## Task 6 - 280936

Determine all numbers $n \geq 3$ for which it is possible to completely divide an $n \times n$ board without the four corner squares (see figure) into parts such that each part can be obtained from one of the areas (a), (b) by translation and rotation!

Hint: It is also allowed that both parts (a) and parts (b) occur in a decomposition.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1094.jpg?height=454&width=874&top_left_y=1035&top_left_x=888)
Since (a) and (b) both have area 4, it follows that for such a solution to exist, \( n^2 - 4 \) must be divisible by 4, i.e., \( n \) must be even. For even \( n \), we can alternately color the rows of the board with white and green (see sketch below). This coloring has the property that no matter how the pieces (a) or (b) are positioned on the board, exactly 1 field of the piece is green and the other 3 are white (Type A) or exactly 1 field is white and the other 3 are green (Type B).

Since the number of white fields is equal to the number of green fields, it follows that in a valid decomposition, the number of pieces of Type A must equal the number of pieces of Type B. Therefore, \( n^2 - 4 \) must be divisible by 8, so \( n \) must be of the form \( n = 4k + 2 \) with \( k \in \mathbb{N} \).

This condition is also sufficient: For \( n = 2 \), a decomposition is clearly possible (from a \( 2 \times 2 \) board, all four fields have been removed).

Moreover, it is clear that \( 2 \times 4 \) rectangles can be divided into exactly two pieces.

If a decomposition of a board of size \( n = 4k + 2 \) has been found, then this can be extended to a decomposition of a board of size \( 4k + 6 \) by positioning four pieces in the corners of the already decomposed board and then filling the two horizontal sides with \( k \) \( 2 \times 4 \) rectangles and the two vertical sides with \( k + 1 \) \( 2 \times 4 \) rectangles, as shown in the sketch:
![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1094.jpg?height=362&width=1250&top_left_y=2234&top_left_x=405)