Problem 5.7. Jerry the mouse decided to give Tom the cat a square cake $8 \times 8$ for his birthday. In three pieces marked with the letter "P," he put fish, in two pieces marked with the letter "K," he put sausage, and in one piece, he added both, but did not mark it (all other pieces are without filling). Jerry also told Tom that in any $6 \times 6$ square, there are at least 2 pieces with fish, and in any $3 \times 3$ square, there is no more than one piece with sausage.

What is the minimum number of pieces of cake Tom needs to eat to ensure that he gets a piece with both fish and sausage?

|  |  |  |  |  |  |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  | P |  |  |  |  |  |  |
|  | K |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  |  |  |  | P | K |  |  |
|  |  |  |  |  |  |  |  |
|  | P |  |  |  |  |  |  |
# Answer: 5.

Solution. Let's call a piece with fish and sausage a coveted piece.

According to the problem, in any $6 \times 6$ square, there are at least 2 pieces with fish. In any such square, at least one known piece with fish is already included; consider the squares that contain only 1 known piece of fish (all of them are adjacent to the right edge of the pie). It can be seen that they have a common rectangle of $4 \times 6$ (gray in Figure 1). Therefore, the coveted piece must be in this rectangle.

![](https://cdn.mathpix.com/cropped/2024_05_06_6da73bfd3e09e8b55e3fg-12.jpg?height=495&width=495&top_left_y=988&top_left_x=479)

Figure 1: Solution to problem 5.7

Next, according to the problem, in any $3 \times 3$ square, there is no more than 1 piece with sausage. Therefore, in any such square that already contains a known piece of sausage, the coveted piece will not be there. Such squares cover areas of $5 \times 5$ centered on each piece of sausage (shaded in the figure).

There remain exactly 5 cells (gray but not shaded). It is not difficult to verify that each of these 5 cases satisfies the conditions of the problem. If Tom eats no more than 4 pieces of the pie, one of the above-mentioned five cells will definitely remain untouched, and the coveted piece may be in it. If Tom eats 5 pieces in these cells, then among them, there will definitely be the coveted one.