8. In the cells of an $8 \times 8$ board, the numbers 1 and -1 are placed (one number per cell). Consider all possible placements of a four-cell figure

$\square$ on the board (the figure can be rotated, but its cells must not go beyond the board's boundaries). A placement is called unsuccessful if the sum of the numbers in the four cells of the figure is not equal to 0. Find the smallest possible number of unsuccessful placements. (M. Antipov)
Answer: 36. Solution: We will show that in each "cross" of five cells on the board, there will be at least one unsuccessful placement. Suppose the opposite; let the numbers in the outer cells of the cross be \(a, b, c, d\), and in the central cell be \(e\); denote the sum of all these five numbers by \(S\). Then, according to our assumption, \(S-a=S-b=S-c=S-d=0\), from which it follows that \(a=b=c=d\). But in this case, \(e=-(a+b+c)=-3a\), which is impossible. Since each placement of the figure lies in no more than one cross, it follows that in 36 crosses (with centers in all non-edge cells), there will be at least 36 unsuccessful placements. On the other hand, the example arrangement on the right shows that the number of unsuccessful placements can be exactly 36 (the sign of the corresponding number is indicated in each cell). Indeed, in any cross, there is exactly one unsuccessful placement, and all placements adjacent to the long side of the board are successful.

![](https://cdn.mathpix.com/cropped/2024_05_06_eb3696f4547e2bb45d25g-2.jpg?height=372&width=377&top_left_y=862&top_left_x=1679)

Criteria: Answer without justification - 0 points. Proved that there are at least 36 unsuccessful placements, no example of 36 - 4 points. Provided an example where there are exactly 36 unsuccessful placements, no proof that there cannot be fewer than 36 - 3 points. Proved that at least one of the four placements of the figure in a "cross" is unsuccessful, no further substantial progress - 2 points.