

$18 - 2x + 15 = 30 + 2x$

$4x = 3; x = 0.75$.

The perimeter is

$30 + 2 \times 0.75 = 31.5$.

4) It is possible to cut off a rectangle

with a cut along

the shorter side,

![](https://cdn.mathpix.com/cropped/2024_05_06_a4325d569d59da29736fg-1.jpg?height=227&width=373&top_left_y=1796&top_left_x=1932)
and then divide the remaining rectangle into two equal parts with a perpendicular cut.

Using an equation, we find the shorter side of the long rectangle:

$2 \times 9 + 2x = 2(15 - x) + 2 \times 9 / 2$

$18 + 2x = 30 - 2x + 9$

$4x = 21; x = 5.25$.

The perimeter is $18 + 2 \times 5.25 = 28.5$.

Note. It is not necessary to require a proof that there are no other partitions, although this can be proven.

## Grading Criteria.

Two examples are given: 1 point.

Three examples are given: 3 points.

Four examples are given: 7 points.