7. Each cell of a $10 \times 10$ board is painted black or white. A cell is said to be out of place if it has at least seven neighbors of a different color than itself. (Neighbors are cells that share a common side or corner.) What is the maximum number of white cells on the board that can be out of place at the same time?
Solution. Answer: 26

Example: 13 dominoes, not touching each other or the edge of the board.

![](https://cdn.mathpix.com/cropped/2024_05_06_4e3584856b1eca6a5ca8g-3.jpg?height=462&width=460&top_left_y=109&top_left_x=798)

Estimate. Let's call a cell that is not in its own place an OOP cell. For each OOP cell, we draw its neighborhood: a square with side 2, centered on it (its sides are not aligned with the grid lines!). White OOP cells are divided into components - single cells, dominoes, and diagonal dominoes. They have neighborhoods of size 4, 6, and 7 cells, respectively. These neighborhoods do not overlap and lie within the central $9 \times 9$ square. Also, it can be noted that the area of the neighborhoods is at least three times the area of the cells themselves. If there are 27 OOP cells, then $27 \cdot 3 = 81$ (exactly three times!), but the ratio can be exactly 3 only if everything is divided into dominoes, and 27 is odd.