Evdokimov M.A.

Dominoes $1 \times 2$ are placed without overlapping on a chessboard $8 \times 8$. In this case, dominoes can extend beyond the board's edges, but the center of each domino must lie strictly inside the board (not on the edge). Place

a) at least 40 dominoes;

b) at least 41 dominoes;

c) more than 41 dominoes.
a) First method. Take a square with vertices $(4, \pm 4)$ and rotate it slightly around its center so that its vertical sides do not intersect the lines with half-integer abscissas. The original square intersected eight such lines. After the rotation, the square cuts off segments longer than 8 on them. Therefore, inside each of them, five centers of vertical dominoes can be placed with a step of 2.

Second method. See the left figure.
![](https://cdn.mathpix.com/cropped/2024_05_06_7a1b7dea474101c41e1dg-26.jpg?height=578&width=1262&top_left_y=530&top_left_x=406)

b) The right figure shows the placement of 42 dominoes. The centers of the outermost dominoes lie on the boundary of a square with a diagonal of 11, which is less than the diagonal of the $8 \times 8$ board. Therefore, the centers of all dominoes will fit on the board. Send a comment

## Problem