9.8. 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 the figure $\square$ on the board (the figure can be rotated, but its cells must not go beyond the board's boundaries). We will call such a placement 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.
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 the number in the central cell be \(e\); denote by \(S\) (M. Antipov) the sum of all these five numbers. Then, according to our assumption, \(S-a = S-b = S-c = S-d = 0\), from which it follows that \(a = b = c = d\). Therefore, \(S-a = e + 3a = 0\), which means \(e = -3a = \pm 3\), which is impossible.

Thus, in each of the 36 "crosses" (with centers in all non-edge cells), there is an unsuccessful placement. Clearly, each placement is contained in no more than one cross; therefore, there are at least 36 such placements.

On the other hand, Figure 2 shows an example of a placement where the number of unsuccessful placements is 36 (in each cell, the sign of the corresponding number is indicated). Indeed, in any cross, there is exactly one unsuccessful placement, and all placements adjacent to the long side of the board are successful.

Comment: Only the answer without justification - 0 points.

Only an example with exactly 36 unsuccessful placements is provided - 2 points.

It is only proven that there must be at least 36 placements, but the corresponding example is missing (or incorrect) - 4 points.

It is only proven that at least one of the four placements of the figure in the "cross" is unsuccessful - 2 points. (These 2 points can be combined with points for a correct example.)