3. In the cells of a $10 \times 10$ table, the natural numbers 1, 2, ..., 99, 100 are arranged. Let's call a corner the figure that results from removing one cell from a 2x2 square. We will call a corner good if the number in its cell, which borders by sides with two other cells, is greater than the numbers in these two other cells. What is the maximum possible number of good corners? (Each corner is counted independently of how it is positioned relative to others, different corners can partially overlap). (A. Gолованов)
Answer: 162. Solution: Let's call the center of a corner the cell that borders by sides with two other cells of the corner. In any $2 \times 2$ square, no more than two of the four corners contained in it can be good: those with the two largest numbers in the center of this square. There are 81 $2 \times 2$ squares in our table, as the top-left cells of all such squares form a $9 \times 9$ square. Therefore, there are no more than $2 \times 81=162$ good corners. Let's provide an example where there are exactly 162. We will color the cells of the table in two colors in a checkerboard pattern, and arbitrarily place the numbers from 51 to 100 on the black cells, and the numbers from 1 to 50 on the white cells. It is easy to see that in this case, in each $2 \times 2$ square, there will be exactly two good corners with centers in the black cells.