## Task 1 - 240911

a) A square field consisting of 25 unit squares (Figure a) is to be divided such that each piece is congruent to one of the figures (each consisting of five unit squares) in Figures b to f, and that each of these figures appears at least once. Provide one such division!

b) Determine all natural numbers \( n > 0 \) for which such a division of an \( n \times n \) field is possible!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1056.jpg?height=177&width=182&top_left_y=782&top_left_x=503)

(a)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1056.jpg?height=100&width=189&top_left_y=858&top_left_x=705)

(b)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1056.jpg?height=100&width=143&top_left_y=864&top_left_x=914)

(c)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1056.jpg?height=97&width=131&top_left_y=862&top_left_x=1088)

(d)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1056.jpg?height=134&width=142&top_left_y=824&top_left_x=1251)

(e)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1056.jpg?height=140&width=146&top_left_y=821&top_left_x=1412)

(f)
a) An example of the desired decomposition is shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1056.jpg?height=314&width=309&top_left_y=1142&top_left_x=902)

b) (I) If for a natural number \( n > 0 \) such a decomposition of the square field is possible and if \( k \) is the number of resulting pieces, then the field consists of \( 5k \) unit squares, so \( n^2 = 5k \). Therefore, \( n^2 \) is divisible by 5, and consequently, since 5 is a prime number, \( n \) is also divisible by 5.

(II) If \( n \) is divisible by 5, then the side length of the square field can be decomposed into segments, each of which has a length of 5 unit segments; thus, the square field can be decomposed into square subfields, each with a side length of 5 unit segments.

Each of these subfields can be decomposed in the required manner according to (a); therefore, such a decomposition also exists for the entire field.

With (I) and (II), it is proven: The desired numbers \( n > 0 \) are exactly those that are divisible by 5.