443. Four chips in a straight line. Before you is a board of 36 squares, on which 4 chips are arranged in a straight line such that any square on the board is on the same row, column, or diagonal as at least one of the chips. In other words, if we consider our chips as chess queens, then every square on the board is under attack by at least one queen. The puzzle is to determine how many ways 4 chips can be arranged in a straight line so that every square is on the same line as one of the chips.

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-154.jpg?height=434&width=439&top_left_y=682&top_left_x=817)

Two positions are considered different if the sets of 4 squares occupied by the chips are at least partially different. For example, in the given example, all chips can be moved to the adjacent column to the right or placed on any of the two central rows. Thus, we have found 4 different solutions that can be obtained from each other by rotations and reflections. Remember that the chips must always be arranged in a straight line. This puzzle is not too difficult and is quite engaging.
443. There are 9 main solutions presented in the figure. Solution \(A\) is the one given in the problem statement. Of these 9 solutions, \(D\), \(E\), and \(J\) each generate 8 solutions through rotations and reflections, as explained earlier, while the others yield only 4 solutions each. Therefore, there are a total of 48 different solutions to this puzzle.

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-310.jpg?height=1246&width=1218&top_left_y=936&top_left_x=430)

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-310.jpg?height=403&width=383&top_left_y=958&top_left_x=431)

D

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-310.jpg?height=368&width=374&top_left_y=1392&top_left_x=430)

G

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-310.jpg?height=369&width=377&top_left_y=1803&top_left_x=431)

E

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-310.jpg?height=365&width=377&top_left_y=1391&top_left_x=842)

H

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-310.jpg?height=371&width=377&top_left_y=1799&top_left_x=842)

\(F\)

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-310.jpg?height=368&width=374&top_left_y=1392&top_left_x=1252)

I

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-310.jpg?height=371&width=374&top_left_y=1799&top_left_x=1258)

The reader may find it interesting to know that on an \(8 \times 8\) chessboard, five tokens can be arranged in a straight line under the same conditions in four main ways, generating 20 different solutions.