## Task 2 - 190912

Some of the 49 fields in the illustration are to be checked. Any two checked fields may share at most one corner point. In each row and each column of the illustration, exactly as many fields are to be checked as indicated by the numbers at the edges.

Determine all the distributions of the fields to be checked that meet these requirements!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1010.jpg?height=380&width=337&top_left_y=2066&top_left_x=1359)

(Use the given letters to describe the solution path! For example, the first field in the top left corner is designated as aA.)
Note: In Figures 1 and 2, the fields that are guaranteed to remain free are marked with a circle.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1011.jpg?height=454&width=443&top_left_y=247&top_left_x=544)

2) 

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1011.jpg?height=456&width=445&top_left_y=246&top_left_x=1111)

(1) Since exactly 4 fields must be circled in column F, these can only be: $a F, c F, e F, g F$.

(2) The neighboring fields thus remain free: $a E, a G, c E, c G, e E, e G, g E, g G, b F, d F$ and $f F$.

(3) Since no fields are to be circled in column C and row f, the following fields remain free:

$a C, b C, c C, d C, e C, f C, g C, f A, f B, f D, f E$ and $f G$.

(4) Since one field is already circled in row c, the remaining fields remain free, i.e., (see Figure 1): $c A, c B$ and $c D$.

(5) Only two fields remain free in column E; therefore, the following must be circled: $b E$ and $d E$.

(6) The neighboring fields to these remain free: $b D$ and $d D$.

(7) Only two fields remain free in column G, so the following must be circled: $b G$ and $d G$.

(8) Two fields must be circled in row a, one of which must be: $a D$.

(9) One field is already circled in column D, so the following remain free: $e D$ and $g D$.

(10) Two fields are circled in row d, so the remaining fields remain free, i.e.: $d A$ and $d B$.

(11) The following must be circled in column A: $g A$ and $e A$, so the neighboring fields remain free (see Figure 2): $g B$ and $e B$.

(12) For the remaining fields, there are exactly two possibilities:

Either $a A$ and $b B$ are circled or $b A$ and $a B$. Thus, there are exactly two distributions of the required type.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1011.jpg?height=457&width=443&top_left_y=1779&top_left_x=544)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1011.jpg?height=454&width=440&top_left_y=1778&top_left_x=1116)